Related papers: The Trace Method for Cotangent Sums

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…

We show that certain sums studied in two recent papers are basically character coordinates (as they are called in the literature). These sums involve values of Dirichlet characters and powers of $\cot(\pi k/n)$, $1\le k\le n-1$. We also…

We use a spectral theory perspective to reconsider properties of the Riemann zeta function. In particular, new integral representations are derived and used to present its value at odd positive integers.

We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran et al,…

The multiple zeta values are generalizations of the values of the Riemann zeta function at positive integers. They are known to satisfy a number of relations, among which are the cyclic sum formula. The cyclic sum formula can be stratified…

We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the…

We outline an approach to proving functoriality of automorphic representations using trace formula. More specifically, we construct a family of integral operators on the space of automorphic forms whose eigenvalues are expressed in terms of…

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…

We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random…

We introduce certain lattice sums associated with hyperplane arrangements, which are (multiple) sums running over integers, and can be regarded as generalizations of certain linear combinations of zeta-functions of root systems. We also…

Let $S_{p,n}$ denote the sample covariance matrix based on $n$ independent identically distributed $p$-dimensional random vectors in the null-case. The main result of this paper is an explicit expansion of trace moments and power-trace…

We consider analytic functions of the Riemann zeta type, for which, if $s$ is a zero, so is $1-s$. We use infinite product representations of these functions, assuming their zeros to be of first order. We use exponential factors to…

We introduce the method of path-sums which is a tool for exactly evaluating a function of a discrete matrix with possibly non-commuting entries, based on the closed-form resummation of infinite families of terms in the corresponding Taylor…

A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…

We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications.…

This work develops an analytic framework for the study of the $\zeta$-function associated with general sequences of complex numbers. We show that a contour integral representation, commonly used when studying spectral $\zeta$-functions…

Traces and their extension called combined traces (comtraces) are two formal models used in the analysis and verification of concurrent systems. Both models are based on concepts originating in the theory of formal languages, and they are…

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an…

We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first is new and involves sums of roots of unity. The second uses contour integration and extends a previous method used by two of…