Computer Science and Mathematics

Abstract: The Riemann Hypothesis (RH) is proved via a new expression of the completed Riemann zeta function $\xi(s)$, obtained through pairing the conjugate zeros $\rho_i$ and $\bar{\rho}_i$ in the Hadamard product while accounting for zero multiplicities (which are uniquely determined, although their specific values remain unknown), i.e. ξ(s)=ξ(0)∏ρ(1−sρ)=ξ(0)∏i=1∞(1−sρi)(1−sρ¯i)=ξ(0)∏i=1∞(βi2αi2+βi2+(s−αi)2αi2+βi2)mi where $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$, $\bar{\rho}_i=\alpha_i-j\beta_i$, with $0<\alpha_i<1$, $\beta_i\neq 0$, $0<|\beta_1|\leq|\beta_2|\leq \cdots$, and $m_i\geq 1$ is the multiplicity of $\rho_i/\bar\rho_i$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have ∏i=1∞(1+(s−αi)2βi2)mi=∏i=1∞(1+(1−s−αi)2βi2)mi which, owing to the divisibility of entire functions, uniqueness of $m_i$, and the irreducibility of each real quadratic polynomial factor, is finally equivalent to αi=12,0<|β1|<|β2|<|β3|<⋯,i=1,2,3,… Thus, we conclude that the RH is true.

Abstract: Black box algebra is a part of computational algebra focused on probabilistic methods of solving problems in very large finite algebraic objects where deterministic approaches simply do not work. The present paper provides a construction of structural proxies for black box rings encrypting rings of 2×2 matrices over finite fields of odd order.

Abstract: We develop a centered geometric and combinatorial framework for the distinct-prime Goldbach problem---the assertion that every even integer $2N \geq 8$ is the sum of two \emph{distinct} primes. The rigorous content is an exact geometric reformulation: for each $N \geq 4$, the problem is equivalent to finding an integer $M \in [1,N-3]$ such that $N-M$ and $N+M$ are both prime, or equivalently such that the L-shaped region between nested squares of side lengths $N$ and $M$ has area $(N-M)(N+M)$ with both factors prime. We define the centered sets $C_N=\{M: N-M \text{ is prime}\}$ and $D_N=\{M: N+M \text{ is prime}\}$ inside $\{1,\ldots,N-3\}$; then $C_N\cap D_N\neq\varnothing$ is exactly the distinct-prime Goldbach assertion for $2N$. We also study the diagnostic set $E_N$ of admissible half-differences arising from straddling prime pairs; its elements correspond to Goldbach partitions of nearby even numbers. A decomposition by upper primes expresses $|E_N|$ in terms of the half-differences generated by each prime $Q\in(N,2N)$ minus explicit collision sets. Under a fixed upper-prime collision hypothesis, explicit estimates of Rosser--Schoenfeld and Dusart imply positivity of the statistic $H(N)=\log^2(2N)-((N-3)-|E_N|)$ for all sufficiently large $N$, and finite computation verifies the remaining tested range. Positivity of $H(N)$ gives a pigeonhole result $C_N\cap E_N\neq\varnothing$. The distinct-prime Goldbach conjecture is not proved here; the remaining gap is a centering theorem, equivalently a uniform lower bound for $R(2N)=|C_N\cap D_N|$.

Abstract: We present a streaming algorithm for the full finite fractional Fourier orbit over a prime field Fp with p≡1(mod4). The construction rests on a projector-orbit identity: once the four eigenspace projections of the input signal under the normalized finite Fourier operator have been computed, every fractional power is recovered by a degree-three scalar recombination, indexed by the orbit parameter. The streaming phase materializes the full orbit of p−1 output vectors in O((p−1)2) field operations, matching the Ω((p−1)2) output-size lower bound up to constant factors. For the subclass of primes with 2 a primitive root --- which we call dyadic-symmetry-complete primes, characterized by the congruence p≡5(mod8) --- a contribution-array reformulation eliminates general modular multiplication from the streaming phase, leaving only modular additions and shift updates. This subclass is structurally distinct from the Fermat-prime NTT setting; only p=5 is common to both. We give self-contained proofs of the four-cycle and projector identities, prove the necessary congruence and the Fermat-prime separation, and report timing experiments validating correctness and the O(n2) streaming scaling against a dense matrix-vector baseline.

Abstract: In this paper, we establish the coincidence conditions for the generalized core-inverse and (b,c)-inverse of a Banach element. We introduce the generalized core-(b,c)-inverse, a unifying extension characterized via quasinilpotents. A novel limit-based approach is developed to investigate this generalized inverse, yielding polar-like properties and image-based representations that extend the core-(b,c) inverse framework.

Abstract:

We reformulate the Growth Theorem criterion for the Riemann hypothesis as a global positivity problem for a modular oscillatory kernel arising from the Riemann xi function. Using the fact that the squared modulus $|\xi(\sigma+it)|^2$ is strictly increasing with respect to $\sigma$ for $\sigma > \frac{1}{2}$, we reformulate this as a positivity condition on a theta-kernel double integral. The half-plane $\sigma > 1$ is closed unconditionally using the symmetric Hadamard product. The remaining obstruction is thereby localized to the critical strip $\frac{1}{2} < \sigma \leq 1$. Introducing diagonal coordinates $(a,b) = (\frac{u+v}{2}, \frac{u-v}{2})$, we decompose the kernel into a positive diagonal sector and an oscillatory off-diagonal sector, and show that the Riemann hypothesis is equivalent to the positivity $ I(x,y) = \iint_{a > |b|} \Phi(a+b)\Phi(a-b)\,K_{x,y}(a,b)\,da\,db > 0 \qquad (x \in R,\ y > 0). $ We give an exact characterization of the positive-amplitude structure of this integral and state the resulting theta-kernel positivity problem in its sharpest form.

Abstract: In this paper, we study when the core inverse of a ring element coincides with its (b,c)-inverse, i.e., core-(b,c)-inverse. To establish a broader framework for generalized inverses, we define core-EP-(b,c)-inverse that serves as a natural extension of the generalized core inverse and (b,c) inverse. We characterize this new generalized inverse by combining the core-(b,c)-inverses and quasinilpotents. This generalized inverse is thereby examined through a novel limit-based approach. The polar-like properties of the core inverse are presented. Finally, we investigate the connection between the core-EP-(b,c) inverse and core-(b,c)-inverse. In particular, we derive the reverse law for the core-EP inverse by means of core-EP-(b,b) inverse.

Abstract: We establish unconditional almost-all ternary results for the restricted weighted Goldbach sum W_(a,q) \( W_{a,q}(n) := \sum_{\substack{p_1+p_2+p_3=n \\ p_1 \equiv a \pmod{q}}} (\log p_1)(\log p_2)(\log p_3), \quad q \geq 1,\ \gcd(a,q)=1 \), together with conditional extensions under the Density Hypothesis and the Generalized Riemann Hypothesis. Main results. (A) Almost-all via prime anchoring [Proved]. For every \( A > 0,\#\{n \leq X,\ n \text{ odd} : W_{a,q}(n) = 0\} \ll_{A,q} X(\log X)^{-A}. \) (B) Almost-all with full asymptotic [Proved]. For all odd \( n \leq X \) outside an exceptional set of size \( O_{A,q}(X(\log X)^{-A}),W_{a,q}(n) = \frac{J_{3,a,q}(n)}{\varphi(q)}\, n^2 + O_{A,q}\!\left(\frac{n^2}{(\log n)^A}\right), \) where the ternary singular series \( J_{3,a,q}(n) > 0 \) is positive and effectively computable. For \( (a,q) = (3,4) \) the main term equals \( (C_2 S(n)/4)\,n^2 \) with \( C_2 \in [0.6601618157,\, 0.6601618160] \). The minor-arc bound \( K_{\min}(q, A) \leq 2.10/\sqrt{\varphi(q)} \) is proved in full, self-containedly, via the exact \( L^2 \)-orthogonality identity \( \|S_{a,q}\|_{L^2}^2 = \|S\|_{L^2}^2/\varphi(q) \) and a Hölder \( (L^2, L^\infty, L^2) \) factorization. The exceptional-set constant is \( C_{\text{ternary}}(A,q) \leq 4.41/\varphi(q) \cdot 2^A \) [Proved]. Articles 1, 2, and 3 by the same author of this work provide all the black box input data. No GRH, no density hypothesis, and no ternary sieve are used in Parts (A) and (B).

Abstract:

In this article, we offer a complete, self-contained, and entirely elementary proof of the mean-square estimate for the Chebyshev function. From this we deduce the convergence of the integral is valid, thus proving the validity of the Riemann hypothesis. The proof primarily employs elementary estimates of the Chebyshev function, the Cauchy-Schwarz inequality, and a dyadic decomposition (with Abel summation applied in the appendix), in which the argument results of this article are already optimal within the elementary framework and sufficient to derive the convergence of the required integral that it is a suffficient condition for the Riemann hypothesis. In particular, the appendix of this paper provides a theoretical complement linking integral convergence, pointwise bounds and analyticity, and concludes that the well-known $o$-bound is valid, thereby reconfirming the validity of the Riemann hypothesis. In other words, we give a self-contained elementary proof for the mean-square estimate that $\displaystyle\int_2^{X} \bigl(\psi(t)-t\bigr)^{2}\,dt = O(X^{2}\log^{2} X),$ where $\displaystyle \psi(x)$ is the Chebyshev function. From this we deduce that $\displaystyle\int_{1}^{\infty}\frac{|\psi(x)-x|}{x^{\frac{3}{2}+\varepsilon}}\,dx < \infty$ holds for every $\varepsilon>0,$ thus concluding the integral $\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ converges absolutely for every $\varepsilon>0,$ so that the integral $\displaystyle \int_1^{\infty} \frac{\psi(x)-x}{x^{\frac{3}{2}+\varepsilon}}\,dx$ converges conditionally for every $\varepsilon>0,$ whereas the integral converges conditionally $\iff \text{RH},$ so then the Riemann hypothesis is true. In particular, the absolute convergence of the integral is equivalent to the conditional convergence of the integral, either of which is equivalent to the $o$-bound: $|\psi(x)-x| = o(x^{\frac{1}{2}+\varepsilon}),$ and all of them imply the $O$-bound: $\psi(x)-x= O(x^{\frac{1}{2}+\varepsilon})$ is also valid for every $\varepsilon>0,$ thus reconfirming the validity of the Riemann hypothesis.

Abstract: In this paper, which is entirely unconditional, we prove a sharpened almost-all theorem with fully explicit effective constants for the restricted weighted Goldbach sum$$R_{a,q}(N) := \sum_{\substack{p_1+p_2=N \\ p_1 \equiv a \pmod{q}}} (\log p_1)(\log p_2), \quad q \geq 1,\ \gcd(a,q)=1,$$whose expected main term is $M_{a,q}(N) = C_2\,S(N)\,N/\varphi(q)$, where $C_2 = 0.6601618\ldots$ is the twin-prime constant and $S(N)$ is the binary singular series.Our results are organised around four pillars. **(I)** We give a complete character-pair decomposition of the second moment of the error $E(N) := R_{a,q}(N) - M_{a,q}(N)$, extracting the exact diagonal constant $G/(2\varphi(q))$, where $G = \prod_{p>2}(1+(p-1)^{-2}) \in [1.41320886, 1.41320899]$ is the Gallagher–Goldston constant. **(II)** We establish a uniform minor-arc $L^4$ bound$$\int_{\mathfrak{m}} |S(\alpha)|^4\,d\alpha \leq \kappa_{\mathrm{safe}} \cdot 2^A \cdot \frac{X^3}{(\log X)^A}, \qquad \kappa_{\mathrm{safe}} = 4.40,$$by combining the complete Vaughan identity with the Bombieri–Vinogradov theorem in integral form, giving an explicit derivation of $\kappa_{\mathrm{explicit}} = C_V^2\,c_{L^2} = 4.004$ before applying a rigorous 10% safety margin. **(III)** We derive the effective almost-all theorem$$\#\{N \leq X\ \text{even} : |R_{a,q}(N) - M_{a,q}(N)| > C(A,q)\,N(\log N)^{-3}\} \ll_{A,q} X(\log X)^{-A},$$with the explicit constant $K := 2C(1,4) \leq 38.82$, obtained from $C(1,4) \leq 19.41$ via a Stechkin-type optimisation. **(IV)** We prove a Pintz-type exceptional-set bound on $\{N \leq X : R_{a,q}(N) = 0\}$.Every statement in the main body carries the tag [PROVED]. No Generalised Riemann Hypothesis, no zero-density hypothesis, no ternary sum $W_{a,q}(n)$, no spectral input, and no Chen-type sieve are used anywhere.

Abstract: We introduce and study square-difference factor absorbing primary hyperideals (sdf-absorbing primary hyperideals) in commutative multiplicative hyperrings. A proper hyperideal I is called sdf-absorbing primary if x²-y² \( \subseteq \) I implies \( x+y\in\sqrt{I} \) or \( x-y\in I \). This class provides a proper common generalization of both primary hyperideals and sdf-absorbing hyperideals. We establish a comprehensive characterization, investigate the behavior under homomorphisms, localization, intersections, ascending chains, and Cartesian products, and identify the precise conditions under which the three notions coincide.

Abstract: Let R be a commutative ring with a nonzero identity and M a nonzero unital R-module. We introduce the concepts of weakly n-submodules and weakly (1, n)-submodules as module-theoretic generalisations of the weakly n-ideal and weakly (1, n)-ideal. A proper submodule N of M is called a weakly n-submoduleif whenever 0 ̸= am ∈N for some a ∈R and m ∈M , then a ∈(N :R M )or m ∈Nil(M )·M , where Nil(M ) = annR(M ). Similarly, N is called a weakly (1, n)-submodule if whenever 0 ̸= abm ∈N for some nonunit elementsa, b ∈R and m ∈M , then ab ∈(N :R M ) or m ∈Nil(M )·M . Every weaklyn-submodule is a weakly (1, n)-submodule, and every weakly (1, n)-submoduleis weakly 1-absorbing primary. We provide a six-fold characterisation, provestructure theorems classifying the rings and modules over which every propersubmodule belongs to these classes in particular, we show that for faithfulnitely generated multiplication modules, every proper submodule is weakly (1, n) if and only if R is a UN-ring or a product of two elds and investigatebehaviour under homomorphisms, localisations, and quotient modules.

Abstract: In this paper, whose main results are conditional on the density hypothesis or the Generalized Riemann Hypothesis, we establish a complete conditional hierarchy for the restricted weighted Goldbach sum $R_{a,q}(N) := \sum_{p_1+p_2=N,; p_1 \equiv a \pmod{q}} (\log p_1)(\log p_2)$, with expected main term $M_{a,q}(N) := C_2 S(N) N/\varphi(q)$, for fixed $q \geq 1$ and $\gcd(a,q)=1$.Under the Density Hypothesis DH($A$) (any $A \geq 2$), the exceptional-set exponent is shown to equal $\theta(A) = 1 - 2/(A+2)$ via a saddle-point argument, correcting prior formulas $\theta = 1 - 1/A$ and $\theta = 1 - 2/A$, which are both wrong. Under the Generalized Riemann Hypothesis (GRH), we prove the pointwise bound $R_{a,q}(N) = M_{a,q}(N) + O_{q,\varepsilon}(N^{1/2+\varepsilon})$ for all even $N \geq N_0(q)$, and derive the explicit threshold $\log N_0(4) = 45.93$ via a fixed-point iteration. We present both normalizations of the effective constant ($C^2_{4,\text{eff}} \approx 529$ and $\approx 2111$) and give a complete account of the discrepancy. A complete constant-chain audit carried out in Section 7 shows all three normalizations (values 529, 2111, and the independently reconstructed 2375) are consistent, and the worst-case certified bound is $\log N_0(4) \leq 46.1$, with no remaining caveat.We further provide:A certified computation showing all 122 primitive real Dirichlet characters of conductor $q \leq 200$ are free of Siegel zeros in the Stechkin critical interval, with global minimum $L_{\text{cert}} = 0.2344$ at $q = 163$. An unconditional restricted Chen-type theorem $N = p + P_2$, $p \equiv a \pmod{q}$, via Bombieri–Vinogradov. A conditional short-interval lower bound under GRH. None of these results proves the binary Goldbach conjecture or GRH. The paper establishes conditional results under explicitly stated hypotheses. This paper is a companion to "An Almost-All Theorem for a Restricted Goldbach Sum over Arithmetic Progressions with Explicit Unconditional Constants", whose results are used here as a black box.

Abstract: Robin's criterion equates the Riemann hypothesis with the inequality $\sigma(n) < e^{\gamma}\,n\,\log\log n$ for every $n > 5040$. By a theorem of Robin, if the Riemann hypothesis is false then infinitely many colossally abundant numbers fail this inequality, so the existence of a counterexample to Robin's criterion is equivalent to the existence of a colossally abundant counterexample. Combining structural properties of colossally abundant numbers with explicit estimates for the Chebyshev theta function and the Mertens product due to Aoudjit, Berkane, and Dusart, we prove an unconditional and effective upper bound on any such counterexample. Specifically, if $n > 5040$ is colossally abundant and violates Robin's inequality, then $n < N_{k}^{Y_{k}}$, where $p_{k}$ is the largest prime factor of $n$, $N_{k} = \prod_{i=1}^{k} p_{i}$ is the primorial of order $k$, and $Y_{k} \to 1^{+}$ is an explicit constant. Together with the lower bound $n \geq N_{k}$ that holds for any Hardy--Ramanujan integer, this confines every hypothetical colossally abundant counterexample to a narrow explicit window above the corresponding primorial. We deduce a conditional reformulation of the Riemann hypothesis localized to this window. This work refines the approach taken in the author's earlier article ``Robin's criterion on divisibility'', published in The Ramanujan Journal.

Abstract: In this paper we first give a new formula of the Liouville function and then by using the method for proving the Bateman-Horn conjecture, we give a parallel proof of the Chowla conjecture.

Abstract: An elementary and self-contained approach to the Euler--Mascheroni constant $\gamma$ is presented, based solely on Simpson's quadrature rule and the convexity of the function $f(x)=1/x$. By introducing Simpson-type weighted harmonic sums, local logarithmic increments are approximated by simple finite linear combinations of reciprocal integers. Sharp two-sided inequalities, derived from monotonicity and convexity, yield explicit control of the quadrature error and provide a purely numerical proof of the classical limit defining $\gamma$, without recourse to the Euler--Maclaurin summation formula.A key structural outcome of this framework is a decomposition $\gamma = ( \log [2] + 1 )/3 + \delta$, where the constant $\delta$ arises naturally as the limit of a rational sequence associated with Simpson-regularized harmonic sums. This formulation isolates a dominant oscillatory component and leads to a sequence with markedly faster convergence.This work highlights an unexpected connection between elementary numerical quadrature and one of the fundamental constants of analysis.

Abstract: ZPIF (Zero Pair Interaction Functional) is introduced as a quadratic spectral operator framework extending the classical explicit formula of the Riemann zeta function. Unlike the standard linear spectral decomposition, ZPIF incorporates second-order interactions between spectral modes within a Hilbert space formulation. The framework includes a rigorous operator definition, spectral expansion, trace-class regularization, and conditional convergence under truncation. A computational scheme based on numerical zeta zeros is also proposed. The novelty of ZPIF lies in introducing a quadratic spectral energy functional consistent with classical spectral heuristics without assuming unresolved conjectures. Numerical experiments demonstrate nonlinear growth behavior and quadratic interaction effects that are absent in classical linear formulations.

Abstract: For over a century, the distribution of prime numbers has been modeled as a stochastic process. This study presents results from a multi-year computational census that challenges this paradigm. Using a deterministic Sequential Reflection Filter implemented on a decentralized architecture, we analyzed a specific four-prime configuration, “The Southern Cross Constellation", across the range 10¹ to 2.2410¹⁴. The method targets twin-prime seeds and applies the symmetric reflection operator to generate the structure. We identified 6,365,871 unique prime quadruples exhibiting a consistent trailing-digit signature [9,1,9,1] with zero observed deviation. Additionally, we observe an “ironing effect,” characterized by a systematic reduction in relative variance η with increasing magnitude. At 10¹⁴, the relative variance η is reduced by a factor of more than 40 relative to 10⁹, at the level of confidence exceeding 7σ, indicating a transition into a highly regular, symmetric topological structure. These findings indicate the existence of a scale-invariant, deterministic lattice, governing prime distribution. This challenges the assumption of high-entropy randomness in prime-based lattices. The study identified the Golden-Gamma constant as the foundational principle governing the Southern Cross Constellation.

Abstract: In the course of extending the obtained regularities of lattice packings corresponding to layered lattices of the “Lambda” series in dimensions 1–24 to groups of higher dimensions, a hypothesis was formulated about the density function of the densest lattice packing of equal spheres in n−dimensional Euclidean space. The hypothesis can be used as an approximate method for solving the problem of lattice packing of equal spheres in n−dimensional Euclidean space.

Abstract: We introduce and study (2, n)-hyperideals in commutative multiplicative hyperrings, a new class of hyperideals that simultaneously generalizes n-hyperideals and 2-absorbing primary hyperideals. Our central result characterizes (2, n)-hyperideals intrinsically: a proper hyperideal I is a (2, n)-hyperideal if and only if it is 2-absorbing primary and √I = √0. As a consequence, a hyperring R admits a (2, n)-hyperideal precisely when R has at most two minimal prime hyperideals—provided all hyperideals are C-ideals, a condition that has no classical analogue and underscores a fundamental departure from ordinary ring theory. We establish a complete hierarchy among the main classes of hyperideals, prove stability under radicals and finite intersections, and characterize those hyperrings in which every proper hyperideal is a (2, n)- hyperideal. We further connect this theory to Krull dimension, Von Neumann regularity, and the structure of quotient hyperfields. Throughout, explicit counterexamples demonstrate where classical ideal-theoretic arguments break down in the multivalued setting, revealing the genuine novelty of the hyperstructure framework.