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We obtain a full asymptotic for the sum of ζ(n)(ρ)Xρ, where ζ(n)(s) denotes the nth derivative of the Riemann zeta function, X is a positive real number, and ρ denotes a nontrivial zero of the Riemann zeta function. The sum is over the zeros with imaginary parts up to a height T, as T →∞.We also specify what the asymptotic formula becomes when X is a positive integer, highlighting the differences in the asymptotic expansions as X changes its arithmetic nature. [ABSTRACT FROM AUTHOR]

We introduce the zeta function of a signed graph and give a determinant expression for it in terms of the signed oriented line graph, and moreover, we obtain some properties of the zeta function of a signed graph. As applications we affirm the conjecture proposed by Sato [Weighted zeta functions of graph coverings, Electronic J. Combin. 13 (2006), #91] and give a method to construct a family of cospectral signed digraphs. Additionally, we show that two connected regular signed graphs have the same zeta function if and only if they are cospectral. [ABSTRACT FROM AUTHOR]

We prove that for any twist rigid compact p-adic analytic group G, its twist representation zeta function is a finite sum of terms n_{i}^{-s}f_{i}(p^{-s}), where n_{i} are natural numbers and f_{i}(t)\in \mathbb {Q}(t) are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If G is moreover a pro-p group, we prove that its twist representation zeta function is rational in p^{-s}. To establish these results we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass. [ABSTRACT FROM AUTHOR]

In this article, we consider classical and Bayesian estimations of some economic measures, specially Bonferroni Curve and Bonferroni Index of the Pareto distribution. We obtain the Maximum Likelihood Estimator and Uniform Minimum Variance Unbiased Estimator in the classical setup, and their properties are studied. Additionally, we conduct Bayesian estimation procedures based on symmetric loss functions and a truncated gamma prior. The precision of the estimators is evaluated under different sample sizes via Monte Carlo simulation. Furthermore, a real dataset is provided to compute all the estimators. [ABSTRACT FROM AUTHOR]

AbstractIn recent three decades, there has been a rapid increasing interest in the mixture models for ordinal responses, and the classical CUB model as a fundamental one has been extended to different preference and uncertainty models. In this article, based on a response style supported by Zipf’s law, we propose a novel mixture model for ordinal responses via replacing the uncertainty component of the CUB model with a truncated Zeta distribution. Parameters estimation with EM algorithm, inferential issues with respect to the approximation of a truncated Riemann Zeta function and estimators’ variance-covariance information matrix are investigated. The advantages of the proposed model over the CUB model have been illustrated with simulations of two sets of Monte Carlo experiments and practical applications of a health survey and a bicycle use. The intention of the article is to distinguish the respondents’ true preference from the response style of “the higher, the less”, so as to understand more reasonably the formation causes of ordinal responses. [ABSTRACT FROM AUTHOR]

We study the function |$\Delta _{k}(x):=\sum _{n\leq x} d_{k}(n) - \mbox{Res}_{s=1} (\zeta ^{k}(s) x^{s}/s)$|⁠ , where |$k\geq 3$| is an integer, |$d_{k}(n)$| is the |$k$| -fold divisor function, and |$\zeta (s)$| is the Riemann zeta-function. For a large parameter |$X$|⁠ , we show that if the Lindelöf hypothesis (LH) is true, then there exist at least |$X^{\frac{1}{k(k-1)}-\varepsilon }$| disjoint subintervals of |$[X,2X]$|⁠ , each of length |$X^{1-\frac{1}{k}-\varepsilon }$|⁠ , such that |$|\Delta _{k}(x)|\gg x^{\frac{1}{2}-\frac{1}{2k}}$| for all |$x$| in the subinterval. In particular, |$\Delta _{k}(x)$| does not change sign in any of these subintervals. If the Riemann hypothesis (RH) is true, then we can improve the length of the subintervals to |$\gg X^{1-\frac{1}{k}} (\log X)^{-k^{2}-2}$|⁠. These results may be viewed as higher-degree analogues of theorems of Heath-Brown and Tsang, who studied the case |$k=2$|⁠ , and Cao, Tanigawa, and Zhai, who studied the case |$k=3$|⁠. The first main ingredient of our proofs is a bound for the second moment of |$\Delta _{k}(x+h)-\Delta _{k}(x)$|⁠. We prove this bound using a method of Selberg and a general lemma due to Saffari and Vaughan. The second main ingredient is a bound for the fourth moment of |$\Delta _{k}(x)$|⁠ , which we obtain by combining a method of Tsang with a technique of Lester. [ABSTRACT FROM AUTHOR]

In this paper, we provide a symmetric formula and a duality formula relating multiple zeta values and zeta-star values. We find that the summation ∑ a + b = r − 1 (− 1) a ζ ★ (a + 2 , { 2 } p − 1) ζ ★ ({ 1 } b + 1 , { 2 } q) equals ζ ★ ({ 2 } p , { 1 } r , { 2 } q) + (− 1) r + 1 ζ ★ ({ 2 } q , r + 2 , { 2 } p − 1) . With the help of this equation and Zagier's ζ ★ ({ 2 } p , 3 , { 2 } q) formula, we can easily determine ζ ★ ({ 2 } p , 1 , { 2 } q) and several interesting expressions. [ABSTRACT FROM AUTHOR]

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. [13] , the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem. [ABSTRACT FROM AUTHOR]

Let p , q ≥ 1 be two real numbers such that 1 p + 1 q = 1 , and let a , b ∈ R be two parameters defined on the domain of a function, for example, f. Based on the well known Hölder inequality, we propose a generic inequality of the form | f (a p + b q) | ≤ | f (a) | 1 p | f (b) | 1 q , and show that many basic special functions, such as the gamma and polygamma functions, Riemann zeta function, beta function and Gauss and confluent hypergeometric functions, satisfy this type of inequality. In this sense, we also present some particular inequalities for the Gauss and confluent hypergeometric functions to confirm the main obtained inequalities. [ABSTRACT FROM AUTHOR]

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z = F 1 2 (1 / 2 , 1 / 2 ; 1 ; x) and z ′ = d z / d x . When a certain parameter in these series is equal to π , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. [ABSTRACT FROM AUTHOR]

We introduce a new method to detect the zeros of the Riemann zeta function, which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few "half-isolated" zeros. By combining this with classical methods, we improve the Ingham–Huxley zero-density estimate under the assumption that the non-trivial zeros of the zeta function are restricted to lie on a finite number of fixed vertical lines. This has new consequences for primes in short intervals under the same assumption. [ABSTRACT FROM AUTHOR]

Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava's general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann's zeta function, Catalan's constant, polylogarithm functions, and so on. [ABSTRACT FROM AUTHOR]

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure. [ABSTRACT FROM AUTHOR]

The twisted Ruelle zeta function of a contact, Anosov vector field, is shown to be equal, as a meromorphic function of the complex parameter ħ ∈ C and up to a phase, to the partition function of an ħ -linear quadratic perturbation of BF theory, using an "axial" gauge fixing condition given by the Anosov vector field. Equivalently, it is also obtained as the expectation value of the same quadratic, ħ -linear, perturbation, within a perturbative quantisation scheme for BF theory, suitably generalised to work when propagators have distributional kernels. [ABSTRACT FROM AUTHOR]

In this paper by using q-numbers and r-shift, the (q, r)-Stirling numbers with level s are studied. One of the main aims is to give several identities in their transforms. We also give some applications to the values of a certain kind of q-multiple zeta functions. [ABSTRACT FROM AUTHOR]

In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the $2k$ th moments of the Riemann zeta function on the critical line. [ABSTRACT FROM AUTHOR]

In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields. [ABSTRACT FROM AUTHOR]

In this work, we estimate the sum ∑0<ℑ(ρ)⩽Tζ(ρ+α)X(ρ)Y(1−ρ)$$\begin{align*} \sum _{0 < \Im (\rho) \leqslant T} \zeta (\rho +\alpha)X(\rho) Y(1\!-\! \rho) \end{align*}$$over the nontrivial zeros ρ$\rho$ of the Riemann zeta function where α$\alpha$ is a complex number with α≪1/logT$\alpha \ll 1/\log T$ and X(·)$X(\cdot)$ and Y(·)$Y(\cdot)$ are Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where ζ(ρ+α)$\zeta (\rho +\alpha)$ is replaced by ζ(m)(ρ)$\zeta ^{(m)}(\rho)$ for all m∈N$m\in \mathbb {N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann hypothesis, we obtain the lower bound ∑0<ℑ(ρ)

Assuming the Riemann hypothesis, we investigate the shifted moments of the zeta function Mα,β(T)=∫T2T∏k=1mζ12+i(t+αk)2βkdt$$\begin{equation*} \hspace*{24.5pt}M_{{\bm \alpha}, {\bm \beta}} (T) = \int _T^{2T} \prod _{k = 1}^m {\left|\zeta \left(\tfrac{1}{2} + i (t + \alpha _k)\right)\right|}^{2 \beta _k} dt\hspace*{-24.5pt} \end{equation*}$$introduced by Chandee Q. J. Math. 62(2011), no. 3, 545–572, where α=α(T)=(α1,...,αm)${\bm \alpha} = {\bm \alpha} (T) = (\alpha _1, \ldots, \alpha _m)$ and β=(β1...,βm)$\bm {\beta } = (\beta _1 \ldots, \beta _m)$ satisfy |αk|⩽T/2$|\alpha _k| \leqslant T/2$ and βk⩾0$\beta _k\geqslant 0$. We shall prove Mα,β(T)≪βT(logT)β12+⋯+βm2∏1⩽j

Let ζ(k)(s) be the k-th derivative of the Riemann zeta function and a be a complex number. The solutions of ζ(k)(s) = a are called a-points. In this paper, we give an asymptotic formula for the sum ... ζ(j) (ρa(k)) as T → ∞ where j and k are non-negative integers and ρa(k) denotes an a-point of the k-th derivative ζ(k)(s) and γa(k) = Im(ρa(k)). [ABSTRACT FROM AUTHOR]

We have derived alternative closed-form formulas for the trigonometric series over sine or cosine functions when the immediate replacement of the parameter appearing in the denominator with a positive integer gives rise to a singularity. By applying the Choi–Srivastava theorem, we reduce these trigonometric series to expressions over Hurwitz's zeta function derivative. [ABSTRACT FROM AUTHOR]

The Z-function is the real function given by Z (t) = e i θ (t) ζ 1 2 + i t , where ζ (s) is the Riemann zeta function, and θ (t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z (t) and its zeros. The sections of Z (t) are given by Z N (t) : = ∑ k = 1 N cos (θ (t) − ln (k) t) k for any N ∈ N . Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z (t) ≈ 2 Z N ˜ (t) (t) for N ˜ (t) = t 2 π . While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z (t) ≈ Z N (t) (t) for N (t) = t 2 , which is Spira's approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira's conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. [ABSTRACT FROM AUTHOR]

Knowledge of the partition and spectral zeta function of a quantum system is fundamental for both physics and mathematics, and the positions these functions occupy in their respective fields share a common philosophy. In this article, we describe the number theoretic structures hidden behind light and matter interaction models, focusing on the partition function and special values of the spectral zeta function, highlighting how modern mathematical research is involved. [ABSTRACT FROM AUTHOR]

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology. In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the generalized zeta function technique to investigate the vacuum energy, temperature corrections and heat kernel coefficients associated with a scalar field under a quasiperiodic condition in a (D + 1) -dimensional conical spacetime. In this scenario, we find that the renormalized vacuum energy, as well as the temperature corrections, are both zero. The nonzero heat kernel coefficients are the ones related to the usual Euclidean divergence, and also to the nontrivial aspects of the quaisperiodically identified conical spacetime topology. An interesting result that arises in this configuration is that for some values of the quasiperiodic parameter, the heat kernel coefficient associated with the nontrivial topology vanishes. In addition, we also consider the scalar field in a (D + 1) -dimensional spacetime formed by the combination of a conical and screw dislocation topological defects. In this case, we obtain a nonzero renormalized vacuum energy density and its corresponding temperature corrections. Again, the nonzero heat kernel coefficients found are the ones related to the Euclidean and nontrivial topology divergences. For D = 3 , we explicitly show, in the massless scalar field case, the limits of low and high temperatures for the free energy. In the latter, we show that the free energy presents a classical contribution. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and λ -Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. [ABSTRACT FROM AUTHOR]

Deep work by Shintani in the 1970's describes Hecke L -functions associated to narrow ray class group characters of totally real fields F in terms of what are now known as Shintani zeta functions. However, for [ F : Q ] = n ≥ 3 , Shintani's method was ineffective due to its crucial dependence on abstract fundamental domains for the action of totally positive units of F on R + n , so-called Shintani sets. These difficulties were recently resolved in independent work of Charollois, Dasgupta, and Greenberg and Diaz y Diaz and Friedman. For those narrow ray class group characters whose conductor is an inert rational prime in a totally real field F with narrow class number 1, we obtain a natural combinatorial description of these sets, allowing us to obtain a simple description of the associated Hecke L -functions. As a consequence, we generalize earlier work of Girstmair, Hirzebruch, and Zagier, that offer combinatorial class number formulas for imaginary quadratic fields, to real and imaginary quadratic extensions of totally real number fields F with narrow class number 1. For CM quadratic extensions of F , our work may be viewed as an effective affirmative answer to Hecke's Conjecture that the relative class number has an elementary arithmetic expression in terms of the relative discriminant. [ABSTRACT FROM AUTHOR]

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function ζ (s) on the critical line s = 1 / 2 + i t. In particular, we show that | ζ (1 / 2 + i t) | ≤ 66.7 t 27 / 164 for t ≥ 3. Combined, our results form the sharpest known bounds on ζ (1 / 2 + i t) for t ≥ exp ⁡ (61). [ABSTRACT FROM AUTHOR]

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b , the set { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a , b) # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log 2 ⁡ x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S = { m 1 , m 2 , m 3 } such that m 1 , m 2 , m 3 , − 3 m 1 m 2 , − 3 m 2 m 3 , − 3 m 1 m 3 , m 1 m 2 m 3 are not squares, there exists a pair of elements a , b ∈ S such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. Further, under the assumption of a level of distribution greater than x 2 3 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S = { m 1 , m 2 } such that m 1 , m 2 , − 3 m 1 m 2 are not squares, there exists a pair of elements a , b ∈ { m 1 , m 2 , − 3 m 1 m 2 } such that # { p ⩽ x : p prime, m a mod p ∈ 〈 b 〉 mod p } ≫ x log ⁡ log ⁡ x log 2 ⁡ x. [ABSTRACT FROM AUTHOR]

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to establish certain generalized Marichev-Saigo-Maeda fractional integral and derivative formulas involving generalized p–extended Hurwitz-Lerch zeta function by using the Hadamard product (or the convolution) of two analytic functions. We then obtain their composition formulas by using fractional integral and derivative formulas and certain Integral transforms associated with Beta, Laplace and Whittaker transforms involving generalized p–extended Hurwitz-Lerch Zeta function. [ABSTRACT FROM AUTHOR]

In this paper, we solve an open problem considered by Steven Finch (Central Binomial Coefficients, 2007, Available from: , p. 5), as far back as 2007, concerning the calculation of a series involving the central binomial coefficient. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the positivity of the real part of the logarithmic derivative of the Riemann &#163l; -function when 1/2 <σ < 1 and t is sufficiently large. We consider explicit upper and lower bounds of R/(α-σ) where the summation runs over the zeros of £ (s) on the line 1/2+it. We also examine the positivity of R £/£ (s) in the strip 1/2 < σ < 1 assuming that there occur non-trivial zeros of £ (s) off the critical line. [ABSTRACT FROM AUTHOR]

Previously the authors proved subconvexity of Shintani's zeta function enumerating class numbers of binary cubic forms. Here we return to prove subconvexity of the Maass form twisted version. The method demonstrated here has applications to the subconvexity of some of the twisted zeta functions introduced by F. Sato. The argument demonstrates that the symmetric space condition used by Sato is not necessary to estimate the zeta function in the critical strip. [ABSTRACT FROM AUTHOR]

In this paper, we proved that ζ itself cannot satisfy any non-trivial algebraic differential equations whose coefficients are polynomials in Γ and its derivatives, which is a conjecture of L. Markus in 2007. We also proved the difference analogue of Markus's conjecture. [ABSTRACT FROM AUTHOR]

Fractional Kinetic equations (FKEs) including a wide variety of special functions are widely and successfully applied in describing and solving many important problems of physics and astrophysics. In this work, the solutions of the FKEs of the generalized Hurwitz-Lerch Zeta function using the Laplace transform are derived and examined. [ABSTRACT FROM AUTHOR]

Secure routing and communication with confidentiality based on encryption of texts in multiple natural languages are challenging issues in wireless sensor networks which are widely used in recent applications. The existing works on Elliptic Curve Cryptography based secured routing algorithms are focused only on the encryption and decryption of single language text encrypted over a Prime finite field. In this article, a new algorithm called Multi‐Language ECC encrypted Secure Routing algorithm with trust management is proposed, in order to ensure confidentiality and integrity which focuses on the encryption of plain text using Riemann's zeta function and Elliptic Curve Cryptography for improving the key strength which is applied for encryption over a range of multi languages namely Tamil, English, Hindi French and German which are supported by Unicode and routing the text security. From the experiments conducted using the proposed multi‐lingual encryption algorithm with network routing, we prove that the suggested method provides greater security than the current secure routing algorithms due to the use of Zeta function and Gamma function with ECC key and trust management. but also boasts reduced complexity compared to other existing multi‐lingual encryption algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we present a smooth version of Landau’s explicit formula for the von Mangoldt arithmetical function. By assuming the validity of the Riemann hypothesis, we show that in order to determine whether a natural number μ is a prime number, it is sufficient to know the location of a number of nontrivial zeros of the Riemann zeta function of order μlog3 2μ. Next we use Heisenberg’s inequality to support the conjecture that this number of zeros cannot be essentially diminished. [ABSTRACT FROM AUTHOR]

We consider a special class of generalized Paley graphs over finite fields, namely the Cayley graphs with vertex set 픽qm and connection set the nonzero (qℓ + 1)th powers in 픽qm, as well as their complements. We explicitly compute the spectrum and the energy of these graphs. As a consequence, the graphs turn out to be (with trivial exceptions) simple, connected, non-bipartite, integral and strongly regular, of pseudo or negative Latin square type. Using the spectral information we compute several invariants of these graphs. We exhibit infinitely many pairs of integral equienergetic non-isospectral graphs. As applications, on the one hand we solve Waring’s problem over 픽qm for the exponents qℓ + 1, for each q and for infinitely many values of ℓ and m. We obtain that the Waring number g(qℓ + 1,qm) = 1 or 2, depending on m and ℓ, thus solving some open cases. On the other hand, we construct infinite towers of integral Ramanujan graphs in all characteristics. Finally, we give the Ihara zeta functions of these graphs. [ABSTRACT FROM AUTHOR]

Statistical characterizations of complex network structures can be obtained from both the Ihara Zeta function (in terms of prime cycle frequencies) and the partition function from statistical mechanics. However, these two representations are usually regarded as separate tools for network analysis, without exploiting the potential synergies between them. In this paper, we establish a link between the Ihara Zeta function from algebraic graph theory and the partition function from statistical mechanics, and exploit this relationship to obtain a deeper structural characterisation of network structure. Specifically, the relationship allows us to explore the connection between the microscopic structure and the macroscopic characterisation of a network. We derive thermodynamic quantities describing the network, such as entropy, and show how these are related to the frequencies of prime cycles of various lengths. In particular, the n-th order partial derivative of the Ihara Zeta function can be used to compute the number of prime cycles in a network, which in turn is related to the partition function of Bose–Einstein statistics. The corresponding derived entropy allows us to explore a phase transition in the network structure with critical points at high and low-temperature limits. Numerical experiments and empirical data are presented to evaluate the qualitative and quantitative performance of the resulting structural network characterisations. [ABSTRACT FROM AUTHOR]

The Hurwitz zeta-function ζ (s , α) , s = σ + i t , with parameter 0 < α ⩽ 1 is a generalization of the Riemann zeta-function ζ (s) ( ζ (s , 1) = ζ (s) ) and was introduced at the end of the 19th century. The function ζ (s , α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ (s , α) is the main example of zeta-functions without Euler's product (except for the cases α = 1 , α = 1 / 2 ), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ (s , α) for some α is universal, i.e., its shifts ζ (s + i τ , α) , τ ∈ R , approximate every analytic function defined in the strip { s : 1 / 2 < σ < 1 } . For needs of effectivization of the universality property for ζ (s , α) , the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ (σ + i t , α) in short intervals. In this paper, we obtain the bound O (H) for that mean square over the interval [ T − H , T + H ] , with T 27 / 82 ⩽ H ⩽ T σ and 1 / 2 < σ ⩽ 7 / 12 . This is the first result on the mean square for ζ (s , α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ (s , α) and other zeta-functions in short intervals. [ABSTRACT FROM AUTHOR]

We study universal quadratic forms over totally real number fields using Dedekind zeta functions. In particular, we prove an explicit lower bound for the rank of universal quadratic forms over a given number field K, under the assumption that the codifferent of K is generated by a totally positive element. Motivated by a possible path to remove that assumption, we also investigate the smallest number of generators for the positive part of ideals in totally real numbers fields. [ABSTRACT FROM AUTHOR]

For k ≥ 1 , let ℛ k (x) denote the reciprocal sum up to x of numbers with k prime factors, counted with multiplicity. In prior work, the authors obtained estimates for ℛ k (x) , extending Mertens' second theorem, as well as a finer-scale estimate for ℛ 2 (x) up to (log x) − N error for any N > 0. In this paper, we establish the limiting behavior of the higher Mertens constants from the ℛ 2 (x) estimate. We also extend these results to ℛ 3 (x) , and we comment on the general case k ≥ 4. [ABSTRACT FROM AUTHOR]

We introduce a new Lie-algebraic approach to explicitly construct the motivic coaction and single-valued map of multiple polylogarithms in any number of variables. In both cases, the appearance of multiple zeta values is controlled by conjugating generating series of polylogarithms with Lie-algebra generators associated with odd zeta values. Our reformulation of earlier constructions of coactions and single-valued polylogarithms preserves choices of fibration bases, exposes the correlation between multiple zeta values of different depths and paves the way for generalizations beyond genus zero. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann's zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann's zeta function. Finally, a heuristic approach to evaluating more general sums is also given. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra B n of order 2 n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ (P) of complement-closed and convex subposets P of B n. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V (P) on P. We then write down the adjacency operator A of the graph Γ (P) as the composition of two linear operators on V (P) , namely, the operator that multiplies elements of V (P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P , then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ (P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property. [ABSTRACT FROM AUTHOR]

We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers–Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated. [ABSTRACT FROM AUTHOR]

The zeta function is like a milestone in analytic number theory. Considering Euler's definition of the zeta function in this article, we will find some recursion at even integral points. We will also discuss the Dirichlet eta function, even Bernoulli numbers, and the gamma–zeta relation. Toward the end, we will define something known as the odd zeta function and find a recurrence for the same. [ABSTRACT FROM AUTHOR]

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For k⩽1/2$k\leqslant 1/2$, our bounds for the 2k$2k$‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros. [ABSTRACT FROM AUTHOR]