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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]

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 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]

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 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]

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]

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]

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]

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]

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]

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

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]

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]

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 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]

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]

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]

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 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]

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]

In recent studies of many-body localization in nonintegrable quantum systems, the distribution of the ratio of two consecutive energy level spacings, |$r_n=(E_{n+1}-E_n)/(E_{n}-E_{n-1})$| or |$\tilde{r}_n=\min (r_n,r_n^{-1})$|⁠ , has been used as a measure to quantify the chaoticity, alternative to the more conventional distribution of the level spacings, |$s_n=\bar{\rho }(E_n)(E_{n+1}-E_n)$|⁠ , as the former makes unnecessary the unfolding required for the latter. Based on our previous work on the Tracy–Widom approach to the Jánossy densities, we present analytic expressions for the joint probability distribution of two consecutive eigenvalue spacings and the distribution of their ratio for the Gaussian unitary ensemble (GUE) of random Hermitian |$N\times N$| matrices at |$N\rightarrow \infty$|⁠ , in terms of a system of differential equations. As a showcase of the efficacy of our results for characterizing an approach to quantum chaoticity, we contrast them to arguably the most ideal of all quantum-chaotic spectra: the zeroes of the Riemann |$\zeta$| function on the critical line at increasing heights. [ABSTRACT FROM AUTHOR]

We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations. [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 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]

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]

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]

An explicit description is obtained for the subspaces of the Hardy space on the right half-plane whose images under the Mellin transform yield a chain of de Branges spaces related to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C 2 defined by means of the Epstein ζ (s ; Q) and Hurwitz ζ (s , α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Classical hypergeometric series are reformulated as analytic functions of their parameters (in both the numerator and the denominator). Then, the coefficient extraction method is applied to examine hypergeometric series transformations. Several new closed form evaluations are established for harmonic series containing multinomial coefficient 4 n n , n , n , n and central binomial coefficient 2 n n . These results exclusively concern the alternating series of convergence rate " − 1 / 4 ". [ABSTRACT FROM AUTHOR]

In this paper, we prove Selberg's announced result on r$r$‐gaps between zeros of the Riemann zeta‐function ζ$\zeta$. Our proof uses a result on variations of argζ$\arg \zeta$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage‐Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments. [ABSTRACT FROM AUTHOR]

In this article we define and study a zeta function ζ G —similar to the Hasse-Weil zeta function—which enumerates absolutely irreducible representations over finite fields of a (profinite) group G. This Weil representation zeta function converges on a complex half-plane for all UBERG groups and admits an Euler product decomposition. Our motivation for this investigation is the observation that the reciprocal value ζ G (k) - 1 at a sufficiently large integer k coincides with the probability that k random elements generate the completed group ring of G. The explicit formulas obtained so far suggest that ζ G is rather well-behaved. A central object of this article is the Weil abscissa, i.e., the abscissa of convergence a(G) of ζ G . We calculate the Weil abscissae for free abelian, free abelian pro-p, free pro-p, free pronilpotent and free prosoluble groups. More generally, we obtain bounds (and sometimes explicit values) for the Weil abscissae of free pro- C groups, where C is a class of finite groups with prescribed composition factors. We prove that every real number a ≥ 1 is the Weil abscissa a(G) of some profinite group G. In addition, we show that the Euler factors of ζ G are rational functions in p - s if G is virtually abelian. For finite groups G we calculate ζ G using the rational representation theory of G. [ABSTRACT FROM AUTHOR]

Yamamoto’s integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto’s integral. We determine the value of infinite series of the form: ∑ m1,...,mn,ℓ1,...,ℓk≥1 H(a1)m1 · · · H(an)mn/mb11 · · · mbnn ℓc11 · · · ℓckk (m1+···+mn+ℓ1+···+ℓkℓk), in terms of a finite sum of multiple zeta values, for positive integers a1, . . ., an, b1, . . ., bn, c1, . . ., ck. [ABSTRACT FROM AUTHOR]

Two algebraic number fields are called arithmetically equivalent if the Dedekind zeta functions of the fields coincide. We show that if a G-extension contains non-conjugate arithmetically equivalent fields and there is an injection from G to another group H inducing an isoclinism between G and H, then there are non-conjugate arithmetically equivalent fields inside an H-extension. [ABSTRACT FROM AUTHOR]

Numerous logarithmic integrals have been extensively documented in the literature. This paper presents an algorithmic evaluation of a specific class of these integrals. Our systematic approach, rooted in logarithmic principles, enables us to extend our findings to other cases within this family of integrals. Furthermore, we explore special cases derived from our main results, thereby enhancing the applicability and significance of our work for a wider audience of researchers. [ABSTRACT FROM AUTHOR]

Sarnak's conjecture in quantum computing concerns when the groups PU2$\operatorname{PU}_{2}$ and PSU2$\operatorname{PSU}_{2}$ over cyclotomic rings Z[ζn,1/2]${\mathbb {Z}}[\zeta _{n}, 1/2]$ with ζn=e2πi/n$\zeta _n=e^{2\pi i/n}$, 4|n$4|n$, are generated by the Clifford‐cyclotomic gate set. We previously settled this using Euler–Poincaré characteristics. A generalization of Sarnak's conjecture is to ask when these groups are generated by torsion elements. An obstruction to this is provided by the corank: a group G$G$ has corankG>0$\operatorname{corank}G>0$ only if G$G$ is not generated by torsion elements. In this paper, we study the corank of these cyclotomic unitary groups in the families n=2s$n=2^s$ and n=3·2s$n={3\cdot 2^s}$, n⩾8$n\geqslant 8$, by letting them act on Bruhat–Tits trees. The quotients by this action are finite graphs whose first Betti number is the corank of the group. Our main result is that for the families n=2s$n=2^s$ and n=3·2s$n=3\cdot 2^s$, the corank grows doubly exponentially in s$s$ as s→∞$s\rightarrow \infty$; it is 0 precisely when n=8,12,16,24$n= 8,12, 16, 24$, and indeed, the cyclotomic unitary groups are generated by torsion elements (in fact by Clifford‐cyclotomic gates) for these n$n$. We give explicit lower bounds for the corank in two different ways. The first is to bound the isotropy subgroups in the action on the tree by explicit cyclotomy. The second is to relate our graphs to Shimura curves over Fn=Q(ζn)+$F_n={\mathbf {Q}}(\zeta _n)^+$ via interchanging local invariants and applying a result of Selberg and Zograf. We show that the cyclotomy arguments give the stronger bounds. In a final section, we execute a program of Sarnak to show that our results for the n=2s$n=2^s$ and n=3·2s$n={3\cdot 2^s}$ families are sufficient to give a second proof of Sarnak's conjecture. [ABSTRACT FROM AUTHOR]

Let S(t)=1πImlogζ12+it$S(t) = \frac{1}{\pi }\operatorname{Im}\log \zeta \left(\frac{1}{2}+it\right)$. We prove an unconditional lower bound on the measure of the sets {t∈[T,2T]:S(t)⩾V}$\lbrace t\in [T,2T] \colon S(t) \geqslant V\rbrace$ for loglogT⩽V≪logTloglogT1/3$\sqrt {\log \log T} \leqslant V \ll \left(\frac{\log T}{\log \log T}\right)^{1/3}$. For V⩽(logT)1/3−ε$V \leqslant (\log T)^{1/3-\varepsilon }$, our bound has a Gaussian shape with variance proportional to loglogT$\log \log T$. At the endpoint, V≍logTloglogT1/3$V \asymp \left(\frac{\log T}{\log \log T}\right)^{1/3}$, our result implies the best known Ω$\Omega$‐theorem for S(t)$S(t)$ that is due to Tsang. We also explain how the method breaks down for V≫logTloglogT1/3$V \gg \left(\frac{\log T}{\log \log T}\right)^{1/3}$ given our current knowledge about the zeros of the zeta function. Conditionally on the Riemann hypothesis, we extend our results to the range loglogT⩽V≪logTloglogT1/2$\sqrt {\log \log T} \leqslant V \ll \left(\frac{\log T}{\log \log T}\right)^{1/2}$. [ABSTRACT FROM AUTHOR]

In recent decades, a growing number of discoveries in mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces. As computers become more powerful, an intriguing possibility arises--the interplay between human intuition and computer algorithms can lead to discoveries of mathematical structures that would otherwise remain elusive. Here, we demonstrate computer-assisted discovery of a previously unknown mathematical structure, the conservative matrix field. In the spirit of the Ramanujan Machine project, we developed a massively parallel computer algorithm that found a large number of formulas, in the form of continued fractions, for numerous mathematical constants. The patterns arising from those formulas enabled the construction of the first conservative matrix fields and revealed their overarching properties. Conservative matrix fields unveil unexpected relations between different mathematical constants, such as K and ln(2), or e and the Gompertz constant. The importance of these matrix fields is further realized by their ability to connect formulas that do not have any apparent relation, thus unifying hundreds of existing formulas and generating infinitely many new formulas. We exemplify these implications on values of the Riemann zeta function Ç (n), studied for centuries across mathematics and physics. Matrix fields also enable new mathematical proofs of irrationality. For example, we use them to generalize the celebrated proof by Apéry of the irrationality of Ç(3). Utilizing thousands of personal computers worldwide, our research strategy demonstrates the power of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science. [ABSTRACT FROM AUTHOR]

In this paper, we solve the open problem posed by Kuba by expressing ∑ j , k ≥ 1 H k (u) H j (v) H j + k (w) j r k s (j + k) t as a linear combination of multiple zeta values. These sums include Tornheim's double series as a special case. Our approach is based on employing two distinct methods to evaluate the specific integral proposed by Yamamoto, which is associated with the two-poset Hasse diagram. We also provide a new evaluation formula for the general Mordell–Tornheim series and some similar types of double and triple series. [ABSTRACT FROM AUTHOR]

We derive explicit asymptotic formulae for the joint moments of the |$n_{1}$| -th and |$n_{2}$| -th derivatives of the characteristic polynomials of Circular Unitary Ensemble random matrices for any non-negative integers |$n_{1}, n_{2}$|⁠. These formulae are expressed in terms of determinants whose entries involve modified Bessel functions of the first kind. We also express them in terms of two types of combinatorial sums. Similar results are obtained for the analogue of Hardy's |$Z$| -function. We use these formulae to formulate general conjectures for the joint moments of the |$n_{1}$| -th and |$n_{2}$| -th derivatives of the Riemann zeta-function and of Hardy's |$Z$| -function. Our conjectures are supported by comparison with results obtained previously in the number theory literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

We study the relations of multiple t -values of general level. The generating function of sums of multiple t -(star) values of level N with fixed weight, depth and height is represented by the generalized hypergeometric function 3 F 2 , which generalizes the results for multiple zeta(-star) values and multiple t -(star) values. As applications, we obtain formulas for the generating functions of sums of multiple t -(star) values of level N with height one and maximal height and a weighted sum formula for sums of multiple t -(star) values of level N with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman's restricted sum formulas for multiple t -(star) values of level N. Some evaluations of multiple t -star values of level 2 with one–two–three indices are given. [ABSTRACT FROM AUTHOR]

We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in P 4 . Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H 3 (W , Z) . It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ 0 (25) , whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]