Farey Sequences and the Riemann Hypothesis

Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.


Introduction
The Farey sequence F x of order x is the ascending series of irreducible fractions between 0 and 1 whose denominators do not exceed x.In this article, the fraction 0/1 is not considered to be in the Farey sequence.Let A(x) denote the number of fractions in F x .A(x) = ∑ x i=1 φ(i) where φ is Euler's totient function.For v = 1, 2, 3, ..., A(x) let δ v denote the amount by which the vth term of the Farey sequence differs from v/A(x).Franel (in collaboration with Landau) [1] proved that the Riemann hypothesis is equivalent to the statement that |δ 1 | + |δ 2 | + ... + |δ A(x) | = o(x 1 2 + ) for all > 0 as x → ∞.Let M(x) denote the Mertens function (M(x) = ∑ x i=1 µ(i) where µ(i) is the Möbius function).Littlewood [2] proved that the Riemann hypothesis is equivalent to the statement that for every > 0 the function M(x)x −(1/2)− approaches zero as x → ∞.Mertens conjectured that |M(x)| < √ x.This was disproved by Odlyzko and te Riele [3].

Shorter Intervals of Farey Points
Let r 1 , r 2 , ..., r A(x) denote the terms of the Farey sequence of order x and let h(ξ) denote the number of r v less than or equal to ξ. Kanemitsu and Yoshimoto [4] proved that each of the estimates ∑ r v ≤1/3 (r v − h(1/3) 2A(x) ) = O(x 1/2+ ) and ∑ r v ≤1/4 (r v − h(1/4) 2A(x) ) = O(x 1/2+ ) is equivalent to the Riemann hypothesis.Let n = 4, 5, 6, ..., and let j = n/2 .Let y x (n) denote the number of fractions less than 1/n and let z x (n) denote the number of fractions greater than 1/n and less than 2/n in a Farey sequence of order x.(If x ≤ n, set y x to 0. If x ≤ j, set z x to 0. If x > j and x < n, set z x to x − j.If x = n, set z x to j − 1 if n is even or j if n is odd.)Franel proved that M(x) = ∑ A(x) v=1 e 2πir v , so there should be some discernible relationship between M(x) and y x (4) − z x (4).The "curve" of y x (4) − z x (4) values resembles that of M(x) in that the peaks and valleys occur roughly at the same places and have about the same heights and depths.See Figure 1 for a plot of M(x) for x = 1, 2, 3, ..., 5000.See Figure 2 for a plot of y x (4) − z x (4) for x = 1, 2, 3, ..., 5000.Let h x (n) denote ∑ x i=1 (z x/i (n) − y x/i (n)).
√ n, n = 1, 2, 3, ....The ratio of this value for n = 100 to the value for n = 1 is approximately 0.338 (0.14/0.4145), but based on the above empirical evidence, the −p 1 (1)x + ∑ x i=1 δ 1 ( x/i )Λ(i) values are "stretched" by a factor of 100 to approximately give the −p 1 (100 , it should be possible to use this technique to find a −p 1 (n)x upper bound.

Conjecture 4.
When k is prime, the ∑ x i=1 c k ( x/i ), x = 1, 2, 3, ..., values fall on the line segments at least one value falls on every line segment.
See Figure 131 for a plot of −2.668x + ∑ x i=1 κ 100 ( x/i )Λ(i) and 5.66(−0.4714x+ ∑ x i=1 100 ( x/i )Λ(i)) for x = 1, 2, 3, ..., 100 (5.66≈2.668/0.4714).The peaks and valleys of the two curves occur at the same places and have almost the same magnitudes.Only 201 approximate limits accurate to about four (or possibly more) decimal places were used for the convolutions.This may account for the erratic y-intercepts in some of the linear least-squares fits.Even when 1001 zeta function zeros are used, the y-intercepts in some of the linear least-squares fits of the κ convolutions are erratic.For a linear least-squares fit of ∑ x i=1 100 ( x/i )Λ(i) for x = 1, 2, 3, ..., 10000 where 101 of the limits corresponding to the imaginary parts of the Dirichlet character modulo 11 are used, p 1 = −1.628and p 2 = 3.075.See Figure 132 for a plot of −2.668x + ∑ x i=1 κ 100 ( x/i )Λ(i) and −1.64(1.628x+ ∑ x i=1 100 ( x/i )Λ(i)) for x = 1, 2, 3, ..., 100 (−1.64 ≈ −2.668/1.628).The peaks and valleys of the two curves occur at the same places and have almost the same magnitudes.

For
Of course, the number of limits used cannot remain fixed as n → ∞ and x → ∞.For the linear least-squares fit of ∑ x i=1 n ( x/i )Λ(i) for x = 1, 2, 3, ..., 1999999, n = 50000, and where 40 elements of the sequence for the principal Dirichlet character modulo 13 are used, the p 1 and p 2 values are 0.5001 and 5.844 respectively.For the linear least-squares fit of ∑ x i=1 κ n ( x/i )Λ(i) for x = 1, 2, 3, ..., 1999999, n = 50000, and where 40 nontrivial zeta function zeros are used, the p 1 and p 2 values are 2.649 and 41.15 respectively.See Figure 133 for a plot of −2.649x + ∑ x i=1 κ n ( x/i )Λ(i) and 2.649 0.5001 (−0.5001x + ∑ x i=1 n ( x/i )Λ(i)) (superimposed on each other) for x = 1, 2, 3, ..., 100.For x up to 2000, the maximum difference between the two curves is 0.1793.

Conjecture 9.
If m is odd and greater than 3, m−3 2 log(x) is greater than or equal to 2 ∑ i|x (1,m) ( x/i )Λ(i) − log(x), m does not divide x.
See Figure 150 for a plot of ∑ i|x δ n ( x/i )Λ(i) and − log(x) where n = 1 for x = 1, 2, 3, ..., 999 (1000 limits accurate to about 6 decimal places were used).Positive values occur only when x has exactly two prime factors (not necessarily distinct).See Figure 151 for a plot of ∑ i|x δ n ( x/i )Λ(i) and − log(x) where n = 2 for x = 1, 2, 3, ..., 1999.Positive values occur only when x has at least three distinct prime factors.

Materials and Methods
The figures referenced above are given at the following link."www.darrellcox.website/Binder6.pdfA C program for computing the , κ, and δ convolutions is at the following link."Include" files are at the subsequent links.The 201 limits for the convolutions are included."www.darrellcox.website/test1g.htm""www.darrellcox.website/zeros1.htm""www.darrellcox.website/table2g.htm"Bibliography