Search Results (87)

We prove that for the odd integers $n\in \{5,7,9,\dots ,25\}$, the Riemann zeta value $\zeta \left(n\right)$ is not a Liouville number. Our method applies a general strategy pioneered by Wadim Zudilin and D.V. Vasilyev. Specifically, we construct families of high-dimensional integrals that expand into rational linear combinations of odd zeta values, eliminate lower-order terms to isolate $\zeta \left(n\right)$, and apply Nesterenko’s linear independence criterion. We verify the required asymptotic growth and decay conditions for each odd $n\le 25$, establishing that $\mu \left(\zeta \right(n\left)\right)<\infty$, and thus that $\zeta \left(n\right)\notin \mathcal{L}$. This is the first unified proof covering all odd zeta values up to $\zeta \left(25\right)$ and highlights the structural barriers to extending the method beyond this point. We also give rigorous upper bounds on $\mu \left(\zeta \right(n\left)\right)$ for all odd integers $n\in \{5,7,\dots ,25\}$, using multiple integral constructions due to Vasilyev and Zudilin, elimination of lower zeta terms, and the quantitative version of Nesterenko’s criterion. Full article

We present novel proofs of the Riemann hypothesis by extending the standard complex Riemann zeta function into a quaternionic algebraic framework. Utilizing λ-regularization, we construct a symmetrized form that ensures analytic continuation and restores critical-line reflection symmetry, a key structural property of the Riemann ξ(s) function. This formulation reveals that all nontrivial zeros of the zeta function must lie along the critical line Re(s) = 1/2, offering a constructive and algebraic resolution to this fundamental conjecture. Our method is built on convexity and symmetrical principles that generalize naturally to higher-dimensional hypercomplex spaces. We also explore the broader implications of this framework in quantum statistical physics. In particular, the λ-regularized quaternionic zeta function governs thermodynamic properties and phase transitions in Bose–Einstein condensates. This quaternionic extension of the zeta function encodes oscillatory behavior and introduces critical hypersurfaces that serve as higher-dimensional analogues of the classical critical line. By linking the spectral features of the zeta function to measurable physical phenomena, our work uncovers a profound connection between analytic number theory, hypercomplex geometry, and quantum field theory, suggesting a unified structure underlying prime distributions and quantum coherence. Full article

By applying the “coefficient extraction method” to the symmetric transformation of hypergeometric series due to Chu and Zhang (2014), an overview is presented systematically for a large class of infinite series of convergence rate “$-1/4$” concerning harmonic numbers. Numerous closed formulae in terms of mathematical constants (such as $\pi$, $ln2$ and the Riemann zeta values) are established. They may serve as a reference source for readers in their further investigations. Full article

In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference $\alpha$, an integer with the Voronoĭ function weight $V_k$. In the case of $V_1\left(x\right)=e^{-x}$, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The $\alpha =0$ case is the divisor function, while the $\alpha =1$ case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan. Full article

This study focuses on an in-depth investigation of the Riemann zeta function. For this purpose, infinite numbers and rotational infinite numbers, which have been introduced in previous studies published by the author, are used. These numbers are a powerful tool for solving problems involving infinity that are otherwise difficult to solve. Infinite numbers are a superset of complex numbers and can be either complex numbers or some quantification of infinity. The Riemann zeta function can be written as a sum of three rotational infinite numbers, each of which represents infinity. Using these infinite numbers and their properties, a correlation of the non-trivial zeros of the Riemann zeta function with each other is revealed and proven. In addition, an interesting relation between the Euler–Mascheroni constant (γ) and the non-trivial zeros of the Riemann zeta function is proven. Based on this analysis, complex series limits are calculated and important conclusions about the Riemann zeta function are drawn. It turns out that when we have non-trivial zeros of the Riemann zeta function, the corresponding Dirichlet series increases linearly, in contrast to the other cases where this series also includes a fluctuating term. The above theoretical results are fully verified using numerical computations. Furthermore, a new numerical method is presented for calculating the non-trivial zeros of the Riemann zeta function, which lie on the critical line. In summary, by using infinite numbers, aspects of the Riemann zeta function are explored and revealed from a different perspective; additionally, interesting mathematical relationships that are difficult or impossible to solve with other methods are easily analyzed and solved. Full article

We supplement the formulas for partial sums of the Hurwitz zeta-function and its derivatives, producing more integral representations and generic definitions of important constants. Then, these are used, coupled with the functional equation for the completed zeta-function to clarify the results of Choudhury, giving rise to closed expressions for the Riemann zeta-function and its derivatives. Full article

In the author’s previous studies, new infinite numbers, their properties, and calculations were introduced. These infinite numbers quantify infinity and offer new possibilities for solving complicated problems in mathematics and applied sciences in which infinity appears. The current study presents additional properties and topics regarding infinite numbers, as well as a comparison between infinite numbers. In this way, complex problems with inequalities involving series of numbers, in addition to limits of functions of x $\in$ ℝ and improper integrals, can be addressed and solved easily. Furthermore, this study introduces rotational infinite numbers. These are not single numbers but sets of infinite numbers produced as the vectors of ordinary infinite numbers are rotated in the complex plane. Some properties of rotational infinite numbers and their calculations are presented. The rotational infinity unit, its inverse, and its opposite number, as well as the angular velocity of rotational infinite numbers, are defined and illustrated. Based on the above, the Riemann zeta function is equivalently written as the sum of three rotational infinite numbers, and it is further investigated and analyzed from another point of view. Furthermore, this study reveals and proves interesting formulas relating to the Riemann zeta function that can elegantly and simply calculate complicated ratios of infinite series of numbers. Finally, the above theoretical results were verified by a computational numerical simulation, which confirms the correctness of the analytical results. In summary, rotational infinite numbers can be used to easily analyze and solve problems that are difficult or impossible to solve using other methods. Full article

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein. In addition, the following series ${\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n(n+1)\left(\genfrac{}{}{0pt}{}{n+z}{n}\right)}},{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n(n+1)(n+2)\left(\genfrac{}{}{0pt}{}{n+z}{n}\right)}},{\sum }_{n=1}^{\infty }{\displaystyle \frac{1}{n(n+1)(n+2)(n+3)\left(\genfrac{}{}{0pt}{}{n+z}{n}\right)}}$, as well as the harmonic and odd harmonic number series associated with them are evaluated. Full article

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function $\zeta \left(s\right)$ (zeros different from $s=-2m$, $m\in \mathbb{N}$) lie on the critical line $\sigma =1/2$. In this paper, combining the universality property of $\zeta \left(s\right)$ with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts $\zeta (s+i{t}_{\tau })$ approximating the function $\zeta \left(s\right)$. Here, $t_{\tau }$ denotes the Gram function, which is a continuous extension of the Gram points. Full article

We discuss applications of the Dirichlet $\eta \left(s\right)$ function in physics. To this end, we provide an introductory description of one-dimensional (1D) ionic crystals, which are well-known in the condensed matter physics literature, to illustrate the central issue of the paper: A generalization of the Coulomb interaction between alternating charges in such crystalline structures. The physical meaning of the proposed form, characterized by complex (in the mathematical sense) ion–ion interactions, is argued to have emerged in many-body systems, which may include effects from vacuum energy fluctuations. We first consider modifications to the bare Coulomb interaction by adding an imaginary component to the exponent of the Coulomb law of the form $s=1+ib$, where b is a real number. We then extend the results to slower-decaying interactions, where the exponent becomes $s=a+ib$, presenting numerical results for values $1/2\le a\le 2$, which include the critical strip relevant to the Riemann hypothesis scenario. Full article

In this paper, the asymptotic behavior of the modified Mellin transform ${\mathcal{Z}}_2\left(s\right)$, $s=\sigma +it$, of the fourth power of the Riemann zeta function is characterized by weak convergence of probability measures in the space of analytic functions. The main results are devoted to probability measures defined by generalized shifts ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$ with a real increasing to $+\infty$ differentiable functions connected to the growth of the second moment of ${\mathcal{Z}}_2\left(s\right)$. It is proven that the mass of the limit measure is concentrated at the point expressed as $h\left(s\right)\equiv 0$. This is used for approximation of $h\left(s\right)$ by ${\mathcal{Z}}_2(s+i\phi \left(\tau \right))$. Full article

In this study, the approximation of a pair of analytic functions defined on the strip $\{s=\sigma +it\in \mathbb{C}:1/2<\sigma <1\}$ by shifts $\left(\zeta \right(s+i\tau ),\zeta (s+i\tau ,\alpha \left)\right)$, $\tau \in \mathbb{R}$, of the Riemann and Hurwitz zeta-functions with transcendental $\alpha$ in the interval $[T,T+H]$ with $T^{27/82}⩽H⩽T^{1/2}$ was considered. It was proven that the set of such shifts has a positive density. The main result was an extension of the Mishou theorem proved for the interval $[0,T]$, and the first theorem on the joint mixed universality in short intervals. For proof, the probability approach was applied. Full article

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function $\ln \sec x=-\ln \cos x$; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function $(2^x-1)\zeta \left(x\right)$ on $(1,\infty )$, they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders. Full article