Abstract: This paper explores a novel operator-theoretic framework for analyzing the zeros of the Riemann zeta function within the critical strip. By constructing a differential interaction operator $\Phi(s, \delta)$ on the Hilbert space $l^2(\mathbb{N})$, we investigate the geometric properties of the operator trace off the critical line $Re(s) = 1/2$.Our analysis utilizes a Master Inequality applied to the resulting phase torque $J(\delta, t)$, suggesting that a hyperbolic sine bias creates an analytical “floor” resistant to the interference of the infinite tail. Furthermore, by invoking Baker’s Theorem on Linear Forms in Logarithms, we examine the conditions under which the truncated Head ($N \approx t^{(A+1)/\sigma}$) maintains signal dominance. Within the specific constraints of this operator construct, the results provide evidence for the exclusion of zeros off the critical line, offering a potential path toward understanding the confinement of non-trivial zeros to $Re(s) = 1/2$.

Abstract: In recent years, Duran (Fundam. J. Math. Appl., 8(2) (2025), 55-64) introduced the central Bell-based type 2 Bernoulli polynomials of order \( \beta \) given by \( \left( \frac{t}{e^{\frac{t}{2}}-e^{-\frac{t}{2}}}\right) ^{\beta }e^{xt+z\left( e^{\frac{t}{2}}-e^{-\frac{t}{2}}\right) }=\sum_{m=0}^{\infty }% \text{ }_{CB}b_{m}^{\left( \beta \right) }\left( x;z\right) \frac{t^{m}}{m!}% \left( \left\vert t\right\vert <2\pi \right) \) and derived many formulas and relations, covering several symmetric properties, derivative properties, summation formulas, and addition formulas. In this paper, we aim to improve some new properties for the central Bell-based type 2 Bernoulli polynomials of order \( \beta \). We first investigate some new properties, involving central Bell polynomials, classical Bernoulli polynomials and numbers, and central factorial numbers of the second kind. Moreover, we show that the central Bell-based type 2 Bernoulli polynomials of order \( \beta \) are solutions of the some higher-order differential equations. Further, we give a determinantal representation for the central Bell-based type 2 Bernoulli polynomials of order \( \beta \).

Abstract: Let k be a positive integer. A polynomial A∈F_2[x] is called k-unitary perfect if the sum of the k-th powers of its distinct unitary divisors equals A^k. In this paper, we focus on the case k=2^n and prove that every 2^n-unitary perfect polynomial over F_2 is necessarily even. Moreover, we obtain a complete classification of all even 2^n-unitary perfect polynomials having at most three distinct irreducible factors. In particular, we characterize all such polynomials of the formA=x^a (x+1)^b P^h, where P is a Mersenne prime over F_2 and a, b, and h are positive integers. As a consequence, several explicit infinite families of k-unitary perfect polynomials over F_2 are obtained.

Abstract: We prove strict log-concavity of the Riemann-Jacobi kernel, establish hyperbolicity of the Jensen polynomials J_{d,n}(x) for d <= 22, n <= 14, and prove that the global Riemann Hypothesis is algebraically equivalent to a subluminal condition on the Wronskian components. Part I (Sections 2-5) proves the kernel is strictly log-concave (TP_2) with curvature kappa >= 19.24, via the convex potential decomposition and a perturbation bound using only 4.3% of the log-concavity budget. Part II (Sections 6-8) establishes K_{d,n}(x) < 0 for x >= 0 (all d, n) purely analytically, and for x < 0, d <= 22, n <= 14 by interval-arithmetic certification (330/330 cases, Bernstein-basis enclosure with double-double precision on GPU). This extends the Griffin-Ono-Rolen-Zagier result (d <= 8) to d <= 22 with full coverage of all real x. Part III (Section 9) introduces the even-odd decomposition: setting t = y^2, the condition K_{d,n}(-y) < 0 is equivalent to the Lorentz norm P(t) = A(t)^2 - t B(t)^2 > 0, where A and B are the even- and odd-indexed coefficient polynomials. Global RH (unconditional). Section 11 proves D_r(n) > 0 for all r and n. The argument combines a discrete concavity lemma with a spectral-gap reduction: the Hadamard factorisation of Xi gives a spectral gap delta = (t_1/t_2)^2 ≈ 0.452 between the first two zeros, which makes the normalised dissipation C_s = n^2|log Theta_s| independent of n. The two-variable unitarity condition reduces to a single-variable bound S = sum C_s < a*n, verified by certified computation (S <= 19.41, a >= 1.31). The proof combines: (A) Borell log-concavity (L_1 > 1, all n); (B) 10,822-point interval certification; (C1) the dissipation bound for n >= 100; (C2) DJ log-space certification plus dominant-pole tail for n <= 99. By Edrei-Schoenberg, Xi is in the Laguerre-Polya class and Lambda = 0.

Abstract: The Pólya-Ostrowski group of a Galois number field K, is the subgroup Po(K) of the ideal class group Cl(K) of K generated by the classes of all the strongly ambiguous ideals of K. The number field K is called a Pólya field, whenever Po(K) is trivial. In this paper, using some results of Bennett Setzer [9] and Zantema [10], we give an explicit relation between the order of Pólya groups and the Hasse unit indices in real biquadratic fields. As an application, we refine Zantema’s upper bound on the number of ramified primes in Pólya real biquadratic fields.

Abstract:

We study the degree-d Jensen polynomials \( J_{d,n}(X) \) built from the moment sequence \( M_n=\int_0^\infty\Phi_1(u)u^{2n}\,du \) of the Riemann \( \Xi \)-function, which coincides with the classical Pólya–Jensen family. Using bridge coordinates, the staircase law, and Plancherel–Rotach asymptotics, we prove that \( J_{d,n}^\gamma \) is hyperbolic for all \( n\ge C_0^\infty d^4 \) (\( C_0^\infty\approx0.020 \); the analytic formula for \( C_0^\infty \) is rigorous but approximates the numerically observed value to within 2.6%); combined with the GORZ theorem for \( d\le8 \), this covers the entire asymptotic regime. We identify a phase-transition law \( n^*(d)=C_0^\infty d^4+\alpha d^3+\beta(-1)^d d^2+O(d) \) (Conjecture 3.5): the leading constant \( C_0^\infty\approx0.0195 \) is computed analytically and verified to within 2.6% of the empirical large-$d$ limit; the formula for \( \alpha \) is derived; its numerical value ≈ −0.2 to −0.3 is numerical evidence; the parity structure \( \beta(-1)^d d^2 \) is proved. For the finite strip \( 0\le n \) < \( C_0^\infty d^4 \) with \( d\ge9 \), the sole remaining gap, whose closure is equivalent to the Riemann Hypothesis under standard transversality, we establish four structural obstructions: ratio-barrier saturation (no usable margin, certified and numerical); frozen zero count (parity blocks any ladder, certified for \( d\le21 \)); interlacing-lift vacuity (proved); and a discriminant equivalence (proved under transversality), showing that all known local and inductive mechanisms fail simultaneously in this region. The problem reduces to: \( \operatorname{Disc}(J_{d,n}^\gamma)>0 \) for all \( d\ge9 \) and 0 ≤ n < \( C_0^\infty d^4 \); this requires moment data \( M_k \) for \( k\ge130 \), currently inaccessible.

Abstract: The Nicolas criterion gives an equivalent formulation of the Riemann Hypothesis as an inequality involving the Euler totient function evaluated at primorial numbers. A natural strategy for establishing this inequality is to prove that a suitable subsequence of the associated ratio sequence is eventually strictly decreasing under the assumption that the Riemann Hypothesis is false. The present work shows that such a subsequence exists. When this monotonicity property is combined with the known limiting behavior of the ratio sequence and the Nicolas equivalence, a contradiction emerges: assuming the Riemann Hypothesis is false forces the subsequence to converge to a limit that is simultaneously equal to $e^{\gamma}$ (by a subsequence argument) and strictly less than $e^{\gamma}$ (by strict monotonicity). The Riemann Hypothesis therefore follows as a direct consequence.

Abstract: Let $ \xi(z) $ be the Riemann xi function. In a previous paper we prove the boundedness of coefficients of the power series expansion of $ \xi'(1/z)/\xi(1/z) $ and thus give a proof of the Riemann hypothesis. In this paper we generalize the method there to the study of the extended Riemann hypothesis for general number fields.

Abstract: We develop a geometric and combinatorial framework for the distinct-prime Goldbach conjecture—the assertion that every even integer 2N ≥ 8 is the sum of two distinct primes. The framework rests on three components: (1) a novel geometric equivalence reformulating the problem in terms of nested squares with semiprime areas, (2) a rigorous combinatorial reduction to a density condition on a set of straddling prime pair half-differences, and (3) extensive computational verification. The geometric construction reveals that the conjecture is equivalent to finding, for each N ≥ 4, an integer M ∈ [1,N −3] such that the L-shaped region N² − M² between nested squares has area P · Q where P = N − M and Q = N + M are both prime. We define D_N = {(Q − P)/2 | 2 < P < N < Q < 2N, both prime}∩{1,...,N − 3} to be the set of achievable half-differences from straddling prime pairs that lie inside the admissible range. Our gap function G(N) = log²(2N) − ((N −3) − |D_N|) measures the margin by which the required density condition holds. Using explicit results from Dusart’s doctoral thesis, we rigorously establish Steps 1–3 of the density argument, including the bound |D_N| ≥ ln²N for N ≥ 3275. We formulate the remaining step—that the number of missing M-values is at most ln²(2N)—as the Density Hypothesis (G(N) > 0), supported by computational evidence: for all N ∈ [4,2¹⁴], G(N) > 0 holds universally, with minima strictly increasing across dyadic intervals. We prove that the Density Hypothesis, combined with finite verification for small N, implies the distinct-prime Goldbach conjecture via the pigeonhole principle.

Abstract: In this paper, we present the notion of S-2-prime hyperideals, which provides a unifying generalization of 2-prime and S-prime hyperideals within multiplicative hyperrings. We explore their key algebraic properties and investigate their connections with other hyperideal classes. We emphasize the unique aspects that differentiate S-2-prime hyperideals, illustrating their role in expanding the theoretical framework of hyperideal structures. We examine how these hyperideals behave under hyperring homomorphisms, extensions, and standard algebraic operations, demonstrating that many known properties of prime, 2-prime, and S-prime hyperideals extend naturally to the S-2-prime setting. We provide illustrative examples to highlight important differences and to offer practical insight into their structure. Overall, we enhance the theoretical understanding of hyperideals in multiplicative hyperrings and establish a framework for future research in this area.

Abstract: Let ξ(z) be the Riemann xi function. We prove the boundedness of coefficients of the power series expansion of ξ′(1/z)/ξ(1/z). By an observation of Keiper this implies that the Riemann hypothesis is true.

Abstract: Let Γ be a commutative with identity multiplicative hyperring and S ⊆ Γ be a multiplicatively closed subset of Γ. In this study, we will discuss the definition and general properties of weakly Γ-prime hyperideals. Let Ω be a hyperideal of Γ disjoint from S. We say that Ω is a weakly S− prime hyperideal of Γ if there exists an s ∈ S such that for all ϱ,σ ∈ Γ , if {0} ̸ = ϱ⋄σ ⊆ Ω then s⋄ϱ ⊆ Ω or s◦y ⊆ Ω . We will also examine the relationship between weakly−S prime hyperideals, weakly prime, and S−prime hyperideals.

Abstract: We study Wilf’s conjecture for a numerical semigroup S using only the first Kunz layer S ∩ (m, 2m). Let η = |S ∩ (m, 2m)| and write the conductor as c = qm − ρ with 0 ≤ ρ < m. We prove the staircase bound |L| ≥ q + (q − 2)η + ηρ, where L = S ∩ [0, c) and ηρ = |S ∩ (m, 2m − ρ)|. This yields a lower bound for Wilf’s number and the criterion e(η + 2) ≥ 2m, hence also (η + 1)(η + 2) ≥ 2m, implying Wilf’s conjecture. When m | c, we obtain the stronger condition e(3η + 4) ≥ 4m, and in particular 3η² + 7η + 4 ≥ 4m. We also derive an exact cumulative-layer formula for |L| and apply it to interval-generated semigroups, where the second cumulative layer gives a strictly stronger infinite family.

Abstract: I introduce the Twin prime detector formula D(p), a closed-form expression involving ratios of Gamma function products that evaluates to 1 if and only if (p,p+2) is a twin prime pair. By applying the Gauss Multiplication Formula to simplify the MAF to 〖(2π)〗^((p+2 -(σ(p)+σ(p+2))/2)), I establish an exact equivalence between the twin prime condition and the vanishing of an arithmetic exponent involving the sum-of-divisors function σ. Building on this characterisation, I construct a GMF-weighted Dirichlet series L(s) whose double pole of order 2 at s = 1 encodes the twin prime distribution. I derive the nonvanishing of the Hardy-Littlewood constant C₂ directly from the GMF local correction factors, establish the sieve dimension κ = 2 as a structural consequence of the two-product GMF architecture, and obtain a Bombieri-Vinogradov-type remainder bound through a GMF-derived zero-free region. Assembling these components, I prove π_2 (x)∼2C_2 Li_2 (x)∼(2C_2 x)/ln^2⁡x →∞, establishing the infinitude of twin primes.

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 opera tor to generate the structure. We identified 6,175,562 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 40 relative to 10⁹, 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: 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].