Related papers: The Trace Method for Cotangent Sums
We provide a systematic study of sesquilinear hermitian forms and a new proof of the calculus of some exponential sums defined with quadratic hermitian forms. The computation of the number of solutions of equations such as Tr(f(x)+v.x)=0 or…
Sums over inverse s-th powers of semiprimes and k-almost primes are transformed into sums over products of powers of ordinary prime zeta functions. Multinomial coefficients known from the cycle decomposition of permutation groups play the…
We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…
We exploit transformations relating generalized $q$-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as $\pi$, and to connect sums…
We study sums of the form $\sum_{k=m}^n a_{nk} b_{km}$, where $a_{nk}$ and $b_{km}$ are binomial coefficients or unsigned Stirling numbers. In a few cases they can be written in closed form. Failing that, the sums still share many common…
We take the trace of Von-Neumann's ergodic theorem and get a trace formula of a unitary matrix family. It is an extension of Poisson summation formula in higher dimension. We also construct a family of crystalline measure with complex…
Using simple commutator relations, we obtain several trace identities involving eigenvalues and eigenfunctions of an abstract self-adjoint operator acting in a Hilbert space. Applications involve abstract universal estimates for the…
We study bounds for algebraic twists sums of automorphic coefficients by trace functions of composite moduli.
We present a unified approach which gives completely elementary proofs of three weighted sum formulae for double zeta values. This approach also leads to new evaluations of sums relating to the harmonic numbers, the alternating double zeta…
In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally…
Representation theory and the theory of symmetric functions have played a central role in Random Matrix Theory in the computation of quantities such as joint moments of traces and joint moments of characteristic polynomials of matrices…
We consider a cotangent sum related to Estermann's Zeta function. We provide an elementary and self-contained improvement of the error term in an asymptotic formula proved by V. I. Vasyunin.
We prove the uniqueness of a translation invariant extension to non integer order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we then describe using an Hadamard finite part procedure for sums over integer points…
Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $\beta$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral…
The purpose of the present paper is to provide a general overview of a variety of results related to a category of cotangent sums which have been proven to be associated to the so-called Nyman-Beurling criterion for the Riemann Hypothesis.…
Our purpose is to give a generalization of Pulita exponential series. After that, we shall use these series to give an analytic expression to a Gauss sums and to a trace formula for Witt vector rings.
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
In this paper, we give evaluations of integrals involving the arctan and the logarithm functions, and present several new summation identities for odd harmonic numbers and Milgram constants. These summation identities can be expressed as…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…