Two finite mirror-image series restrict the non-trivial zeros of

Abstract: Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤ 1 Euler’s zeta diverges. Dirichlet’s eta function ( ), negates alternate terms of zeta, permitting convergence when ∈ C and Re( ) < 1, and its non-trivial zeros { }, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re( ) = . This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re( ) ≠ . Analytical continuation generates a modified eta series ( ), in which every term is multiplied by (1 − ). If the integer is appropriately determined by the Im( ), similar paired finite vector series have a difference that closely follows ( ) and their terminal vectors intersect in a unique way permitting zeros only when Re( ) = . Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re( ) > .


Introduction
Proposition 30 of Book VII of Euclid's Elements, known as Euclid's lemma, states that if is a prime number and | , where and are integers then | or | [1]. This lemma with propositions 31 and 32 give us the Fundamental Theorem of Arithmetic, namely that every integer > 1 is the product of a unique set of primes = … , where = 2, = 3, = 5, …. .
The Fundamental Theorem of Arithmetic is encoded in the zeta function equating to Euler's product formula over the primes thus To see this clearly the right hand side of Equation (1) is multiplied out ( ) = 1 + + + + + + + + + + ⋯ .
As the positive integers increase, the primes thin out with ( ), the number of primes less than , being asymptotic to /ln( ). This asymptotic distribution, known as the Prime Number Theorem was conjectured by Gauss in the 1790's and proved independently by Hadamard [2] and de la Vallée Poussin [3] in 1896 using the Riemann zeta function. In November 1859 Riemann published his paper "On the Number of Primes Less Than a Given Magnitude" [4]. Riemann only refers to the imaginary component with the variable and does not assign a symbol to the real component. We follow convention and let = Re( ). Riemann's paper contains an explicit formula for ( ), given in terms of the related function Π( ) which counts the primes and prime powers up to , counting a prime power , as 1/ of a prime, thus Π( ) = ( ) + 1 2 + 1 3 + 1 4 + 1 5 + ⋯.
The Möbius function ( ), is related to the inverse of ( ) which enables the number of primes to be recovered, Riemann's formula then becomes Riemann believed that Re( ) was always but did not demonstrate this as it was not central to his paper. There is ample empirical evidence [5], but no accepted proof that Re( ) = for all . In the language of the zeta function, is limited to the "critical-line" in the middle of the "critical-strip" (0 < Re( ) < 1). In 1914 Hardy [6] proved the infinitude of { } with = , which we will call { } indexing those on the critical-line with , but Hardy's proof does not preclude there being other real values in the critical-strip off the line. Importantly, the functional equation imposes a symmetry about the critical-line and the real axis becomes a line of reflection [7].
Euler's zeta function is classically confined to the real axis and diverges when ≤ 1. If in place of we have ∈ ℂ with = + and = √−1 then Euler's zeta has a pathway of diminishing vectors in the Argand plane. After a countable set of individual vectors, that often seem to lack overt structure, in what can be called the proximal pathway the vectors of Euler's zeta series enter a countable set of superstructures in its distal pathway. The superstructures, here designated ℛ , are paired pseudo-spirals whose principal-axes can be equated to vectors, here designated ℛ , which are objects in their own right. The ℛ , form their own series of diminishing vectors running counter to the pathway of Euler's zeta and soon form superstructures of their own, with principal-axes which can be equated to the proximal vectors of Euler's zeta. The largest pair of pseudo-spirals ℛ , has a final pseudo-convergence that precedes the slowly growing spiral of inevitable divergence. The cardinality of the set of proximal vectors equates to that of the set of distal superstructures. The location of the final pseudo-convergence can be remarkably close to zero. Apart from cardinality, there is a striking symmetry between the arguments of neighbouring vectors of the paired sets under all real domains but a symmetry between their magnitudes only when = .
Analytical continuation in Dirichlet's eta function ( ), abolishes the spiral of divergence through negation of alternate terms. Dirichlet's eta locates the non-trivial zeros { }, which are remarkably similar to the minima of an appropriately terminated Euler's partial zeta series, but the relationships between the proximal and distal vectors in Dirichlet's eta are less clear.
This work characterises the overt symmetries in the pathway of the partial Euler's zeta vector series preceding its final divergent expanding spiral. A similar analysis applies to the covert but easily exposed symmetries in a family of analytically continued modified Dirichlet eta functions. Once exposed, the symmetries in the modified eta function can be tracked by two short finite vector series whose behaviour, under changes in the Re( ), confirms Riemann's Hypothesis through a simple symmetry breaking argument. The implications for the derivative easily follow and reinforce that confirmation.

Resources
Microsoft Excel 2010 with Visual Basic was used for repetitive calculations and GraphPad Prism 5 for Windows was used to create figures. A published table of non-trivial zeros was obtained [8].
If , ( ) is a vector series to terms influenced by , then P , ( ) is used to mean the pathway of sequential vectors representing the terms of that series when plotted in the Argand plane. In this way P , ( ) is richer than , ( ) which on its own identifies a point in the plane, or implies a vector to that point. If , ( ) is used as a vector this is implied by context and not notation. Each series has a specific index using different fonts/scripts; care is needed to avoid confusion.
This work describes four divergent vector series and their pathways P( ( )), P ℎ , ( ) , P ℎ , ( ) and P ℓ , ( ) , these summate using the indices , , and respectively to final terms of , , and . The vectors of these series are , ℛ , ℛ and ℒ respectively. The pathway of the converging P , ( ) has index , runs to n terms and summates vectors designated .
Each divergent pathway passes through a number of pseudo-convergences before entering a final pseudo-convergence which precedes the infinite spiral of divergence. In any series the term corresponding to the vector preceding the growing spiral of divergence is designated tau, . These pathways also have proximal and distal parts separated by a vector at a term kappa, . The converging P , ( ) has a related proximo-distal separation at its vector when > , and the function ( ), can be considered as ( ) with fixed, or ( ) with fixed, since curves with one parameter fixed are informative.
The known non-trivial zeros are the set { }, each having Re( ) = ½ and Im( ) ∈ { }. An unknown non-trivial zero off the critical-line is designated with Re( ) ≠ ½ and = with ∉ { }. The functional equation, Equation (2), demands that if ( + ) = 0 then − = 0 when the restriction = 1 − applies. Since the sign of can be ignored without weakening our conclusions we will consider that if there is a = + then there is a = + , see The notation means the nearest integer to ∈ ℝ, with floor and ceiling functions meaning the nearest integers below and above respectively. The notation is used to mean the nearest element to in an ordered set; so if the set were then 4 = 3.1 . Additional objects are detailed in the text and listed in Appendix B Notation.

Salient points in this work
Whilst outlining the strategies of this work this section introduces a number of mathematical objects, including the five vector series ( ), , ( ), ℓ , ( ), ℎ , ( ) and ℎ , ( ). Some objects, common to more than one function, are introduced before their associated functions. The Results section expands upon the enumerated points and illustrates the anatomy and behaviour of the vector series and their pathways. 1. A partial Euler's zeta series, without analytical continuation is defined as; with the vectors having | | = and arg( ) = − ln( ).
2. The series ( ) has a transition when = , at a vector = in P ( ) which signals the end of the final pseudo-convergence and the start of spiraling divergence. Tau is a function of alone in Euler's zeta and in the series ℓ , ( ) which is described below. Tau is also a function of and for the diverging vector series ℎ , ( ) and ℎ , ( ) both of which summate a series of vectors designated ℛ which are also described below. There are two related formulations; = − 1 + 1 − for Euler's zeta ( ) and ℓ , ( ), = − 1 + 1 − if ≥ 2 for ℎ , ( ) and ℎ , ( ).
The function , ( ) needs no tau vector since it converges.
3. The vector series ( ) , ℓ , ( ) , ℎ , ( ) and ℎ , ( ) each have proximal and distal parts separated at a term designated kappa. For example, in ( ) an for which = , separates proximal values of for which < , from distal values of for which > . Proximal and distal parts of P ( ) are similarly separated by the vector for which = , and which is designated . Kappa ∈ ℕ is defined as with a random tie-breaking rule applied. There are similar kappa vectors at the end of the proximal series ℓ , ( ) having vector ℒ , and the proximal series ℎ , ( ) having vector ℛ . In , ( ) an for which = , separates proximal values of for which < , from distal values of for which > , when > .

4.
A residual, kappa dot written ∈ ℝ, with − < < is simply, Kappa dot has a relationship with the fractional intersection of the kappa vectors ℒ and ℛ of at the non-trivial zeros. The fractional intersection of the kappa vectors is 0 < ≤ 1.

5.
A function ( ) represents a "focal point" in the final pseudo-convergence of an ultimately diverging series. Using tau, the point of final pseudo-convergence in Euler's zeta is considered to be at either This average is implied when the formulation ( ) is used, but this averaging does not apply when ( ) entertains values of ≠ . Related, but more involved, averaging is applicable for ℎ , ( ), but once more the formulation ℎ , ( ) is considered to imply that "focal point" in the final pseudo-convergence rather than the point ℎ , ( ) itself.
6. An integer identifies every term in a vector series. This allows a different rule to apply to the magnitude or argument of the vectors. Euler's zeta is assigned a value of = 1, since every term is treated the same as its neighbours. 8. The partial series to terms is specifically considered to be When > the symmetries in P ( ) are overt, and when ≈ and ∈ { } the term ( ) ≈ 0. It is argued in this paper that when ≈ and ∈ { } then the term ( ) ≈ 0.

In relation to
10. An ordered set R has elements ∈ ℕ. Using "\" to mean the set theoretic difference, R is defined for ≥ 2 and has elements indexed by , This rule generates R = when → ∞, a position we require for Euler's zeta, which is in many ways appropriate, however, Euler's zeta is already associated with = 1.
13. An ℛ structure is a set of vectors { } such that with a principal-axis which runs between ̂ (s) and ̂ (s). The principal-axis of an ℛ structure preserves its orientation under changes in . The final and largest structure in the pathway is ℛ , whose principal-axis has magnitude √ when = .
14. The tangent to a smooth curve following the distal pathway of P ( ) within an ℛ has a point-of-inflection when the curve changes from a clockwise to an anticlockwise progression. This tangent bisects the principal-axis crossing it at an acute angle of .
17. A vector ℛ , which represents the magnitude of the principal-axis of an ℛ structure is an object in its own right and has magnitude ℛ being and after taking the nearest integer , the vector has an argument 18. A series ℎ , ( ) summing the ℛ with ̅ ∈ to tau terms for all ≥ 2 has a relationship with ( ), and is defined as; At a non-trivial zero the function ℎ , ( ) after its kappa vector ℛ follows the proximal part of P( ( )). The final pseudo-convergence can be refined from the point ℎ , ( ) to an average of points near ℎ , ( ) using ( ) again. Once more this averaging is a nicety and of no real material importance other than for calculations and illustrations; ( ) = ℎ , (s) + ℎ , (s) + ℎ , (s) + ℎ , ( ) with = τ/ and = τ/ .
19. The term proximal refers to the early vectors of a pathway and distal to the later. Since P ℎ , ( ) runs counter to P( ( )) at zeros, what is proximal for one pathway is distal for the other.
20. When < a vector ℛ has a simpler magnitude since = giving and a simpler argument arg ℛ = − ln .
23. Likewise the series ℎ , ( ) to kappa terms is applicable to Euler's zeta and to ( ) if < ; The notation for the final term is analogous to and for other series in this paper. However, this series is principally needed to kappa terms and so ℓ , ( ) is; The function ℓ , ( ) has a final pseudo-convergence before diverging that can be considered to be at as defined for Euler's zeta.
25. In the case of Euler's zeta ( ), matched vectors from P ( ) , and ℛ from P ℎ , ( ) have mirror-image arguments about a common line of reflection but their magnitudes, which equate when = differ when ≠ .
26. In the case of ( ), if > then matched vectors ℒ from ℓ ( ), and ℛ from ℎ ( ) have mirror-image arguments as far as the kappa vectors ℒ and ℛ . Once more, the magnitudes, which equate when = differ when ≠ .
27. If > and the overlap of the kappa vectors ℒ and ℛ is accounted for with 0 < < 1 then The symmetries between P ℓ , ( ) and P ℎ , ( ) are now overt, but ( ) introduces a small asymmetry. Fortunately, near the non-trivial zeros the term ( ) can be ignored leaving the symmetries fully exposed. This is best satisfied with = when ( ) ≈ 0.
28. The line of reflection , has an unspecified magnitude but an argument of and arg( ) = − ln( ) if = 1, for Euler's zeta.
29. A function ( , ), which is essentially a modified eta function that is agnostic of ( ), has zeros which are remarkably close to the non-trivial zeros of Riemann's zeta function and with a minor adjustment to the magnitudes of the kappa vectors through = ( ) those zeros can be understood to equate with { }.
30. The symmetry breaking in ( , ) either side of Re( ) = limits the non-trivial zeros to the critical-line in support of RH. It then follows by elementary calculus that the pathways for the zeros of the derivative are limited to the right of the critical-line adding further support.

31.
A P( ( )) capable of producing a and a would have to be able to do so far all . An asymmetry is required to produce a loop with a double-point in ( ), and those loops are necessarily sensitive to changes in . The sensitivity of loops in ( ) to changes in is yet further support for RH.

The graphical context for this work
The modified eta function ( ) can be plotted as ( ) for fixed , and as ( ) for fixed , and provides a context for imagining the mechanisms required if RH is not true.  plots of curves related to for a , and . If there is a which disproves RH there has to be a loop in ( ) for fixed = , something like the loop shown in black. Crossing this loop at right angles, as prescribed by the partial differentials ( ) = − ( ) , and passing in almost opposite directions are the two curves ( ) for fixed and ( ) for fixed . The red curve is the true cycloid-like curve in ( ) for fixed , which would meet a true cycloid-like curve ( ) for fixed , which is not shown. The cycloids meet at ( + ) , indicated with a red square, and ( + ) = 0 with − > − . This is the challenge for the differential requiring it to have a zero to the left of the critical-line. The graph also gives context to that which seems impossible: if RH is to fail then a loop in ( ) must exist that would be stable under changes in .  It is important to appreciate how pathway changes secondary to a rising Re( ) can bring about a curve in ( ) and so a loop in ( ). A loop which satisfies Equation (2) is required for RH to fail. Curves capable of forming loops in ( ) as Re( ) rises are solely a consequence of pathway metamorphosis secondary to the differential change in the magnitude of the vectors along the pathway in a proximo-distal direction. In Figure 3(a) three pathways for P( ( )) are shown for = in black, = in red and =1 in blue, with the curve ( ) in green. The curve ( ) for passes through zero when = and then heads towards (1,0) for larger values of . There is no loop in Figure 3  higher values of as rises. A plot of ( ) in green is shown for fixed . There is no loop or double-point but the mechanism underlying looping is evident. (b) and (c) hypothetical loops for a formed by the same proximo-distal "shrinkage" gradient see text for detail. Figure 3 panels (b) and (c) show two hypothetical loops in ( ) in green for two . Pathway (a) in black is for = + , pathway (b) in blue is for = + (shown in the centre panel only), and pathway (c) in red is for a value of satisfying < < . Pathway (d) in light blue is for a value of satisfying < . Figure 3(b) and Figure 3(c) illustrate how the proximo-distal gradient of "shrinkage" of the pathway on rising can create a loop.
It is not clear how a double-loop with a triple-point could be created by such a mechanism since there are only two phenomena acting: (1) pathway layout and (2) the proximo-distal gradient. A double-loop and triple-point have to be capable of being generated if RH is not true. This corollary of RH will be discussed below.
The behaviour of ( ), and therefore the existence of its non-trivial zeros, is intimately linked to the non-trivial zeros of ( ), and has a deterministic link to a set of sequential integers whose cardinality rises gently with the Im( ). The mechanics of this determinism follow directly from the relationship between neighbouring vectors and + 1 as far as the vector.

Results
The results are presented in 8 sections with further results and illustrations in Appendix A. it can then be seen that, since The pathway P , ( ) is then the operation of plotting those vectors in the Argand plane.
3.1.1. The pathway ( ( )), the index , ( ( )) and other series The orientation of neighbouring in P( ( )) going forward sees a vector at ( + 1) of length ( + 1) lying in relation to an vector of length at an angle of = ln with a reduced form , with chosen so that 0 < ≤ 2 . In Figure 4(a) a cartoon shows a pathway a little after the point-of-inflection when = 1 and = at a value a little less than . As rises, the path encloses irregular polygons with a gradually reducing number sides before triangles are formed (see Figure 4(b)) which narrow and shorten as the vectors repeatedly double-back on themselves. In Figure 4(c) the cartoon is used to illustrate the final pseudo-convergence of P( ( )). This pathway has passed the point-of-inflection and = is a little less than 2 . When = falls to the tau vector is found, and thereafter the final spiral of divergence is entered.
The cartoon in Figure 4(c) would also apply if the were replaced with ℛ or ℒ and we had their superstructures ℳ and ℒ indexed by .
Consideration is also given to ln , which looks backwards, since this can be important at structural changes at low values of and can affect some approximations. If in P( ( )) the associated with ln and that associated with ln differ by more than 1 then that will be in the proximal pathway. That is if ln > 2 then < 1 − / and this of course is kappa, see Equation (5). The same applies to ℎ , ( ) and ℓ , ( ).
The distal pathway has a set of vectors forming a superstructure for each since neighbouring pairs of share the same value of . These superstructures have a stability under changes in . A superstructure, seen as rising, has a smooth clockwise growing pseudo-spiral which unwinds before entering a straighter region which crosses the principal-axis at , before entering a smooth anti-clockwise pseudo-spiral which winds in towards a pseudo-convergence or a final convergence in the case of ( ).
Three examples using ( ) are shown in Figure (5). The proximal pathways have a smaller number of much larger vectors than the distal pathways which have very many small vectors which form the superstructures. The ℛ structures all have a principal-axis of magnitude √ , here sitting on a circle of radius √2. Note the paired pseudo-spirals have an anticlockwise unwinding followed by a clockwise pseudo-convergence. For P( ( )) when > 2 the mechanism of convergence is subtly different and is illustrated in Figure 4, for = 5 and = 23. One consequence of the modification to Dirichlet's eta is that as rises the final pseudo-spiral occurs at higher values of . The final pseudo-spiral contains near regular polygons with one side missing. vectors that lie between each of the negated 5 th vectors. (c) With = 23, the mechanism of convergence can be appreciated as 22 vectors preceding convergence form smooth arcs (in red), followed by the larger negated returning vectors (in black): the arcs gradually reduce their curvature as infinity approaches.
In ( ) when > 2 there are near regular polygons and near regular star-polygons along the pathway P( ( )) whose number of sides relates to . The polygons lie between the vectors where | . The rules that apply when > 2 are generalizable to = 2. The index falls as rises. We proceed in stages using first ( ) = ( , , ) to identify the points in a pathway at which significant geometric changes take place to the polygons. These changes lie near the pseudo-convergences and near the points-of-inflection that would be found in a smooth curve approximating to the pathway. Appendix A1 expands on the geometry at structural changes in pathways.
The difference in arguments between neighbouring is ln looking forward and ln looking backwards. Important structural changes occur when neighbouring differences lie immediately either side of specific integral multiples of . This follows from the geometry of the pathway. The integer distinguishes significant vectors, thus A function ( ) determines important values of , with ∈ ℕ such that ≠ 0; We let designate a structural unit containing all points in the pathway P( ( )) between = ( + 1) and = ( ) . Although we should not strictly included = 0 we can give meaning to by briefly allowing ∈ ℝ when 0 < ≤ 1 since we recognise that the lim → − 1 = ∞. To this end we can allow ∈ ℕ without loss of rigor or argument conscious of the limitations. Temporary meaning is given to as a set thus The set M = { : = ( ) } are important points in the pathway P( ( )). The structural unit may be a substantial part of a paired pseudo-spiral ℛ or a region that makes no substantial progress in the Argand plane but rather contains oscillations about a pseudo-convergence.
We now consider a partial series , ( ) ending near the centre of a pseudo-spiral with ∈ M .
The end point of the final vector , with = in the partial series , ( ) may not best represent the centre of the pseudo-spiral. This is evident in smaller pseudo-spiral pairs at low values of and especially evident when ≫ 2 , since | | when | is much greater than the magnitude of the preceding ( − 1) vectors and the subsequent ( − 1) vectors. For example if = 5 the 100 vector is longer than vectors 96 to 99 and longer than vectors 101 to 104.
An appropriate average to represent the region of the pathway P( ( )) at (for a specified ) An approximate start to is the point ̂ ( ) and an approximate end of is ̂ ( ). The point ̂ ( ) approximates to a point best representing the transition from one structure to the next.
Our next task is to recognise the geometric requirement that when 2 | the neighbouring structures and will collapse and become pseudo-convergences within the region of a pseudo-spiral of ℛ . We can also state the following equivalence relation when |2( + 1) or |2( − 1).
In practice convergence can be placed at = 1 since ( ) = ̂ ( ) ≅ ̂ ( ) and pursuing , ( ) any further merely adds computational burden without benefit, see Figure 7. We can now comfortably revert to ∈ ℕ as we no longer require = 0. We now clarify the relationship between and which will depend on and the geometry near each when ∈ M . The mapping of an onto a specific is captured in the set R = { , , … … } and is easily derived. The set R is not to be confused with the structure indicated with the scripted ℛ . For any value of ≥ 2 we take odd values of , and the even values of which satisfy 2 | . We then remove the odd values of which lie either side of the values of for which 2 | . We could include = 0 since this specifies convergence, but are happy to accept = 1 in its place. We order this set as rises and index the elements by ∈ ℕ and so generate R = { , , … … }. We can state R formally using "\" to mean the set theoretic difference, thus A few sets are illustrated below. To incorporate Euler's zeta as a special case (rather than letting → ∞) we add that if = 1 then R = {1, 3, 5, 7 … } = and notice that the elements of R for ≥ 2 follow this pattern until the term. With = 1 Equation (39) gives ( ) = for Euler's zeta which is appropriate.
The collapse of structures either side of 2 | is shown in Appendix A2.
In ( ) each ℛ has a principal-axis which have arguments and magnitudes related to the vector ℛ . The principal-axes have end points ̂ ( ) and ̂ ( ) with ∈ R .
We now have all the vectors in ℛ with in the specified interval using the floor function The principal-axis of an ℛ has a magnitude Δ ̂ ; Appendix A3 tabulates some ratios of Δ ̂ / Δ ̂ . The centre or point-of-inflection in any ℛ structure can be located with the element of the ordered set = { ̅ }, which contains even values of but for ≥ 2 excludes those where 2 | ; and = for Euler s = 1. (46) The bar in ̅ , clarifies that the object carrying the subscript relates to the inflection or "middle" of the ℛ structure and not to a pseudo-convergence. The elements are in a similar format to those of R ; = { , , , … ̅ , … }. For ≥ 2 we can also define ̅ inductively as = 2 and then = ̅ + 2 if 2 ∤ ( ̅ + 2) and The elements in the ordered set rise in increments of 2 until − 1 terms have been reached, and so if = − 1 we have The section above principally refers to ( ) and so ≥ 2. To allow application in Euler's zeta we have = {2, 4, 6, 8, 10 … ̅ … }.

An example of an ℛ structure and the arg ℛ
To proceed we need nu, the indicator of the point-of-inflection in ℛ for any t and ; Using = 740 Figure 8 shows that the orientation of the vectors, for which | , follow the principal part of the structure. It is noted that their magnitudes are ( − 1) . There are gaps between neighbouring vectors for with | which are filled by the near-regular polygons with one missing side.

Comment on nu
Nu locates a vector which is nearly tangential to the point-of-inflection on a smooth curve following the pathway. The associated integer must satisfy | else the vector could lie in the near-regular open polygon in P( ( )) and its relationship to the ℛ would be challenging to determine. Importantly, the larger becomes the smoother the pathway P( ( )) is in the region and the closer the ( ) becomes to the tangent when | , and it is pretty good already at low values of . Hence, the tangential vector is located by and the ℛ becomes − . ln ; noting that the addition of for when | , cancels with the associated with ℛ being directed against the progress of the principal-axis. Nu also separates proximal and distal parts within the ℛ pathway. In ℛ there are 3 times as many in the distal part as in the proximal part. The proximal part has − 1 < < − 1 (48) and in the distal part has and In sequence, the ratios for the number of vectors in the distal to proximal parts of well recognisable ℛ structures will be very close to , , , … , with the approximation being best with large and small .
3.1.6. The function ℎ , ( ): a partial series to terms driven by A vector ℛ , with < , shares a relationship with the principal-axis of an ℛ superstructure made of many vectors. Using nu, the vector ℛ has magnitude Turning to the arguments of ℛ , we have for ≥ 2 Because ̅ , has a subscript bar, and this is the distinguishing parameter for the vector, the bar is re-used as an accent in the function ℎ , ( ). The function ℎ , ( ), now summates a series of vectors ℛ , driven by the sequential terms ̅ 2 ⁄ , with ̅ ∈ χ up to = terms and defined as 3.1.7. The function ℎ , ( ): a partial series to terms driven by This is simpler than ℎ , ( ). In appropriate circumstances Equations (51) and (52) can be simplified with important consequences.
Firstly, in Euler's zeta we have = 1, and substitute = to give with the arguments of ℛ now being, The index is now allowed to rise through the important point = to reach = . In Euler's zeta ( ), the index runs to the important point = at the end of the vector . In the region of a non-trivial zero the vector crosses , the line of reflection (see below). If the pathway of Euler's zeta P ( ) is considered after , the small ℛ vectors soon form superstructures designated ℳ before reaching = which precedes overt commitment to divergence. Secondly, for ≥ 2, we have = for all < giving with the arguments of ℛ having a phase shift of in relation to Equation (55), In both instances, i.e. for all , we can define ℎ , ( ) ∶= ℛ for = 1 in Euler s and for all ≥ 2 when < .
It is now clear that if an l is selected satisfying < ≤ and = , the symmetries will be overt.
If there is a non-trivial zero and = and > then the pathway P ℎ , ( ) after ℎ , ( ), which is its distal part, lies superimposed upon the proximal part of P ( ) namely P , ( ) . It is important to appreciate that P ℎ , ( ) after ℎ , ( ) does not track P ℓ , ( ) , but the pathway P ℎ , ( ) after ℎ , ( ) does track P ℓ , ( ) and vice versa. Even if ≯ there is of course a relationship between Δ ̂ and Equation (51) (see Appendices A3, and A8).

Lines of reflection Psi,
There are two lines of reflection , with indeterminate lengths but prescribed orientations and which pass through the origin. Psi is a line of reflection for the arguments for all paired vectors, and therefore the line of reflection for the difference between all neighbouring vectors. It is the line of reflection for the pathways P ℎ , ( ) and P ℓ , ( ) or P( ( )) when = since the paired vectors have equal magnitudes when = and paired pathways set of from the origin. The first vector of ℓ , ( ), ℒ has an orientation of −( ln( ) + ) and the orientation of the first vector of ℎ , ( ) appears in Equation (22). The arg( ) is half the sum of the two arguments.

A partial Euler's zeta function with a complex domain has two roles
This section illustrates ( ) and its P( ( )) having zeros or minima at an Im( ) very close to Im( ) and symmetry arguments keep these minima on the critical-line. When becomes large the function heads to unity.
When analytical continuation is applied in ( ), the series ( ) plays a dual role, (1) as a function which vanishes when ∈ { } and, (2) as the driver that brings ( ) to unity when ≫ 1.
Furthermore, if ( ) were to play a role when ≠ , in helping to generate a non-trivial zero off the critical-line, then it would need to do so at all values of and yet ( ) is insensitive to whilst ℓ , ( ) and ℎ , ( ) are affected by . This reassures us that if there were there a ( ) = 0 there would also be ( ) = 0.
3.2.1. In Euler's zeta, the final pseudo-convergence ends where divergence starts at the tau vector A point which best represents the centre of the final pseudo-convergence of ( ) is the output of a function (s) which employs the tau vector. The point (s) does not sit at ( ) but at a point which can be specified by a variety of averaging formulae involving neighbouring vectors. For ( ) the point ( ) is very close to zero and gets asymptotically closer to zero as Im( ) rises. The function (s) could generate a point in P ( ) half-way along , or generate one based on an average of with an immediate neighbour but a refined mechanism for (s), would be to use the diverging spiral when > , see Figure 9. Displacing from { } can bring the vector of Euler's zeta to cross zero, but these are not non-trivial zeros. This exercise emphasises why small changes in or away from a , that bring a to sit exactly at zero are not elements of { }. Given that at low the vector sits a little away from zero there are two scenarios to illustrate. Firstly, passes through zero for 14.1347 at = 0.55 see Figure 10   Furthermore, only when = do the paired kappa vectors meet at in symmetry, but a plot of the divergent spiral would cross a circle centred at the origin twice. These show that there is no zero for ( ) off the critical-line at = 0.55. Secondly a change in to = 14.23 brings the vector to zero but it is clear in this example that this is far from the middle of , see Figure 10(b).
Analytical continuation, as in Dirichlet's eta, obviates the need for a (s), but the points above are made to emphasis that: (1) there is no reason to believe that a P ( ) would have a vector that did not cross close to zero, nor (2) a divergent spiral whose ( ) was not centred at zero. In these respects P ( ) would share the features shown in Figure 9(a) that follow for all of { }. For these reasons, it is axiomatic that for high values of , where RH might fail we can say that, if for Riemann's zeta = 0 then for ( ) we have a = 0 and we lose no force of argument by considering that if tau is the subscript we can understand that = 0.

Symmetry in Euler's zeta
The pathway of Euler's zeta, without analytical continuation is shown for the 901 non-trivial zero in Figure 11(a) in black. The first five of the proximal pathway are emphasized with dark arrows on the right-hand side. Overall, the symmetries are striking since arg( ) is close to . The pathway has 14 blue proximal vectors, and many distal vectors, with 9 primitive superstructures and 5 obvious superstructures each with paired pseudo-spirals ending back at the origin. The distal pathway, in black, has 14 red ℛ vectors laid upon it. These mirror the blue vectors and the first five of P ℎ , ( ) are shown with red arrow heads. The black pathway of Euler's zeta ultimately diverges as implied by the expanding grey clockwise spiral. Figure 11(a) also illustrates the approximation In Figure 11(a) divergence of P( ( )), shown as a dashed grey spiral, commences immediately after two easily located vectors which almost exactly oppose one another, as shown in Figure 11(b). Divergence starts after the tau vector , which for Euler's zeta is at In Figure 11(c), the junction of proximal and distal parts of the pathway are redrawn with the vector of ( ), when = indicated and here shown in blue. Kappa is defined as Since the point-of-inflection in the last superstructure ℛ , will be near we know that and indeed ( + 1) will be located in this region and so will be sited only a little distance from the final pseudo-convergence of ( ). For very large the product ( + 1) will reference a vector that is asymptotically close to the point-of-inflection. The lowest value of which reveals the symmetries yields = ( + 1). These two locations are shown in Figure 12. Figure 12. The partial Euler zeta series: P( ( )) for the 901 st non-trivial zero. The red circle and arrow are at the point-of-inflection, the blue circle is at ( + 1) which bears a relationship to when is the lowest value which exposes the symmetries in P ( ) .
The lowest value of which exposes the symmetries is useful but it should be appreciated that when = + 1 though ( ) is small it does not completely vanish at the non-trivial zeros.
However, for all the non-trivial zeros of Riemann's zeta function, ( ) = 0 , we expect the un-extended Euler zeta series ( ) ≈ 0. Likewise, if there exists an as yet unknown non-trivial zero of an analytically continued zeta function ( ) = 0 with ≠ then we expect Euler's zeta series ( ) ≈ 0. Later we will see that ( ) ≈ ( ) + ( , ) and so manipulations that allow ≈ in the ( ) series are helpful since they permit ( ) ≈ ( , ) near the non-trivial zeros. It will be seen that if = ⁄ the those requirements are met.

Unique intersections at in Euler's zeta: the function ( , )
The final vectors of the paired pathways P( ( )) and P ℎ , ( ) are and ℛ , and over certain short intervals in they meet and cross at the line . These intersections occur when Im( ) is close to Im( ), but all intersections have to be with = to be in symmetry. Without a symmetrical intersection there is no overlap of proximal vectors with complimentary distal superstructures for either pathway. The fraction along the kappa vector at which intersection takes place is designated with 0 < ≤ 1 so that = 0 is no intersection.
The difference between the two pathways can then be defined as ( , ); yielding an approximation to Euler's zeta of ( ) ≈ ( , ).
Calculations suggest that the relationship = ( ) is an oscillation whose frequency decreases as rises, with mostly satisfying < < (see Appendix A4). In the absence of a deeper understanding of = ( ) the value = is used in Figure 13 to illustrate ( , ) tracking the non-trivial zeros of Riemann's zeta function over an interval in from 14 to 50.  Figure 13 shows that minima in the magnitude of ( , ), over an interval embracing the first 10 known non-trivial zeros of the Riemann zeta function, coincide well with those zeros. The function ( , ) behaves well in this interval and it is of interest to identify the upper and lower bounds that could host non-trivial zeros if 0 < ≤ 1, rather than fixing = .
3.2.4. The short finite series ( ) and ℎ , ( ) specify intervals in which host non-trivial zeros The short finite series ( ) and ℎ , ( ) can be used to determine the upper and lower bounds of an interval in , which could host a non-trivial zero. The interval in is that for which there is overlap of the kappa vectors, either with each other, or with . These are formally the same but the latter is computationally simpler. An example is shown in Table 1. The intervals in in Table 1 are conservative since the values of = 0 and = 1 lie just inside the Lower and Upper bounds. This example shows that a short finite series to kappa terms of only 425 can identify narrow intervals in relatively high up the imaginary axis that are able to host non-trivial zeros, however, there is no relationship between the prime composition of kappa and the number of non-trivial zeros in an associated region.

Kappa and the number of non-trivial zeros covered
Each sequential value of kappa makes a growing imaginary region accessible to the lambda function such that each value hosts more non-trivial zeros. Figure 14 shows on the left-hand axis the total number of non-trivial zeros and on the right axis the number of zeros associated with that specific value of kappa.   It is noted that Figure 15(a) has plots of ( ) for fixed which track the end of and the end of ℛ and not the movement of the points and ℛ , nonetheless the figure serves to show that ℎ , ( ) + ℛ + ( ) = ( ) + . Figure 15(a) also shows that for Euler's zeta when < the P ℎ , ( ) is bigger than the matching P( ( )), and when > the pathway P ℎ , ( ) is smaller than the matching P( ( )). This difference in behaviour is important and will be shown to be similar for ( ) with ≥ 2; when < a P ℎ , ( ) is bigger than the matching P ℓ , ( ) , and when > a P ℎ , ( ) is smaller than the matching P ℓ , ( ) .
An alternative approach is to reverse ℎ , and start a related vector series , ( ) from a point of overlap with the end of ( ), this complimentary pathway is , ( ) and is defined as In Figure 16, the pathway P( ( )) is shown in blue for = 0.35. This is the pathway of Euler's zeta preceding the overt commitment to divergence. The relationship with ℎ , ( ) is clear, and the two vectors in black representing ( ) can be seen to have the same magnitudes and arguments. The first 14 vectors of P( ( )) up to ( ) appear with blue arrows. The distal part of P( ( )) after the vector is tracked by the dark red pathway of P , ( ) in which the index falls from to 1 so that ( ) = ( )+ ( ).
The handling of is a trivial refinement that could be added if desired. This exercise is merely for illustration as the function , ( ) adds no new information. It may however lend appreciation to the importance of the superposition of vectors from one pathway on the superstructures from a complimentary pathway.
3.2.7. The pathway for Euler's zeta, P( ( )) is the Bauplan for P ( ) A "Bauplan" is a biological term for a collection of morphological features which are shared among many members of a group. Euler's zeta is the Bauplan for P( ( )) This work expects Euler's P , ( ) zeta ( ) ≈ 0 when ∈ { } and for this to be more challenging at low values of than at high values. However, even at the first non-trivial zero 14.1347, for which = 5 and = 2, which we met in section 3.2.1. and in Figure 9(a) it is still pretty good. This zero is shown again in Figure 17, with the symmetrical pathway P ℎ , ( ) , which also has 2 vectors. The line of reflection is . and its predecessor appears as a black circle. It is close to zero but not at zero.
The expectation is that all subsequent non-trivial zeros of ( ) have the middle of their vectors, or an appropriate average of neighbours, lying even closer to zero. Moving to ( ) in Figure 18(b), plots of = 2 to = 5 are shown with Euler's zeta for comparison, there is one P( ( )) shown for each vector of P( ( )). The images in Figure 18 show the first non-trivial zero is preserved on altering . Even at this low value of near-regular polygons with one missing side are discernible in the principal-axis of the rudimentary ℛ structures; a triangle ( = 3) , a square ( = 4) , and a pentagon ( = 5) , are highlighted. For completeness it is noted that when = 2, at larger values of the equivalent of the near-regular polygon with a missing side is a single vector between two nearly co-linear negated vectors.

The pathway to convergence for Dirichlet's eta function
In Figure 19, the pathway P( ( )) for a zero is shown in blue. The proximal pathway has in sequence, a set of starting with 1 and proceeding with diminishing magnitudes > ( + 1) . These proximal vectors do not form an obvious superstructure unless their reduced arguments Equation (32) form an opportunistic sequence (see Appendix A5).
Most importantly the vector 1 is unaffected by and introduces an asymmetry which is absent when we consider the differentials. The influence of the vector 1 on how the pathway shrinks when rises varies when is altered since the rest of the P( ( )) rotates and is magnified.
Once ln − ln < sets of sequential vectors form the paired pseudo-spiral superstructures, ℛ . A finite sequence of these superstructures runs as falls until = 1 and these are also shown in blue. The last, and largest pair of pseudo-spirals is ℛ whose final pseudo-spiral embraces convergence.
The principal-axis of an ℛ structure has a relationship with an ℛ . Importantly ℛ has an identity independent of the ℛ structure and is described by the complimentary pathway P ℎ ( ) , such that when is large the vector ℛ has an existence even when ℛ structures are not formed by P( ( )). are shown in bold in blue. In red the series of ℛ vectors, is plotted with ℛ , ℛ , ℛ and ℛ in bold with = {2, 6, 10, 14 … } . In pink part of the encircling clockwise divergent pathway of P ℎ , ( ) is indicated for an interval in > .
In Figure 19 P ℎ , ( ) is shown in red alongside P( ( )). Proximally P ℎ , ( ) has in sequence, a finite set of vectors, and since = 2 they have diminishing magnitudes ℛ > ℛ . The series ends distally in paired pseudo-spirals ℳ before divergence. The magnitude ℛ = √ when = . Divergence is indicated with a growing pink arc when > . The covert symmetry is that between the arguments of 1 , 3 , 5 and 7 and the arguments of ℛ , ℛ , ℛ and ℛ . Figure 20 shows a rotated version of P ℎ , ( ) with its first 24 vectors in bold. In black the first 24 odd vectors of P( ( )) are placed in sequence, so omitting the alternating or negated even vectors. It is clear that for these matched vectors their arguments are mirror-images and there is a line of reflection. The magnitudes however are not equal.  Figure 19 shows a rotated P ℎ , ( ) in red with the odd early vectors of P( ( )) extracted as a novel series and shown with black arrows. The mirror-imaging of arguments is inescapable.
In the context of Figure 20 this work aimed to extract the symmetries of P( ( )) by allowing to rise.

The ℳ structures in the distal P ℎ , ( ) and the ℒ structures in the distal P ℓ , ( )
This work is principally interested in the proximal P ℎ , ( ) when > but it would be remiss not to illustrate the distal P ℎ , ( ) when > . The next example is for < , with = 1230.58 giving = 14 but for which we set = 4.
The ℳ superstructures are a coherent set of small ℛ in the distal P ℎ , ( ) . An integer determines each ̅ that identifies an important ℛ , in an ℳ . The integer carries subscripts, or , with ∈ and ∈ referring to a pseudo-convergence or a point-of-inflection respectively. In like manner, the integer was used to determine an which identifies an important , with similar ∈ and ̅ ∈ referring to a pseudo-convergence or a point-of-inflection in an ℛ superstructure. The vector , unlike ℛ needs no subscript since it is its own index and the ordered sets and unlike and are independent of . The final pseudo-convergence of P ℎ , ( ) has a point ℎ , ( ) at = close to the start of divergence. An averaging function in this region would be appropriate for calculations but is of no material importance.
The centre of the pseudo-convergences and the points-of-inflection in the ℳ are generated by the diminishing vectors of the distal P ℎ , ( ) when > and occur near ln = when neighbouring differences lie immediately either side of specific integral multiples of . This follows from the geometry of P ℎ , ( ) . The integer distinguishes significant vectors. Sequential ̅ , rise in in increments of 2 or 4. For convenience we first assume the rise is 2 and then determine the nearest element of the set, either or , using the brackets to mean the nearest element of the appropriate set. The inequality but the nearest we can establish will be to choose ̅ ∈ as in; and It is important to be clear that each ̅ relates to the mid-part of an ℛ and a pair of (without the bar) represent the ends of the ℛ . Here we are looking at the whose centres are related to , each determined from a ̅ , and whose ends are related to the which are also determined from a ̅ . This is illustrated in Figure 21 (a).  In Figure 21 the superposition of the ℳ (made of many ℛ ) upon the is clear. Importantly, the figure shows the relationship between 4 having a larger magnitude than its neighbours and the mechanism by which the ℳ superstructure is able to match this change in magnitude through simple geometry. In Table 2 the relationship between , , ̅ and is detailed. To be explicit an specifies a since = /2, then Equation (71) gives ̅ = ( ) and then from the ̅ yields the value , which is of course simply its position in the ordered set.  Figure 21( The final pseudo-convergence of ℳ from Equation (71) is at ̅ = 3134 which relates to = 1176 and is illustrated in Figure 23. is ̅ = 3133.65 = 3134 which locates ℛ . This vector with its 5 neighbours is able to provide an average pseudo-convergence remarkably close to zero.

Label in
In like manner the distal part of P ℓ , ( ) is illustrated with = in Figure 24. This section has illustrated the mechanics of superposition of distal superstructures of one pathway upon the proximal vectors of another and shows how these can be complimentary.

Symmetry-breaking in ( )
We start by remembering that , ( ) to terms, using scripted as a unique index for the second summation, is as follows; This can be restated as , ( ) = ζ ( ) − ( ) ( ).
This gives us which is important, because we know that ζ ( ) can be close to ζ ( ) = 0. We now have which can be refined by to accommodate the intersection of the kappa vectors in lambda; with ( ) = ζ ( ) + ( , ).
3.5.1 Setting > : An example of a pathway ( ) when > The number of proximal vectors in the pathways of P ℎ , ( ) and P ℓ , (s) which have mirror-image arguments about , rises as increases. When > , all sequential arguments between neighbouring vectors of the finite P ℎ , ( ) are mirror-images of those arguments between the corresponding neighbouring vectors of the finite P ℓ , (s) . The pathway P ( ) with > is illustrated for P + for = 107.1686, the 33 non-trivial zero, in Figure 25. In Figure 25 the pathway P( ( )) has proximal and distal parts separated by the vector where = . This vector is shown in black crossing the line . The proximal part under the solid blue arrow will be tracked by P ℓ , ( ) as far as the line of reflection and as far as its kappa vector ℒ . The distal part of P ( ) lies under the dark dashed blue arrow which will be tracked in the forward direction by P ℓ , ( ) after the line of reflection and after its kappa vector ℒ . There is one obvious superstructure ℛ in this P ( ) shown with a black dashed line over the green pathway. The function ( ) also makes a weak attempt at an ℛ shown with a dashed purple line. Figure 26 repeats plots for the 33 zero altering either or .  The difference between P ℎ , ( ) in Figure 26(a) and P ℎ , ( ) in Figure 27(b) after the kappa vectors are clearly illustrated.

The separation of proximal and distal parts of P( ( )) for = + 1 when ℛ ≅ | |
We will consider = + 1. The vectors of P , ( ) and P ℎ , ( ) get smaller as and rise, but since they are superimposed complementary pathways at non-trivial zeros, there must be a crossover-point where ℛ ≅ | |.
If we let this occur when = and when In Figure 28 for a fixed , it can be appreciated that over the same interval in , the point ℎ , ( ) has moved much further than ℓ , ( ). Indeed this is no surprise since , ( ) > ℓ , ( ) . Non-trivial zeros occur in the vicinity of the crossing of these arcs. The next non-trivial zero occurs when ℎ , ( ) catches up with ℓ , ( ).

The function ( , ) identifies short intervals in t capable of hosting non-trivial zeros
The short finite series ℓ , ( ), locates intervals in which can host non-trivial zeros between upper and lower bounds on the critical-line. Half of the lambda function, here ℎ , ( ), is not required since the intersection between ℒ and is sufficient. Since specifies both and and we let = + 1 surprisingly few calculations are required. This is similar to section 3.2.4. Importantly the interval in associated with intersection of the kappa vectors gets narrower as rises, and since and grow slowly with the computational burden rises very slowly with .
In this exercise an Excel spread sheet followed rising from 1,131,941.5 to = 1,131,947.5 in increments of 0.0001. Values of were noted when ℒ intersected with and those values immediately preceding intersection (Lower bound) and those immediately following intersection (Upper bound) appear in Table 2. The bounds are therefore conservative as they lie outside of = 0 and = 1. In this interval = 424 and = 425. In Table 2, if the column contains [1→0] it indicates that the Lower bound is just outside an intersection of = 1 and the Upper bound is just outside an intersection starting after = 0. The next exercise looked at the same interval of covered by Table 2. In this interval each magnitude is of ( , ) with = . In Figure 29 the 11 minima of the logarithm of the magnitudes fall very close to the 11 non-trivial zeros, and the tightness of the upper and lower bounds is evident. The exercise in section 3.2.4 above was repeated for Im( ) over the interval = 226 to = 242 with results appearing in Table 3. In this interval = 6 and = 7. Intersection equates to the maximum interval in for which ( , ) could host a non-trivial zero. In Figure 30 the magnitudes of ( , ) with = are shown with their upper and lower bounds and the location of published non-trivial zeros. In Figure 30 two non-trivial zeros are hosted in the same interval in . In Figure 31 the paired pathways are illustrated at four values of in this interval to show the underlying mechanism. In Figure 31 it can be appreciated how two non-trivial zeros can be hosted in the same interval of . This insensitivity in ( , ) does not weaken the symmetry arguments in their support of the validity of RH.
Appendix A4 describes the empirical relationship between , and the angle of intersection of the kappa vectors at a number of non-trivial zeros. Appendix A4 illustrates two extremes of angles of intersection of the kappa vectors at non-trivial zeros and gives an example of "opportunistic" superstructures in a proximal pathway preceding the kappa vector in Appendix A5.  In Figure 32(a) the arcs are similar but equate only at the non-trivial zero. The other places where the arcs cross are not at the same value of . Pathways are shown for = 7187.70269 showing that the difference between ( ) = ℓ , ( ) − ℎ , ( ) and ( ) is ( ) as indicated with a blue vector. The difference between ( ) and ( ) is ( ), which is plotted in Figure 33 below for = 7187.70269 taken from Figure 32 above in which the blue vector is illustrated.  We should let = ⁄ to justify ignoring the partial Euler's zeta term when ( ) ≈ 0.

Simultaneous zeros for ( ): a metaphor for a
This section illustrates how a loop in ( ) for fixed , which allows a double-point + = 0 ≈ (1 + ), can be appreciated in terms of ( ) = ( ) + ℓ , ( )−ℎ , ( ). In Figure 34   value than an integral power of . It is for this simple reason that simultaneous zeros for some hypothetical within the critical-strip, could not be generated by a similar mechanism of collapse of a P( ( )) over an interval embracing and . To illustrate the effect of a change in the exercise was repeated with = 22 and the results are shown in Figure 35. Figure 35. Parameters are as in Figure 34 above but with = 22. P( ( )) is unchanged but P ℓ , ( ) in blue and P ℎ , ( ) in pink are both rotated through different angles. The summation ( ) + ℓ , ( ) − ℎ , ( ) reaches the green line of ( ) for = 1 at the point in the back circle but not at zero. The black, blue and red arrows represent the same vectors as in Figure  34. There is still a double-point in ( ) but this is not at zero. The loop in green is seen to pass through zero for = as expected.
3.8. The derivatives of ( ) and the related series ℎ , ( ), ℓ , ( ) and , ( ) If RH is false, and the functional equation applies, then there must be an for which ′ ( ) = 0 with Re( ) < . This is equivalent to saying that if Re( ) > for all which satisfy ′ ( ) = 0 then RH is true. In this section it will be apparent why, when Re( ) < we cannot have ′ ( ) = 0.
Consider first the simple thought experiment in which we ignore ( ) and let ( ) ∶= ( , ) with = and = 1 such that the behaviour of ( ) becomes a surogate for the behaviour of ( ) and we ask about the non-trivial zeros of ( ) and the zeros of ′ ( ). When Re( ) < and falls then P ℎ , ( ) grows larger more rapidly than P ℓ , ( ) grows. When Re( ) > and rises then P ℎ , ( ) shrinks more rapidly than P ℓ , ( ) shrinks. These behaviours explain why the kappa vectors separate one way on one side of the critical-line and the other way on the other side. When changes in magnitude occur these are not a uniform scaling along the pathway but have a proximo-distal gradient. In P ℎ , ( ) the effects are greatest proximally and in P ℓ , ( ) the effects are more marked distally. These differences in behaviour account for why the separated kappa vectors lie on one side of when < and on the other side when > . The derivatives provide a means of counteracting these two behavioural differences. Differentiation reverses the direction of every vector but more importantly it multiplies each ℛ by ln and each ℒ by ln ( ), and so since ≡ it can be appreciated how both behavioural differences outlined above are overcome only when > . The non-trivial zeros of ( ) lie on the critical-line and the zeros of ′ ( ) to its right, but ( ) is not ( ). We have no anxieties over ignoring = ( ), since this parameter is a refinement that breaks no symmetry but merely displaces the values of the non-trivial zeros of lambda a little from each neighbouring . However, we do have anxieties over ( ) but these are easily dismissed. The The series are then simply ℎ , ( ) = ℛ (84) and ℓ , ( ) = ℒ . (85) The differential of the partial zeta function is ( ) = − ln ( ) (cos( ln( )) − sin( ln( ))). Clearly however, is less helpful than Equation (78) since the term ( ) does not vanish at the zeros of the differential.
3.8.1. An example of ( ) = 0 In Figure 36 three pairs of pathways are plotted for the matched finite series ℎ , ( ) and ℓ , ( ) each with 34 terms. Added on to the end of P ℓ , ( ) in blue are shown the P( ( )) in green. The index of negation is set at = 64 , with the justification for that value explained below. When  In Figure 37 it can be seen that adjusting upwards narrows the blue loop and moves the double-point from zero towards the apex of the loop to produce in black the cycloid-like curve of ( ) at fixed = with its apex at ( + ) where it meets the cycloid-like curve of ( ) for fixed = . Curves for ( ) over short intervals embracing are shown for = in green and for = 1 in red passing through the origin in roughly opposite directions. Each crosses ( ) at /2. The loop in ( ), shown in blue, is with fixed at ≈ ( ) (Roman and not Greek ) so placing a double-point close to zero and providing a mental image of the type of curve that would have to exist if RH were false. We imagine that = rather than ≈ ( ) and the two values of at the double-point would be and . The following would still apply: the differentials would relate as follows ( ) = ( ) , and this is seen with the two curves of ( ), in green and orange, crossing the two regions of ( ) at /2 -note the directions indicated by the arrows. The blue loop has a minimum in its differential at the apex, at a value that lies closer to than ; note the blue dots making rapid progress before slowing at the apex and then speeding up before finally slowing at high values of .
Adjustment of allows the loop to shrink, and the two values of at the double-point approach one another until they meet at a value we call at which point we designate as . Two cycloid-like curves meet at ( + ); in red ( ) for = over a short interval in and in black ( ) for = over an interval in . Just as in Figure 37 we have (1 − ) > ( − ), in our mental image ( − ) > ( − ). Since = 1 − if RH is false there must be a zero of the differential with < . We now take part of Figure 36 and illustrate the directions, and schematically the magnitudes, of the partial differentials to produce Figure 38. For values of significantly removed from the vicinity of a ∈ { } the kappa vectors of the pathways P ℎ , ( ) and P ℓ , ( ) do not cross one another. In the vicinity of a non-trivial zero the vectors may cross. Figure 36 shows the meeting of the pathways of P ℎ , ( ) and P ℓ , ( ) , there is symmetry but no "distal superposition" since ∉ { }. If we were to continue a little way further towards P ℎ , ( ) and P ℓ , ( ) by allowing and to rise above we would soon find paths sepatating and eventually ℎ , ( ) ≠ ℓ , (s).
In Figure 39 there is a zero of the differential and it can be seen that P , ( ) meets P  The pathway P ℓ , ( ) does not meet P , ( ) unless P ( ) is added in, see Figure 40 . In Figure 40 the ( ) vector in purple runs between the ends of the kappa vectors but could have been placed to reflect and would then have ended in the red circle.
Appendix A6 tabulates some example calculations in relation to the magnitudes of the vectors for the differentials.

Discussion
In 1826 Niels Abel wrote to his friend Bernt Holmboe saying it was a disgrace to base any proof on divergent series, "Les séries divergentes sont en général quelque chose de bien fatal et c'est une honte qu'on ose y fonder aucune démonstration." which is often misquoted as "divergent series being an invention of the devil". In the work presented here, a partial Euler's zeta series to tau terms, designated ( ) with = − 1 foreshortens the divergent series at a final pseudo-convergence such that in the critical-strip ( ) ≈ 0. Euler's zeta has a proximal pathway of kappa terms, with = − 1 and a distal pathway thereafter which forms superstructures whose principal-axes can be considered to be vectors of a complementary finite series. Euler's zeta acquires convergence through negation of every term which is multiplied by − 1. This is the function ( ); a modification of Dirichlet's eta. The non-trivial zeros of ( ) are also the non-trivial zeros of ( ).
The infinite series ( ) can be closely followed by the sum of three finite series in that ( ) = ( ) + ℓ , ( ) − ℎ , ( ). For any given , as long as ≫ , the matched vectors of the two series of ( ) share mirror-image arguments about a line of reflection for all , but their magnitudes equate only when = . These geometric constraints limit the non-trivial zeros to the critical-line as hypothesised by Riemann and force the differentials to the right. If for some with ( ) ≠ 0 there was also a ( ) = ℎ , ( ) − ℓ , ( ) this zero of ( ) would not to be independent of , such instances can only arise at the trivial zeros when is an integral multiple of ( ) and = 1. However, if in the same way a hypothetical loop in ( ) for fixed were imagined to generate a double-point, and that double-point were to be imagined at zero, then a change in to + 1 would clearly preclude its existence in ( ) and since all non-trivial zeros have to exist for all values of the imagined ∉ { }. Loops are only preserved if rises to where ∈ ℕ. The two pathways P ℎ , ( ) and P ℓ , ( ) start from the origin and set off in different directions. If their terminal vectors ℛ and ℒ intersect over an interval in there can be a zero of ( ) in that interval. Either function alone, ℎ , ( ) or ℓ , ( ), can be used to identify those intervals, and every element of the set { } for = lies in such an interval. The pathways have vectors with matching orientations that are invariant under changes in but their magnitudes are not invariant. In this analysis there is no room for a zero off the critical-line. If one were hypothesised to exist above or below then its partner could not come into existence by changing since ℛ and ℒ will either not intersect or if they do intersect they will not intersect with symmetry and that intersection will not be at . When there is asymmetry a pathway's distal superstructures will not overlie the proximal vectors of the appropriate complimentary pathway.
A model for simultaneous zeros at the same is possible in ( ) when ≅ ( ) since there will be a loop in ( ) for fixed with a double-point near the origin however the mechanics of such simultaneous convergence are sensitive to and are driven by the principal orientation of the pathways in relation to their final collapse to (1,0) as Re( ) → ∞. Were there a mechanism, capable of generating ( + ) = 0 = + which would disprove RH it would have to be a mechanism whose output is unaffected by changes in .
No mechanism is capable of generating loops in ( ) which are stable under changes in and also generate similar loops in ( ) that only appear at certain values of e.g. when ≅ ( ) . Furthermore, if there were a ( + ) = 0 = + and we then allowed ≅ there would be a requirement for two loops in ( ) for = and a triple-point formed by the addition of 0 ≅ (1 + ). No pathway to convergence could be envisaged to behave in such a way though a rigorous proof along these lines has not been found, see Figure 41. The blue curve (a) has no zero, the red curve (b) has one zero, and the black loop has two zeros. A mechanism to create three zeros seems a challenge too far but one for which a rigorous proof has not been found.
Any loop in ( ) for some with a double-point for two real parts < , not necessarily at zero, has two limbs, a primary one from reaching the apex of the loop and a secondary one from the apex to . At the apex, ( ) will be a minimum and the apex will have a value of closer to than . If is adjusted in the correct direction the interval between and will alter with rising to and falling to . It is easy to find examples of ( ) = 0 when > but the geometry of convergence of the pathway of the differential necessarily precludes ( ) = 0 for Re( ) < . This is a corollary of RH.
The lambda function ( , ) has perfect bilateral symmetry, and it limits its own zeros to the critical-line and those zeros have a strict link to the location of each . Lambda gives some small uncertainty in the Im( ), with each zero hosted in a narrow imaginary interval with the uncertainty related to ( ), but the Re( ) is limited to . There is a relationship between and that is evident from calculations at low values of . That relationship has not been fully characterised and has been set aside as it is of no material consequence for the arguments of this paper.
When Euler's zeta function is added to ( ) it breaks the symmetry since , ( ) = ( ) + ℓ , ( ), and it is this symmetry breaking which permits the generation of loops in ( ) for fixed .
A loop in ( ) for fixed has the double-point ( + ) = ( + ) and raises the possibility of hypothetical simultaneous zeros. It is this same asymmetry which allows pathway collapse on rising outside of the critical-strip to drive all pathways towards their final demise at (1,0). Any such mechanism which generates loops in ( ) for fixed would be highly sensitive to arg ℛ and ( ) and so to changes in . The clarity obtained by the function ( ) is to realize that it is the symmetry breaking of ( ) that permits + = (1 + ) = 0 when = ( ) .
Once > the symmetries are evident and when = the term ( ) = 0 allows ( ) = 0 to reliably specify short regions of the imaginary axis which can host the non-trivial zeros of Riemann's zeta function. Importantly ( ) = 0 confines the hosting regions to Re( ) = .

Conclusions
Two finite vector series ℎ , ( ) and ℓ , ( ) are described whose paired terms have mirror-image arguments about a single line of reflection but whose magnitudes equate only when Re( ) = . The intersection of the final vector of either one of these series with the line of reflection allows identification of a small region of capable of hosting a non-trivial zero. All non-trivial zeros have such a relationship with the line of reflection at the meeting of their final vectors. The cardinality of the sets rises slowly with Im( ) allowing confidence that, no matter how high a region of is contemplated, a comparatively short finite computation would allow identification of those hosting regions.
The intersection of the final vectors of the series is such that the non-trivial zeros of the Riemann zeta function can only come into existence when there is perfect symmetry. Symmetry breaking restricts the non-trivial zeros to Re( ) = . The asymmetry of the differentials of the two series when Re( ) ≠ is such that it immediately follows that the zeros of the differential can only be found when Re( ) > . This relationship has implications for the nature of loops in ( ) for fixed that together with the functional equation preclude all non-trivial zeros from having Re( ) ≠ .
This section illustrates the structures that lie between the vectors near the point-of-inflection of a series of ℛ superstructures. The notation is { / } with being the number of sides in a regular polygon or the number of vertices in a regular star polygram, the is the neighbouring vertex in a clockwise direction to which vector moves to.  In Figure A1 the larger vectors (11| ) which neighbour the structures are not shown, but would be roughly co-linear with each other and in-line with the missing segment of each near regular structure. The value of ̅ is such that the change in argument at each vector is so that from an vector to the next vector we have roughly equal arguments that when summated bring us full circle to give the collinearity seen at inflection. The { /0} structures are collapsed sections that essentially oscillate along one line making no advance in the Argand plane. These are located at pseudo-convergences along the pathways. Details appear in Table A1.

A2. A Note on the collapse of structures either side of 2 |
Inherent in the specification of R is the absence of the odd values of either side of the where 2 | . The is illustrated in Figure A2. In Figure A3 an early part of a pathway is shown and in the table the ratios of interest.
Part of a pathway to convergence is shown to illustrate four relatively early vectors in a distal pathway where inadequacies of approximations should be most evident; reassuringly the errors are small. If we look closer to convergence we find much smaller errors, see Table A3.
In this exercise was determined from the intersection of ℒ and ℛ at known non-trivial zeros. This is formally the same as determining to satisfy for each ℓ , ( ) + ℒ = ℎ , ( ) + ℛ .
In Figure A.2 is plotted against for all the in an interval in .  Figure A4 shows the frequency of oscillation relates to and Figure A5 adds in a plot for as a function of .  t Figure A5. These plots illustrate the relationship between in red on the (right-hand axis) representing the intersection of ℒ and the ℛ vector for all the non-trivial zeros in the interval = 100 to = 630 and in blue (left-hand axis) which is the residual associated with after rounding to the nearest integer.
The relationship between and without appears in Figure A6. Figure A6 uses = + 1 but the kappa vector is independent of l and so this does not affect the plot. Kappa dot is by definition limited to − < < and 0 < ≤ 1, however appears to have bounds of and 1 − which occur at = and = − respectively. Figure A7 shows an example of intersection when ≈ 0. If at a non-trivial zero the difference in arguments were exactly then the ℓ vector and the ℛ vector would be collinear and would not be defined.

An example of a non-trivial zero with nearly collinear kappa vectors
In the following example a non-trivial zero with = + 1 is illustrated with the kappa vectors nearly collinear to each other where they meet. (b) Both pathways in full; the bottom right hand corner shows the associated pathway P( ( )) which nearly vanishes at the non-trivial zero. (c) The partial Euler's zeta, is taken from (b) and enlarged with P ℎ , ( ) in red and P( ( )) in blue. The black circle represents the end of the partial series with for = lying near the point-of-inflection in the ℛ of ( ). In Figure A9 it can be seen that the kappa vectors can be almost collinear. It is likely that true collinearity would not occur for any . The figure also illustrates that the partial Euler's zeta function nearly vanishes when = + 1. In Figure A10 the two kappa vectors are heading almost in the same direction which seems counter intuitive but this is possible since subsequent vectors soon fall in line with complimentary superstructures.

An example of an intricate pathway and kappa vectors at an extreme angle
A5. An example showing failure of superposition either side of a zero and showing "opportunistic superstructures" in the proximal pathway.
In Figure A10 a short region of the mirror-image pathways for values of slightly below a in (a), for the in (b) and for a slightly above the in (c). In (b) there is symmetry and perfect superposition at the zero either side of such that the light blue dashed line is the overlap extending (ℓ ( )) beyond terms. extended beyond the ℓ vector to show the congruence of the pathways. It is evident that this non-trivial zero of ( ) is easily located to a small interval in by the finite series using = 425 and = 426.
In Figure A11 a small interval in is illustrated to show that the superposition of the more distal pathways on the proximal pathways provides a mechanism for isolating non-trivial zeros and narrowing down the region of intersection that could host a non-trivial zero. It can also be appreciated how slight changes in would separate pathways and preclude superposition. Interestingly, there appear to be ℛ superstructures in P ℓ , ( ) before its kappa vector yet the vectors of P ℎ , ( ) clearly do not equate to what might be imagined to be the principal-axes of these superstructures. This is simply because the vectors of P ℓ , ( ) do not satisfy Equation (37). These structures appear because sequential ln , which is the reduced form, generate a closely falling sequence, however since ln

A6. A comparison of the magnitudes of the principal-axes of ℛ structures by summation, and the complementary vectors ℛ by single calculation
A comparison was made between ℛ for P(ℎ ( )) at low and the magnitude of the principal-axis of the related ℛ in ( ), here designated Δ ̂ . The chosen parameters were = 5 and = 40433.69 over a range of values of . The ℛ was determined by Equation (17)  The magnitude of the principal-axis of the ℛ was determined from the location of the start and finish points of the axis after summation of the partial series. Averaging was applied as in Equation (13) The results appear in Table A5 with rising from 0 to 1.1. The same exercise was repeated for the differentials. A comparison being made between the The magnitude of the principal-axis of the ℛ which represents the superstructure in P( ′ (s)) was determined from the location of the start and finish points of the axis after summation of the partial series. Averaging was applied as in a modification of Equation (13) The results appear in Table A6 with rising from 0 to 1.1. value of at a cycloid like curve where the velocity will be zero | and ∤ | means divides whilst ∤ means does not divide Pathways notation: a summary The notation for the four principal pathways appears in Table B1 with the pathway formulation, the name of the associated vector and the name of an indexed superstructure formed by a set of small vectors in that pathway forming the paired pseudo-spirals. The index of a superstructure indicates its position in the pathway with the superstructure that embraces convergence or the final pseudo-convergence preceding divergence having an index of 1. Superstructures exist in the distal part of a pathway after the kappa vector. has no index because is its own index Each pathway has a named vector, an associated distal superstructure, an index and vector that separates proximal from distal pathways. A distal superstructure has a principal-axis upon which a vector from another pathway may be superimposed with meaning. Thus ℛ can be superimposed on an ℛ and may be superimposed on an ℳ . Some ℒ may have an ℛ superimposed upon them.