Abstract: The Riemann Xi function admits the representation \( \Xi(t) = \int_0^\infty \Phi(u)\cos(tu)\,du \) where \( \Phi \) is a positive, even, integrable function. By a classical theorem of P\'olya (1927), if \( \log\Phi \) is concave on \( [0,\infty) \), then \( \Xi \) has only real zeros, which is equivalent to the Riemann Hypothesis. We prove that the dominant term of \( \Phi \) has strictly negative second logarithmic derivative for all \( u \geq 0 \), reducing the full log-concavity to a quantitative tail estimate. We verify this estimate by rigorous interval arithmetic (5000 certified subintervals on \( [0, 1/2] \) at 80-digit precision, with the complement handled analytically). The entire argument is formalised in the Lean~4 proof assistant with the Mathlib library.

Abstract: Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for \( 1/\pi \) and obtain some Ramanujan-like series for \( 1/\pi^2 \) and~\( 1/\pi^3 \).

Abstract: The Extended, Generalized, and Grand Riemann Hypotheses are proved under a unified framework, which is based on the general properties of L-functions, i.e., the divisibility of entire functions contained in the symmetric functional equation, where the uniqueness of zero multiplicities (although their specific values remain unknown) of a given non-zero entire function plays a critical role. Consequently, the existence of Landau-Siegel zeros is excluded, thereby confirming the Landau-Siegel zeros conjecture.

Abstract: Hudzik and Landes [Math. Ann., 1992] derived a major generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was extended to finitely many nonzero elements independently by Dragomir [Bull. Aust. Math. Soc., 2006] and by Kato, Saito and Tamura [Math. Inequal. Appl., 2007]. We derive a finite field version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.

Abstract: Let F be a sub-modulus field such that 2̸ = 0. Let X be a sub-normed linear space over F. Then we show that ∥x∥ − ∥y∥ ≤ 2 |2| ∥x + y∥ + 2 |2| max{∥x − y∥, ∥y − x∥} − (∥x∥ + ∥y∥) and ∥x∥ − ∥y∥ ≤ ∥x∥ + ∥y∥ − 2 |2| ∥x + y∥ + 2 |2| max{∥y − x∥, ∥x − y∥}. Above inequalities are finite field versions of important Tarski-Maligranda inequalities obained by Maligranda [Banach J. Math. Anal., 2008].

Abstract: We advance the study of the Collatz conjecture through an analysis of the binary representations of positive integers via fractional parts. We introduce a direct, non-recursive relation for the intermediate mantissas\( ~\sigma_j \) and prove the exact bound \( \sigma_j \le 1/(\ln 2\,(2^{m+1}-1)) \) after a run of \( m \)consecutive ones (Theorem 4), establishing that every such run must terminate. Using Weyl's equidistribution theorem applied to \( \{n \log_2 3\} \) (averaged over \( n \), not applied to any individual binary expansion), we establish that \( h(n) \le \frac{1}{2}L_n + o(L_n) \) holds for almost all \( n \), i.e.\ outside a set of natural density zero (Theorem 5); we explicitly identify this as the boundary of what the density method achieves. A complete mod-\( 8 \) transition table analysis proves \( Q \ge 2M - 2 \) for every Collatz window (tight constant \( C_0 = 2 \), Lemma 6), giving an exponential compression factor\( (3/4)^M \) deterministically. We identify two remaining open sub-problems: (1) extending the density bound from almost all \( n \) to every \( n \); and (2) ruling out non-trivial cycles, which is required to complete the descent argument. These open points are stated precisely rather than papered over. Numerical experiments confirm all theoretical bounds.

Abstract: This paper clarifies and extends the restricted-variable framework introduced in Drake (2026). Restriction does not alter the underlying analytic object, provided the restricted and unrestricted presentations agree on a nonempty open overlap domain; it changes only the regime of admissible access. In the model case of the completed Riemann zeta function, the paper shows that functional symmetry acts on the argument of the function but does not by itself induce an admissible zero-indexing rule in the restricted Hadamard product. As a result, reflected zeros remain analytically present through symmetry, yet do not appear as separately admissible geometric indices unless they lie on the symmetry boundary. This yields the Hadamard representation dilemma that consistency between canonical Hadamard factorization and the restricted admissible indexing mechanism forces the nontrivial zeros onto the critical line. The same mechanism is then extended to entire functions of order one satisfying a reflection symmetry under an optimal half-plane restriction. The paper also shows that symmetric linear combinations do not constitute counterexamples, since their zeros arise through cancellation rather than through the restricted zero-indexing mechanism.

Abstract: An elementary and self-contained approach to the Euler–Mascheroni constant γ is presented, based solely on Simpson's quadrature rule and the convexity of the function \( f(x)=1/x \). Using Simpson--type weighted harmonic sums, we approximate local logarithmic increments by simple finite linear combinations of reciprocal integers. Exploiting the monotonic and convex nature of \( 1/x \), sharp two-sided inequalities are established that relate these numerical approximations to the exact logarithmic increments. These inequalities imply that the accumulated quadrature errors form a convergent series, yielding a simple proof of the classical limit defining γ without recourse to the Euler–Maclaurin formula.A central structural observation of the paper is that γ admits a Simpson--type decomposition as a sum of two irrational numbers. More precisely, we show that \( \gamma = ( \log{[2]} + 1 ) / 3 + \delta \), where both \( ( \log{[2]} + 1 ) / 3 \) and δ are irrational. The constant δ arises naturally as the limit of a rational sequence derived from a Simpson--type approximation, and its irrationality is established by an elementary rigidity argument based on prime divisibility in rational approximations.

Abstract: The Riemann Hypothesis, one of the most celebrated open problems in mathematics, asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line \( \Re(s) = \frac{1}{2} \) and has profound consequences for the distribution of prime numbers. Since Riemann's original 1859 paper, a vast body of work has attempted to settle the question, frequently by examining the asymptotic behavior of arithmetic functions such as Chebyshev's prime-counting function \( \theta(x) \). In this work we introduce a new criterion that links the Riemann Hypothesis to the comparative growth of \( \theta(x) \) relative to primorial numbers. More precisely, we study the ratio \( R(N_k) = \Psi(N_k)/(N_k\log\log N_k) \), where \( N_k \) is the \( k \)-th primorial and \( \Psi \) is the Dedekind function, and show that the Riemann Hypothesis follows from intrinsic monotonicity properties of this ratio. The argument combines Mertens' theorem, the prime number theorem, and an explicit error analysis of the relevant asymptotic expansions to produce a self-contained proof by contradiction. Beyond its implications for the hypothesis itself, the result offers a fresh framework for understanding how the multiplicative structure of primorials governs the analytic behavior of \( \zeta(s) \), thereby casting new light on one of mathematics' most enduring mysteries.

Abstract: By using Abel's transformation we study the correlation of generalized divisor functions d_k(n) and obtain the correct main term order of the asymptotic estimate for the correlations.

Abstract: Around 1637, Pierre de Fermat famously wrote in the margin of a book that he had a proof showing the equation aⁿ + bⁿ = cⁿ has no positive integer solutions for exponents n greater than 2. This statement, now known as Fermat’s Last Theorem, remained unproven for centuries despite the efforts of countless mathematicians. Andrew Wiles’s work in 1994 finally provided a rigorous proof of Fermat’s Last Theorem. However, Wiles’s proof relied on advanced mathematical techniques far beyond the scope of Fermat’s time, raising questions about whether Fermat could have truly possessed a proof using only the methods available to him. Wiles’s achievement was widely celebrated, and he was awarded the Abel Prize in 2016; the citation described his proof as a “stunning advance” in mathematics. Combining short and elementary tools, we prove the Beal conjecture, a well-known generalization of Fermat’s Last Theorem. The present work potentially offers a solution closer in spirit to Fermat’s original idea.

Abstract: In 1992, Hudzik and Landes [Math. Ann.] derived a breakthrough generalization of the triangle inequality for two nonzero elements in normed linear spaces, which was generalized to finitely many nonzero elements independently in 2006 by Dragomir [Bull. Aust. Math. Soc.] and in 2007 by Kato, Saito and Tamura [Math. Inequal. Appl]. We derive a non-Archimedean version of Hudzik-Landes-Dragomir-Kato-Saito-Tamura inequality.

Abstract: Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version of Massera-Schaffer-Maligranda-Pecaric-Rajic inequality.

Abstract: In this article, we introduce a hypercomplex algebra based on a binary superposition structure. Each algebraic unit is defined by a pair (ƒ; S) where ƒ ∈ {0; 1} encodes the logical presence of a base component, and S ∈ {−1; 1} encodes a geometric phase or orientation. This framework allows us to define an imaginary product that is both commutative and associative, properties rarely combined in higher-dimensional algebras. We demonstrate the consistency of this product through a binary and superposed formalism. This result provides a solid foundation for representing multi-level logic states, with potential applications in quantum computing processing.

Abstract:

We study the structural and analytic aspects of the B\'{a}ez--Duarte approximation problem within the Nyman--Beurling framework, which furnishes a functional-analytic equivalent of the Riemann Hypothesis (RH). Our work studies structural features of this framework; it does not prove RH. First (Rank-one collapse and Hilbert-space theory). The integer-dilate Gram matrix \( G_M=\frac{1}{3}\textbf{dd}^\top \) is rank-one, giving \( span\{r_1,\ldots,r_M\}=span\{x\} \) and fixed distance \( d_M=\frac12 \) for all M. We give the explicit Moore–Penrose pseudoinverse \( G_M^+ \) and the one-dimensional collapse of the optimisation problem. Second (Exact Gram matrix formula). We prove a fully rigorous closed-form expression for the inner products of the correct sawtooth basis: using the Bernoulli polynomial representation of the fractional part, \( \int_0^1\{jx\}\{kx\}\,dx = \frac{\gcd(j,k)^2}{jk}\Bigl(\frac{1}{12} + \frac{B_2(0)}{2}\Bigr) + \frac{1}{4}\Bigl(1-\frac{\gcd(j,k)}{j}\Bigr)\Bigl(1-\frac{\gcd(j,k)}{k}\Bigr) + E_{jk}, \) where \( E_{jk} \) is an explicit correction from higher Bernoulli terms, expressed via the Hurwitz zeta function. The arithmetic role of \( \gcd(j,k) \) is made precise. Third (Hardy-space bounds). Using the \( H^2(\Pi^+) \) reproducing kernel and the Mellin isometry, we prove: (a) the distance identity \( d_M^2=\|1/s-F_M^*(s)\zeta(s)/s\|_{H^2}^2 \); (b) an explicit lower bound \( d_M^2\ge\sum_\rho\frac{|F_M^*(\rho)|^2|\zeta'(\rho)|^{-2}}{|\rho|^2}\cdot c(\rho) \) from the zeros of \( \zeta \); and (c) a pointwise Hardy-space inequality relating \( d_M \) to the supremum of \( |1-F_M^*({\tfrac12}+it)\zeta({\tfrac12}+it)/({\tfrac12}+it)| \) on the critical line. Fourth (Kalman filtration stability). Under the observation model \( z_M=d_M+\varepsilon_M \) with $\varepsilon_M$ sub-Gaussian of variance \( \sigma^2 \), the Kalman estimator satisfies a rigorous oracle inequality \( \mathbf{E}|d_M^{KF}-d_M|^2\le \sigma^2 K_\infty(2-K_\infty)^{-1} \), with an almost-sure bound \( |d_M^{KF}-d_M|\le CM^{-\alpha} \) whenever \( |d_M-d|=O(M^{-\alpha}) \). Fifth (Möbius sparsity). We prove \( |c_k^*|=O(k^{-1+\varepsilon}) \) via Dirichlet series techniques and show that the coefficient sequence is bounded in \( \ell^2 \), with connections to the Möbius function made precise through the optimality conditions. Sixth (Structural Mellin theorem). We identify a hidden structural observation in the Mellin identity: the Gram kernel \( K_G(s,w)=\zeta(s+\bar w)/(s+\bar w) \) appears as the reproducing kernel of the Hardy space \( H^2(\Pi^+) \) restricted to the approximation subspace \( W_M \), and its singularity at \( s+\bar w=1 \) encodes the pole of \( \zeta \) while the zeros of \( \zeta \) in the critical strip contribute exactly as spectral obstructions. Disclaimer. This paper does not prove RH. All results are structural, computational, and analytic observations within the equivalent framework.

Abstract: This work presents a complete arithmetic framework resolving 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 Inverse map R(n; k) = (2^kn − 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 normal–state lattice z(n) reveals a second, purely affine enumeration: each live odd n seeds a unique 4-adic rail 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 bijectively from admissible lifts above minimal anchors produced by n ≡ 1, 5 (mod 6). The locked Forward–Inverse equivalence T(m) = (3m + 1)/ 2^ν2(3m+1) and R(T(n); k) = m then implies that Forward trajectories cannot branch or diverge: each Forward iterate lies on a single admissible rail descending toward its origin at 1. Because the residue–phase automaton is finite and every rail has a uniquely determined Forward parent, no infinite runaway is possible and no nontrivial odd cycle can exist. Together they provide a complete, closed arithmetic description of the Collatz dynamics and establish that every Forward trajectory converges to 1.

Abstract: This study aims to prove the Riemann Hypothesis and the Generalized Riemann Hypothesis by ex-tending the Riemann zeta function and Dirichlet L -functions to the elliptic complex domain, based ona newly constructed system of elliptic complex numbers Cλ(λ < 0) . The core challenge addressed is theinherent difficulty in resolving these conjectures within the traditional ”circular complex domain” frame-work (λ = −1); the author posits that a complete proof is unattainable strictly within this conventionalsetting.The primary innovation of this work lies in the formulation of the theory of elliptic complex numbers,specifically identifying the limiting case as λ → 0− as the key to the proof. Through rigorous deduction,a bijective correspondence between zeros across different complex planes is established. By employingproof by contradiction and leveraging the correspondence between Cλ (as λ → 0) and the circle complexplane C, the Riemann Hypothesis and the Generalized Riemann Hypothesis are ultimately proven. Thispaper is organized into three parts:(1) Construction and Geometric Properties: The first part details the construction of elliptic complexnumbers and their fundamental geometric properties, laying the necessary foundation for subsequentanalysis and the proof of the conjectures.(2) Analytic Extension: The second part introduces elliptic complex numbers into mathematical anal-ysis, deriving numerous results analogous to those in classical complex variable function theory.(3) Proof of Conjectures: The final part presents the formal proofs of the Riemann Hypothesis and theGeneralized Riemann Hypothesis.

Abstract: What are the numbers made of? More precisely, what are prime numbers made of? I posed this question to myself on the evening of August 19, 2025, which prompted prolonged introspection and profound contemplation. Then, I began constructing a numerical pyramid with prime numbers. The number one took the place of the central axis. Therefore, it is possible that large prime numbers could be surrounded by prime numbers on either side of one. However, this property extends to all even and odd non-prime numbers, but without one. The Goldbach ternary conjecture, which was proven by Harald Helfgott and is now recognized as the Goldbach-Helfgott theorem, is applicable to the observation that all odd non-prime numbers can be expressed as a sum of at least three prime numbers. This is due to the fact that non-prime numbers are a subset of all numbers greater than five. Once Goldbach's binary conjecture is proven, it will likely lead to the proof of Riemann's conjecture because we will be able to detect the structure of even numbers preceding prime numbers. For now, we can visualize this in the numerical structure of the first one trillion numbers and even further up to the largest known prime number. Let 3 203 431 780 337 be our number, which is verified as prime. If we subtract another prime number, 3 333 977 , from it, we obtain 3 203 428 446 360$. Subtracting one from the product verifies that 3 203 428 446 359 is prime. If so, then the sum of the two prime numbers plus one equals the proposed prime number above. This study has two objectives. First, it aims to present prime numbers as more than just their primality property. Second, it seeks to define the numbers 2 and 3 as a set of authentic prime numbers.

Abstract: We develop the central identities of the theory of automorphic forms centering on the Jacobi theta constants \( \vartheta_2, \vartheta_3, \vartheta_4 \), the weight-4 Eisenstein series \( E_4 \), the discriminant \( \Delta \), the j–invariant, and the modular \( \lambda \)–function. The classical theory is organized around a single minimality theorem: the pair \( (\vartheta_3(\tau), \vartheta_3(2\tau)) \) suffices to recover every primary automorphic invariant at level \( \leq 2 \) as an explicit polynomial or rational function.Building on this foundation, we derive four new structural observations. \( \textbf{(I)} \) The shifted invariant \( J(\tau) := j(\tau) - 744 \) satisfies \( J(2\tau) = J(\tau)^2 - 2 \cdot 196884 + O(q^2) \) (and in fact \( J(2\tau)=J(\tau)^2-2\cdot 196884-2\cdot 21493760\,q^2+O(q^4) \)), placing the first Monster moonshine coefficient as the \emph{leading deviation from perfect squaring} under the doubling isogeny; the corresponding quadratic fixed-point polynomial has discriminant 1575073. \( \textbf{(II)} \) The sequence \( \vartheta_3(2^n\tau)^2 \) is the arithmetic-mean sequence of the arithmetic-geometric mean (AGM) iteration initialized at \( (\vartheta_3(\tau)^2, \vartheta_4(\tau)^2) \); the unique AGM fixed-point symmetry \( \vartheta_3(\tau) = \vartheta_4(\tau) \) identifies \( j(\tau) = 1728 \) (\( \tau = i \)) as the self-dual elliptic curve. \( \textbf{(III)} \) The \( \lambda \)-ODE \( d\lambda/dt = -\pi\lambda(1-\lambda)\vartheta_3(it)^4 \) approaches a logistic regime for \( t\gg 1 \); matching the exact midpoint value \( \lambda(i)=1/2 \) produces the explicit sigmoid approximation \( \lambda(it) \approx (1 + e^{\pi(t-1)})^{-1} \) for large t. \textbf{(IV)} The quantity \( R(\tau) := 2\vartheta_3(2\tau)^2-\vartheta_3(\tau)^2 = \vartheta_4(\tau)^2 \) satisfies the square-root recursion \( R(2\tau) = \vartheta_3(\tau)\sqrt{R(\tau)} \) under doubling; equivalently, \( \vartheta_4(2^n\tau) \) lies in a nested-radical (generically quadratic) extension tower over the dyadic \( \vartheta_3 \)-field, growing by one quadratic layer at each step---an algebraic obstruction distinct from the polynomial j-isogeny ladder.

Abstract: Define the perrank of an m×n matrix M to be the size of a largest square submatrix with nonzero permanent; if this is equal to m or n, then we say M has full perrank. We derive numerous results on the perrank of a matrix by methods of commutative algebra.