Showing papers on "Riemann zeta function published in 2023"

$$ \mathcal{N} $$ = 2* Schur indices

Abstract: A bstract We find closed-form expressions for the Schur indices of 4d $$ \mathcal{N} $$ N = 2 * super Yang-Mills theory with unitary gauge groups for arbitrary ranks via the Fermi-gas formulation. They can be written as a sum over the Young diagrams associated with spectral zeta functions of an ideal Fermi-gas system. These functions are expressed in terms of the twisted Weierstrass functions, generating functions for quasi-Jacobi forms. The indices lie in the polynomial ring generated by the Kronecker theta function and the Weierstrass functions which contains the polynomial ring of the quasi-Jacobi forms. The grand canonical ensemble allows for another simple exact form of the indices as infinite series. In addition, we find that the unflavored Schur indices and their limits can be expressed in terms of several generating functions for combinatorial objects, including sum of triangular numbers, generalized sums of divisors and overpartitions.

ON q-ANALOGUES OF ZETA FUNCTIONS OF ROOT SYSTEMS

Abstract: . Komori, Matsumoto and Tsumura introduced a zeta function ζ r ( s , (cid:49)) associated with a root system (cid:49) . In this paper, we introduce a q -analogue of this zeta function, denoted by ζ r ( s , a , (cid:49) ; q ) , and investigate its properties. We show that a ‘Weyl group symmetric’ linear combination of ζ r ( s , a , (cid:49) ; q ) can be written as a multiple integral over a torus involving functions ψ s . For positive integers k , functions ψ k can be regarded as q -analogues of the periodic Bernoulli polynomials. When (cid:49) is of type A 2 or A 3 , the linear combinations can be expressed as the functions ψ k , which are q -analogues of explicit expressions of Witten’s volume formula. We also introduce a two-parameter deformation of the zeta function ζ r ( s , (cid:49)) and study its properties. ,

Joint Discrete Approximation of Analytic Functions by Shifts of the Riemann Zeta Function Twisted by Gram Points II

Abstract: Let θ(t) denote the increment of the argument of the product π−s/2Γ(s/2) along the segment connecting the points s=1/2 and s=1/2+it, and tn denote the solution of the equation θ(t)=(n−1)π, n=0,1,⋯. The numbers tn are called the Gram points. In this paper, we consider the approximation of a collection of analytic functions by shifts in the Riemann zeta-function (ζ(s+itkα1),⋯,ζ(s+itkαr)), k=0,1,⋯, where α1,⋯,αr are different positive numbers not exceeding 1. We prove that the set of such shifts approximating a given collection of analytic functions has a positive lower density. For the proof, a discrete limit theorem on weak convergence of probability measures in the space of analytic functions is applied.

On the spectral zeta function of second order semiregular non-commutative harmonic oscillators

Abstract: In this paper we give a meromorphic continuation of the spectral zeta function for second order semiregular Non-Commutative Harmonic Oscillators (NCHO). By “semiregular systems” we mean systems with terms with degree of homogeneity scaling by 1 in their asymptotic expansion. As an application of our results, we first compute the meromorphic continuation of the Jaynes-Cummings (JC) model spectral zeta function. Then we compute the spectral zeta function of the JC generalization to a 3-level atom in a cavity. For both of them we show that it has only one pole in 1.

A note on the zeros of the derivatives of Hardy's function Z(t)$Z(t)$

Abstract: Using the twisted fourth moment of the Riemann zeta‐function, we study large gaps between consecutive zeros of the derivatives of Hardy's function Z(t)$Z(t)$ , improving upon previous results of Conrey and Ghosh (J. Lond. Math. Soc. 32 (1985) 193–202), and of the second named author (Acta Arith. 111 (2004) 125–140). We also exhibit small distances between the zeros of Z(t)$Z(t)$ and the zeros of Z(2k)(t)$Z^{(2k)}(t)$ for every k∈N$k\in \mathbb {N}$ , in support of our numerical observation that the zeros of Z(k)(t)$Z^{(k)}(t)$ and Z(ℓ)(t)$Z^{(\ell )}(t)$ , when k and ℓ have the same parity, seem to come in pairs that are very close to each other. The latter result is obtained using the mollified discrete second moment of the Riemann zeta‐function.

Unified Theory of Zeta-Functions Allied to Epstein Zeta-Functions and Associated with Maass Forms

Abstract: In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series.

Hypergeometric type extended bivariate zeta function

Abstract: Based on the generalized extended beta function, we shall introduce and study a new hypergeometric-type extended zeta function together with related integral representations, differential relations, finite sums, and series expansions. Also, we present a relationship between the extended zeta function and the Laguerre polynomials. Our hypergeometric type extended zeta function involves several known zeta functions including the Riemann, Hurwitz, Hurwitz-Lerch, and Barnes zeta functions as particular cases.

Expressions of content-parametrized Schur multiple zeta-functions via the Giambelli formula

Abstract: In this article, we consider the expressions for content-parametrized Schur multiple zeta-functions in terms of multiple zeta-functions of Euler-Zagier type and their star-variants, or in terms of modified zeta-functions of root systems. First of all, we focus on the Schur multiple zeta-function of hook type. And then, applying the Giambelli formula and induction argument, we obtain the expressions for general content-parametrized Schur multiple zeta-functions.

Zeta limits for the spectrum of quantum Rabi models

Abstract: The quantum Rabi model (QRM), one of the fundamental models used to describe light and matter interaction, has a deep mathematical structure revealed by the study of its spectrum. In this paper, directly from the explicit formulas for the partition function we derive various limits of the spectral zeta function with respect to the systems parameters of the asymmetric quantum Rabi model (AQRM), a generalization obtained by adding a physically significant parameter to the QRM. In particular, we consider the limit corresponding to the growth of the coupling strength to infinity recently studied by Fumio Hiroshima using resolvent analysis. The limits obtained in this paper are given in terms of the Hurwitz zeta function and other $L$-functions, suggesting further relations between spectral zeta function of quantum interaction models and number theory.

Categorifying Zeta Functions of Hyperelliptic Curves

Abstract: The zeta function of a hyperelliptic curve $C$ over a finite field factors into a product of $L$-functions, one of which is the $L$-function of $C$. We categorify this formula using objective linear algebra in the abstract incidence algebra of the poset of effective $0$-cycles of $C$. As an application, we prove a collection of combinatorial formulas relating the number of ramified, split and inert points on $C$ to the overall point count of $C$.

Riemann Hypothesis on Grönwall's Function

Abstract: Grönwall's function \(G\) is defined for all natural numbers \(n > 1\) by \(G(n)=\frac{\sigma(n)}{n \cdot \log \log n}\) where \(\sigma(n)\) is the sum of the divisors of \(n\) and \(\log\) is the natural logarithm. We require the properties of extremely abundant numbers, that is to say left to right maxima of \(n \mapsto G(n)\). We also use the colossally abundant and hyper abundant numbers. A number \(n\) is said to be colossally abundant if, for some \(\epsilon > 0\), \(\frac{\sigma(n)}{n^{1 + \epsilon}} \geq \frac{\sigma(m)}{m^{1 + \epsilon}}\) for all \(m > 1\). Let us call hyper abundant an integer \(n\) for which there exists \(u > 0\) such that \(\frac{\sigma(n)}{n \cdot (\log n)^{u}} \geq \frac{\sigma(m)}{m \cdot (\log m)^{u}}\) for all \(m > 1\). The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part \(\frac{1}{2}\). It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. We state that the Riemann hypothesis is true if and only if there exist infinitely many consecutive colossally abundant numbers \(3 \leq N < N'\) such that \(G(N) \leq G(N')\). In addition, we prove that the Riemann hypothesis is true when there exist infinitely many hyper abundant numbers \(n\) with any parameter \(u \gtrapprox 1\). We claim that there could be infinitely many hyper abundant numbers with any parameter \(u \gtrapprox 1\) and thus, the Riemann hypothesis would be true.

On a Weighted Series of the Hurwitz Zeta Function

Abstract: In this note we prove that for all $a \in \mathbb{N}$, $x \in \mathbb{R}_+ \cup \{0\}$, and $s \in \mathbb{C}$ with $\Re(s) > a + 2$, the (alternating) weighted series of the Hurwitz zeta function, $$ \sum_{k \geq 1} (\pm 1)^k (k + x)^a\zeta(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zena\"idi's theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.

An explicit sub-Weyl bound for $\zeta(1/2 + it)$

Abstract: In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $\zeta(s)$ on the critical line $s = 1/2 + it$. In particular, we show that $|\zeta(1/2 + it)| \le 66.7\, t^{27/164}$ for $t \ge 3$. Combined, our results form the sharpest known bounds on $\zeta(1/2 + it)$ for $t \ge \exp(61)$.

Zeta Function for Commuting Matrix Shift of Finite Type

Abstract: In this paper we show two-dimensional shifts of finite type whose transition matrices commutes and such that the entries of their product also consist of 0’s and 1’s. In particular, the form of associated zeta function of the two-dimensional shift was obtained. Unlike in the one-dimensional shifts of finite type; we gather that zeta functions for them have not closed form. This result derived by obtaining a formula for the number of periodic points of the subshift of finite type. This formula is then incorporated in the corresponding zeta function.

On zeros of bilateral hurwitz and periodic zeta and zeta star functions

Abstract: We show that (1) the periodic zeta function Li ⁡s(e2πia) with 0

On Several Results Associated with the Apéry-like Series

Abstract: In 1979, Apéry proved the irrationality of ζ(2) and ζ(3). Since then, there has been much research interest in investigating the Apéry-like series for values of Riemann zeta function, Ramanujan-like series for π and other infinite series involving central binomial coefficients. The purpose of this work is to present the first 20 results related to the Apéry-like series in the form of 4 lemmas, each containing 5 results. The Sherman’s results are applied to attain this. Thereafter, these 20 results are further used to establish up to 104 results pertaining to the Apéry-like series in the form of 4 theorems, with 26 results each. These findings are finally been described in terms of the generalized hypergeometric functions. Symmetry occurs naturally in the generalized hypergeometric functions.

The Riemann Zeta function using the Method of Stationary Phase Approximation of the Van der Pol Fourier Integral

Abstract: <p>The Van der Pol Fourier representation of the Riemann Zeta function in the critical strip is evaluated using the Method of Stationary Phase. The approximation allows us to propose new representation for the zeta function. </p>

Jacob's ladders and vector operator producing new generations of $L_2$-orthogonal systems connected with the Riemann's $\zeta(\frac 12+it)$ function

Abstract: In this paper we introduce a generating vector-operator acting on the class of functions $L_2([a,a+2l])$. This operator produces (for arbitrarily fixed $[a,a+2l]$) infinite number of new generation $L_2$-systems. Every element of the mentioned systems depends on Riemann's zeta-function and on Jacob's ladder.

Relation Between the Golden Ratio Phi and Zeta Function SUM

Abstract: This paper will explain the relation between the golden ratio Phi and Zeta function SUM. First, we will introduce why we used Phi and its functional properties then we will go through some of Phi Properties in a complex plane. Finally, we will use this Golden ratio Phi functional formula, to find the sum of the Prime numbers in Zeta function infinite series in relation with the Golden ration Phi, and pi.

A note on odd zeta values over any number field and Extended Eisenstein series

Abstract: In this article, we have studied transformation formulas of zeta function at odd integers over an arbitrary number field which in turn generalizes Ramanujan's identity for the Riemann zeta function. The above transformation leads to a new number field extension of Eisenstein series, which satisfies the transformation $z \mapsto -1/z$ like an integral weight modular form over SL$_2(\Z)$. The results provide number of important applications, which are important in studying the behaviour of odd zeta values as well as Lambert series in an arbitrary number field.

Correlations of the Riemann zeta function

Abstract: Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function \[ M_{\alpha,{\beta}}(T) = \int_T^{2T} \prod_{k = 1}^m |\zeta(\tfrac{1}{2} + i (t + \alpha_k))|^{2 \beta_k} dt \] introduced by Chandee, where ${\alpha} = {\alpha}(T) = (\alpha_1, \ldots, \alpha_m)$ and ${\beta} = (\beta_1 \ldots , \beta_m)$ satisfy $|\alpha_k| \leq T/2$ and $\beta_k\geq 0$. We shall prove that \[ M_{{\alpha},{\beta}}(T) \ll_{{\beta}} T (\log T)^{\beta_1^2 + \cdots + \beta_m^2} \prod_{1\leq j < k \leq m} |\zeta(1 + i(\alpha_j - \alpha_k) + 1/ \log T )|^{2\beta_j \beta_k}. \] This improves upon the previous best known bounds due to Chandee and Ng, Shen, and Wong, particularly when the differences $|\alpha_j - \alpha_k|$ are unbounded as $T \rightarrow \infty$. The key insight is to combine work of Heap, Radziwi{\l}{\l}, and Soundararajan and work of the author with the work of Harper on the moments of the zeta function.

Quantum entanglement & purity testing: A graph zeta function perspective

Abstract: We assign an arbitrary density matrix to a weighted graph and associate to it a graph zeta function that is both a generalization of the Ihara zeta function and a special case of the edge zeta function. We show that a recently developed bipartite pure state separability algorithm based on the symmetric group is equivalent to the condition that the coefficients in the exponential expansion of this zeta function are unity. Moreover, there is a one-to-one correspondence between the nonzero eigenvalues of a density matrix and the singularities of its zeta function. Several examples are given to illustrate these findings.

Amplitude-like functions from entire functions

Abstract: Recently a function was constructed that satisfies all known properties of a tree-level scattering of four massless scalars via the exchange of an infinite tower of particles with masses given by the non-trivial zeroes of the Riemann zeta function. A key ingredient in the construction is an even entire function whose only zeroes coincide with the non-trivial zeroes of the Riemann zeta function. In this paper we show that exactly the same conclusions can be drawn for an infinite class of even entire functions with only zeroes on the real line. This shows that the previous result does not seem to be connected to specific properties of the Riemann zeta function, but it applies more generally. As an application, we show that exactly the same conclusions can be drawn for L-functions other than the Riemann zeta function.

A heuristic for discrete mean values of the derivative of the Riemann zeta function

Abstract: Shanks conjectured that $\zeta ' (\rho)$, where $\rho$ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem, including a generalisation to all higher-order derivatives $\zeta^{(n)}(s)$, for which the sign of the mean alternatives between positive for odd $n$ and negative for even $n$. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of $\zeta^{(n)}(\rho)$.

On the splitting conjecture in the hybrid model for the Riemann zeta function

Abstract: Abstract We show that the splitting conjecture in the hybrid model of Gonek, Hughes and Keating holds to order on the Riemann hypothesis. Our results are valid in a larger range of the parameter X which mediates between the partial Euler and Hadamard products. We also show that the asymptotic splitting conjecture holds for this larger range of X in the cases of the second and fourth moments.

Riemann Hypothesis on Grönwall's Function

Abstract: Grönwall's function \(G\) is defined for all natural numbers \(n > 1\) by \(G(n)=\frac{\sigma(n)}{n \cdot \log \log n}\) where \(\sigma(n)\) is the sum of the divisors of \(n\) and \(\log\) is the natural logaritm. We require the properties of extremely abundant numbers, that is to say left to right maxima of \(n \mapsto G(n)\). We also use the colossally abundant and hyper abundant numbers. A number \(n\) is said to be colossally abundant if, for some \(\epsilon > 0\), \(\frac{\sigma(n)}{n^{1 + \epsilon}} \geq \frac{\sigma(m)}{m^{1 + \epsilon}}\) for all \(m > 1\). Let us call hyper abundant an integer \(n\) for which there exists \(u > 0\) such that \(\frac{\sigma(n)}{n \cdot (\log n)^{u}} \geq \frac{\sigma(m)}{m \cdot (\log m)^{u}}\) for all \(m > 1\). The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part \(\frac{1}{2}\). It is considered by many to be the most important unsolved problem in pure mathematics. There are several statements equivalent to the famous Riemann hypothesis. We state that the Riemann hypothesis is true if and only if there exist infinitely many consecutive colossally abundant numbers \(3 \leq N < N'\) such that \(G(N) \leq G(N')\). In addition, we prove that the Riemann hypothesis is true when there exist infinitely many hyper abundant numbers \(n\) with any parameter \(u \gtrapprox 1\). We claim that there could be infinitely many hyper abundant numbers with any parameter \(u \gtrapprox 1\) and thus, the Riemann hypothesis would be true.