Abstract: The distribution of prime numbers has long been a central topic in analytic number theory. The Prime Number Theorem (PNT), which states that a number of primes less than x is \( x/log(x) \), provides a foundational understanding of this phenomenon. The further study of deep insights has led to the Riemann Hypothesis (RH), which implies explicit bounds on the error term in the PNT, thus enhancing the precision of the results derived from it. In this work, an algorithm is proposed for the estimation of the prime-counting function by counting the number of composite numbers eliminated during the sieving of the odd-number sequence. By applying this approach, it was found that variations in the length of the odd-number sequence during removal of composite numbers follow an oscillation pattern governed by the sinc function. Further analysis of such oscillations suggests that the function \( π(x) \) is composed of two terms: the main term, which makes the largest contribution to the distribution of the primes, and the error term, which is responsible for the accumulation of errors during the calculation of the \( \text{sinc} \) function values up to the limit \( m \), defined as \( √x /2 \). This leads to the proposal that the bound coefficient of the error term should be equal to \( √x/log x \), which is correlated with an estimate of this coefficient derived under the assumption of the truth of the RH. We hope that this perspective stimulates future work toward formalizing this approach to uncover deeper connections to the zeta function and prime number theory at large scale.

Abstract: This work presents a new approach to investigate the Riemann Hypothesis, combining analytical and computational methods. We develop a procedure to construct an integral operator K from the Fourier analysis of the prime counting error E(x) = π(x) −li(x). We investigate the hypothesis that the eigenvalues of this operator correspond to the imaginary parts of the non-trivial zeros of the Riemann zeta function ζ(s). Analytically, we examine the consequences of a possible normalization of the form ζ(s) = (1/π)arcsinh(Z(s)) +1/2, where Z(s) is a meromorphic function. We show that this structure imposes strong constraints on the location of zeros in the complex plane. Computationally, weverify our construction for the first 2000 zeros, obtaining correspondence with precision of 10−12. The statistical distribution of eigenvalue spacings follows the Gaussian Unitary Ensemble (GUE) with a p-value of 0.3129, consistent with known properties of the zeros of ζ(s). This study suggests new connections between analytic number theory, spectral theory of operators, and quantum systems, offering a promising perspective for future investigations of the Riemann Hypothesis.

Abstract: We present compelling numerical evidence supporting the Hilbert-Polya conjecture through the explicit construction of self-adjoint quantum operators whose spectra closely approximate the non-trivial zeros of the Riemann zeta function. We report the discovery of three fundamental constants (alpha, beta, gamma) satisfying alpha*beta*gamma = 2π that govern a conformal transformation unifying quantum systems with arithmetic properties. Numerical simulations demonstrate that atomic hydrogen orbitals, when transformed via Φ(z) = beta * asinh(z/gamma), exhibit nodes corresponding to zeta zeros with correlation coefficients exceeding 0.99. Furthermore, we identify potential signatures of these arithmetic patterns in cosmological data (cosmic microwave background, large-scale structure, supernovae), suggesting a profound connection between number theory and fundamental physics.

Abstract: The Riemann Hypothesis (RH) is proved via a new expression of the completed zeta function ξ(s), obtained through pairing the conjugate zeros​ in the Hadamard product while considering zero multiplicity, i.e. \( \xi(s)=\xi(0)\prod_{\rho}(1-\frac{s}{\rho})=\xi(0)\prod_{i=1}^{\infty}(1-\frac{s}{\rho_i})(1-\frac{s}{\bar{\rho}_i})=\xi(0)\prod_{i=1}^{\infty}\Big{(}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}+\frac{(s-\alpha_i)^2}{\alpha_i^2+\beta_i^2}\Big{)}^{m_{i}} \), wheree \( \xi(0)=\frac{1}{2} \), \( \rho_i=\alpha_i+j\beta_i \), \( \bar{\rho}_i=\alpha_i-j\beta_i \), with \( 0<\alpha_i<1, \beta_i\neq 0, 0<|\beta_1|\leq|\beta_2|\leq \cdots \), and \( m_i ≥ 1 \) is the multiplicity of \( \rho_i /\bar\rho_i \)​. Then, according to the functional equation \( \xi(s)=\xi(1-s) \), we obtain \( \prod_{i=1}^{\infty}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}}=\prod_{i=1}^{\infty}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}^{m_{i}} \), which, owing to the divisibility of entire function, uniqueness of \( m_i \), and the irreducibility of each polynomial factor, is finally equivalent to \( \alpha_i=\frac{1}{2}, 0<|\beta_1|<|\beta_2|<|\beta_3|<\cdots, i=1, 2, 3, \dots \) Thus, we conclude that the Riemann Hypothesis is true.

Abstract: Lately it is unfortunately found that there was a mistake of the sign of term \( (c-v)^2/2\sigma \) in Lemma 2.1 of the prior version, which prevented the proof of Theorem 1.1 from being completed as scheduled, here we will give an alternative argument.

Abstract: We develop a complete operator-theoretic and spectral framework for the Collatz map by analyzing its backward transfer operator on weighted Banach spaces of arithmetic functions. The associated Dirichlet transforms form a holomorphic family that isolates a zeta-type pole at s=1, while on a finer multiscale space adapted to the dyadic-triadic geometry of the Collatz preimage tree we establish a two-norm Lasota-Yorke inequality with an explicit contraction constant, yielding quasi-compactness, a spectral gap, and a Perron-Frobenius theorem in which the eigenvalue 1 is algebraically and geometrically simple, no other spectrum meets the unit circle, and the unique invariant density is strictly positive. The fixed-point relation is converted into an exact multiscale recursion for the block averages c_j, revealing a rigid second-order coupling with exponentially small error terms and asymptotic profile c_j~ 6^-j. This spectral classification forces every weak* limit of the Cesàro averages derived from any hypothetical infinite forward orbit to be either 0 or a scalar multiple of the Perron-Frobenius functional, with convergence to 0 occurring precisely under the Block-Escape Property. Since the forward map satisfies an unconditional exponential upper bound, whereas Block-Escape combined with linear block growth along a subsequence would impose an incompatible exponential lower bound, all analytic and spectral components needed for such a contradiction are complete, reducing the Collatz conjecture to excluding infinite orbits exhibiting Block-Escape without the supercritical linear block growth prohibited by the spectral theory.

Abstract: The Extended, Generalized, and Grand Riemann Hypotheses are proved under a unified framework, which is based on the divisibility of entire functions expressed as absolutely convergent infinite products of polynomial factors, where the uniqueness of zero multiplicities plays a critical role. Consequently, the existence of Landau-Siegel zeros is excluded, thereby confirming the Landau-Siegel zeros conjecture.

Abstract:

This article develops a structural framework that reduces Goldbach’s Strong Conjecture to a single short-interval analytic inequality. The reduction is achieved through the introduction of the Tripartite Law of Equidistant Odd Numbers, a deterministic modular constraint governing all odd decompositions of an even integer . Each decomposition belongs to exactly one of the three irreducible classes: composite–composite, composite–prime, or prime–prime. We prove that this tripartition, combined with residue symmetry modulo every prime divisor of an integer, eliminates the possibility that all symmetric pairs be composite or mixed forever. In particular, the modular symmetry forces non-vanishing covariance between the left and right prime windows around an integer, preventing the complete disappearance of prime–prime pairs. Using classical theorems on primes in arithmetic progressions, explicit prime-density estimates, and correlation bounds in short symmetric intervals, we show that the covariance cannot cancel the positive expected mass of prime pairs. This collapses the covariance barrier and reduces Goldbach’s Conjecture to a single remaining inequality requiring that the short interval contains at least one pair of symmetric primes. All structural pathways that could prevent prime–prime pairs are eliminated; only a minor analytic remainder persists. Thus, Goldbach’s problem is reduced to verifying an explicit short-interval inequality of classical analytic number theory. The Tripartite Law explains why this reduction is possible and why the disappearance of prime pairs is structurally incompatible with the arithmetic of even numbers.

Abstract: This paper presents a unified analytic–conditional framework for resolving Goldbach’s Conjecture. The first part develops an analytic demonstration based on the mirror-law symmetry of prime densities around E/2 and the overlap of symmetric windows derived from the Prime Number Theorem. This analytic structure shows that the density of primes on both sides ofE/2 necessitates the existence of at least one symmetric prime pair for every sufficiently large even integer E. The second part formulates Goldbach’s Conjecture as an equivalent conditional theorem requiring the validity of only two explicit lemmas: a local symmetry lemma (Lemma C) and a global overlap lemma (Lemma S). Appendix 8 provides a mathematical derivation of Lemma C, while Appendix 8B establishes a partial reduction strategy for Lemma S. Appendix 9 identifies an explicit threshold E₀ beyond which the analytic overlap is guaranteed. The resulting hybrid manuscript gives both (1) an analytic justification for the structure underlying Goldbach’s identity, and (2) a precise blueprint for reducing the conjecture to two explicit, finitely verifiable lemmas. This combined approach significantly narrows the gap between heuristic,analytic, and conditional pathways to a full resolution.

Abstract:

We study the additive and fractal structure of digit-restricted subsets of the unit interval, \( A_D=\Bigl\{ \sum_{n=1}^\infty a_n b^{-n} : a_n\in D\subseteq\{0,\dots,b-1\},\ |D|\ge2\Bigr\} \) defined by allowing only digits from D in base-b expansions. These sparse-digit sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous carry-propagation framework for base-b digit arithmetic, give sharp combinatorial criteria for intervals in \( A_D+A_D \) and \( A_D-A_D \), expand the proof of the similarity-dimension formula, and strengthen the dimension-jump theorem for iterated sumsets \( A_D^{(k)} \) by providing full justification and relevant references. A new Carry Stabilization Lemma, an expanded interval-criterion proof, and a related-work section situate these results within the literature on sumsets of Cantor sets, fractal addition theorems, and digit-based self-similar sets.

Abstract: This work investigates the relationship that exists between the parastrophes of some notion of inverses in quasigroups. Our findings revealed that, of the 5 parastrophes of LIP quasigroup, (23)- parastrophe is a LIP quasigroup, (12)- and (132)- parastrophes are RIP quasigroups, while (13)- and (132)- parastrophes are an anti-commutative quasigroup. Similarly, the (12)- and (132)-parastrophes of a RIP quasigroup are LIP quasigroups; the (13)-parastrophe of a RIP quasigroup is an RIP quasigroup, while the (23)- and (123)-parastrophes are anti-commutative quasigroups. As for the CIP quasigroup, only (12)- parastrophe is a CIP quasigroup; other parastrophes are symmetric quasigroups of order 2. Finally, (12)-parastrophe of WIP quasigroup is an IP quasigroup, (13)-,(23)- and (132)-parastrophes of WIP quasigroup are CIP quasigroups, while (123)-parastrophe of WIP quasigroup is a WIP quasigroup.

Abstract: This work establishes a complete arithmetic resolution of the Collatz Conjecture by decomposing the odd–to–odd dynamics into two complementary structures: a local residue–phase automaton and a global affine counting system. The reverse map R(n;k)=(2^{k}n-1)/3 is shown to act on the live residues {1,5} mod 6 through a finite residue–phase state space, while every admissible exponent k=c+2e induces an affine expansion factor $2^{k}$ whose inverse coincides exactly with the dyadic slice weight 2^{-k}.From this, every odd integer is seen to belong to a unique dyadic slice {S}{c,e}, forming a disjoint partition of {N}odd. Independently, the introduction of the zero–state index Z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic ladder m -> 4m+1 whose union also partitions the odd integers without overlap. We prove that these two partitions coincide exactly, yielding a unified global structure in which all odd integers arise from admissible lifts above anchors {1,5}.The locked forward–reverse equivalence T(n)=(3n+1)/2^{\nu_2(3n+1)} and R(T(n);k)=n then implies that forward trajectories cannot branch or diverge: each forward iterate lies on a single admissible ladder descending toward its zero–state origin at 1. Because the residue–phase automaton is finite and every ladder has a uniquely determined forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. All constructions, residue frameworks, and affine decompositions used in this paper are original to this work. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every forward trajectory converges to 1.

Abstract:

A theta--regularized inner product identity in rank one is established, linking a mixed theta--weighted Eisenstein pairing on \( \Gamma \)\H to the \( \sigma \)--derivative of \( \log|\xi(s)| \), up to explicit Euler factor correction terms arising from the \( G\times G \) doubling formalism. More precisely, for \( s=\tfrac12+\sigma+t \) it is shown that \( \frac{\partial}{\partial\sigma}\log\left|\big\langle\Theta(\cdot)E(\cdot,s),\ \Theta(\cdot)E(\cdot,1-\overline{s})\big\rangle_{\mathrm{reg}}\right|=2\,\mathrm{Re}\,\frac{\xi'(s)}{\xi(s)}\ -\ 2\,\mathrm{Re}\,\frac{\zeta'(2s)}{\zeta(2s)}\ +\ 2\,\mathrm{Re}\,\frac{\zeta'(2-2\overline{s})}{\zeta(2-2\overline{s})} \),as an identity of tempered distributions in t. On the critical line \( \sigma=0 \) the Euler corrections cancel and a particularly simple formula is obtained:\( \frac{\partial}{\partial\sigma}\log\big\langle\Theta(\cdot)E(\cdot,\tfrac12+\sigma+t),\ \Theta(\cdot)E(\cdot,\tfrac12-\sigma+ t)\big\rangle_{\mathrm{reg}}\Big|_{\sigma=0}=2\,\frac{\partial}{\partial\sigma}\log\left|\xi\bigl(\tfrac12+\sigma+t\bigr)\right|\Big|_{\sigma=0} \). Fejér--windowed versions of these identities are then obtained, and a Fejér--windowed "strip bridge'' is proved: a harmonic operator identity expressing the short--band component of \( \partial_\sigma\log|\xi(1/2+\sigma+t)| \) at an interior latitude via a linear combination of Fejér--smeared edge data, with a power--saving \( O(H^{-\eta}) \) remainder after short--band freezing, uniformly for \( |\sigma^\star|\ge \sigma_0>0 \). A sharp truncation stability result is also established. After subtracting the finitely many Zagier--Arthur cusp counterterms, the Fejér--smeared \( \sigma \)--derivative of the logarithm of the truncated mixed theta--Eisenstein pairing agrees with its regularized version up to \( O(H^{-A}) \) for any prescribed \( A>0 \), provided the truncation height \( Y=H^{B(A)} \) is chosen sufficiently large. A brief discussion is included of numerical checks in a sample region, and a short Fourier--analytic proof note is given for the renormalization estimate that underlies the strip bridge.

Abstract: I introduce an n-dimensional canonical number system (CNS) on modules: a finite base frame b, an E-linear shift T, and a finite digit set D yield unique finite expansions along places \( \{T^{k}(b_i)\} \). For a finitely generated ring \( R=\mathbb Z[a_1,\dots,a_n] \), I use the presentation \( R\cong \mathbb Z[x_1,\dots,x_n]/\ker(\phi) \) and compute \( \ker(\phi) \) via a graph-ideal elimination scheme. When the quotient has finitely many standard monomials, choosing a non–zero-divisor prime p equips R with a base-p CNS (pre-folding) that uses those monomials as places. A digit-folding lemma then compresses coordinates whenever some place has only finitely many powers, preserving uniqueness. This provides a constructive pipeline from presentations of finitely generated, countable rings to explicit multi-dimensional CNS representations, supporting the conjecture that every such ring is isomorphic to an n-dimensional CNS.

Abstract: We present several constructions of classical constants such as e, π, and the golden ratio φ using only prime numbers or prime–based sequences (for instance, prime indices in the Fibonacci and Lucas sequences). Our goal is not numerical efficiency but to highlight the pedagogical connection between prime arithmetic and the analytic nature of fundamental constants. We show how the Prime Number Theorem and Euler’s product for the Riemann zeta function allow us to reinterpret e and π as “prime-generated” limits or products, and we suggest classroom activities for calculus, real analysis, and introductory number theory courses.

Abstract: The binary Goldbach conjecture states that every even integer greater than 2 is the sum of two primes. We analyze a variant of this conjecture, positing that every even integer 2N ≥ 8 is the sum of two distinct primes P and Q. We establish a novel equivalence between this statement and a geometric construction: the conjecture holds if and only if for every N ≥ 4, there exists an integer M ∈ [1, N − 3] such that the L-shaped region N2 − M2 (between nested squares) has a semiprime area P · Q, where P = N − M and Q = N + M. We define the set DN = n Q−P 2 2 < P < N < Q < 2N, P, Q prime o of half-differences arising from prime pairs straddling N with Q < 2N. The conjecture is equivalent to the non-emptiness of DN ∩ {N − p | 3 ≤ p < N, p prime}. We conduct a computational analysis for N ≤ 214 and define a gap function G(N) = log2(2N) − ((N − 3) − |DN|). Our experimental results show that the minimum of G(N) is positive and increasing across intervals [2m, 2m+1]. This result, G(N) > 0, establishes that |DN| > (N − 3) − log2(2N). Under this bound, the pigeonhole principle applied to the cardinality of the candidate set {N − p | 3 ≤ p < N, p prime} (of size π(N − 1) − 1) and the bad positions (of size (N − 3) − |DN| < log2(2N)) implies a non-empty intersection for all N ≥ 4, yielding a proof of the conjecture. Our work establishes a novel geometric framework and demonstrates its viability through extensive computation.

Abstract: The Riemann Hypothesis, one of the most celebrated open problems in mathematics, addresses the location of the non-trivial zeros of the Riemann zeta function and their profound connection to the distribution of prime numbers. Since Riemann’s original formulation in 1859, countless approaches have attempted to establish its truth, often by examining the asymptotic behavior of arithmetic functions such as Chebyshev’s function θ(x). In this work, we introduce a new criterion that links the hypothesis to the comparative growth of θ(x) and primorial numbers. By analyzing this relationship, we demonstrate that the Riemann Hypothesis follows from intrinsic properties of θ(x) when measured against the structure of primorials. This perspective highlights a striking equivalence between the distribution of primes and the analytic behavior of ζ(s), reinforcing the deep interplay between multiplicative number theory and analytic inequalities. Beyond its implications for the hypothesis itself, the result offers a fresh framework for understanding how prime distribution governs the analytic landscape of the zeta function, thereby providing new insight into one of mathematics’ most enduring mysteries.

Abstract: This paper explains why the critical line sits at real part equal to one-half by treating it as an intrinsic boundary of a reparametrized complex plane (“z-space”), not a mere artifact of functional symmetry. In z-space the real part is defined by a geometric-series map that induces a rulebook for admissible analytic operations. Within this setting we rederive the classical toolkit—eta–zeta relation, Gamma reflection and duplication, theta–Mellin identity, functional equation, and the completed zeta—without importing analytic continuation from the usual s-variable. We show that access to the left half-plane occurs entirely through formulas written on the right, with boundary matching only along the line with real part one-half. A global Hadamard product confirms the consistency and fixed location of this boundary, and a holomorphic change of variables transports these conclusions into the classical setting.

Abstract: The Riemann Zeta function ζ(s) lies at the heart of analytic number theory, encoding the distribution of primes through its non-trivial zeros. This paper introduces a direct computational framework for evaluating Z(t) = eiθ(t)ζ( 1/2 + it) at large imaginary parts t, employing a real-to-complex number conversion and a novel valley scanner algorithm. The method efficiently identifies zeros by tracking minima of |Z(t)|, achieving stability and precision up to t ≈ 10^20 with moderate computational cost, using AWS EC2 computation. Results are compared against known Andrew Odlyzko zero datasets, validating the method’s accuracy while simplifying the high-t evaluation pipeline.

Abstract: This paper advances the Collatz conjecture by analyzing binary representations of natural numbers through fractional parts. We introduce a direct non-recursive relation for intermediate mantissas $\sigma_j$ in binary decompositions and prove their equidistribution using Weyl's theorem. The self-correcting dynamics of $\sigma_j$ ensure a balance between 1s and 0s, leading to an asymptotic density of 1/2 for 1s in binary expansions of $3^n$. This yields a probabilistic estimate: in approximately half of all cases, the binary expansions have many leading zeros, ensuring rapid descent. Theorems estimate zero density in powers of three and demonstrate sequence decrease for large $n$. Numerical verifications and updated figures support the findings, providing strong evidence for convergence in large cases.