Related papers: Mean values of some Hecke characters

With the enourmous popularity of large language models, many researchers have raised ethical concerns regarding social biases incorporated in such models. Several methods to measure social bias have been introduced, but apparently these…

Let $x$ be a positive integer. We give an asymptotic result for $\omega(\operatorname{lcm}(m,n))$ summed over all positive integers $m$ and $n$ with $mn \le x$. This answers an open question posed in a recent paper.

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $GL_n$ modules. Our main result gives three…

We obtain a spectral decomposition of shifted convolution sums in Hecke eigenvalues of holomorphic or Maass cusp forms.

Matrix representations of Hecke operators on classical holomorphical cusp forms and corresponding period polynomials are well known. In this article we define Hecke operators on period functions and show that they correspond to the Hecke…

For a nonprincipal character $\chi$ modulo $D$, when $x\ge D^{\frac56+\varepsilon}$, $(l,D) = 1$, we prove a nontrivial estimate of the form $\sum_{n\le x}\Lambda (n)\chi (n-l)\ll x\exp\left(-0.6\sqrt{\ln D}\right)$ for the sum of values of…

The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.

Building upon the work of Stucker, Vennos, and Young we derive generalized Dedekind sums arising from period integrals applied to holomorphic Eisenstein series attached to pairs of primitive non-trivial Dirichlet characters. Furthermore, we…

The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which…

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit…

We consider a short sequence of hermitian vector bundles on some arithmetic variety. Assuming that this sequence is exact on the generic fiber we prove that the alternated sum of the arithmetic Chern characters of these bundles is the sum…

Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The…

In this note, we introduce arithmetic Heilbronn supercharacters that generalize the notions of arithmetic Heilbronn characters and Heilbronn supercharacters and discuss several properties of them.

Let $\psi$ be a function such that $\psi(x) \rightarrow \infty$ as $x \rightarrow \infty.$ Let $\lambda_{f}(n)$ be the $n$-th Hecke eigenvalue of a fixed holomorphic cusp form $f$ for $SL(2,\mathbb{Z}).$ We show that for any real valued…

Positive polynomials arising from Muirhead's inequality, from classical power mean and elementary symmetric mean inequalities and from Minkowski's inequality can be rewritten as sums of squares.

We prove a mean value theorem for the traces of Hecke operators acting on cusp forms of GL(n) over imaginary quadratic number fields together with an upper bound for the error term depending explicitly on the Hecke operator.

In the article [PV] a general procedure to study solutions of the equations $x^4-dy^2=z^p$ was presented for negative values of $d$. The purpose of the resent article is to extend our previous results to positive values of $d$. On doing so,…

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…

The average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n->infty. The same is true for primitive cubic characters. It is therefore surprising that, as n->infty, the average number of primitive…

The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: {\sigma}(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of…