Related papers: On Menon-Sury's identity with several Dirichlet ch…

In this short note, we give a new Menon-type identity involving the sum of element orders and the sum of cyclic subgroup orders of a finite group. It is based on applying the weighted form of Burnside's lemma to a natural group action.

Given a general symmetric elliptic operator $$ L\_{a} := \sum\_{k,,j=1}^d \p\_k (a\_{kj} \p\_j) + \sum\_{k=1}^d a\_k \p\_k - \p\_k(\overline{a\_k} .) + a\_0$$we define the associated Dirichlet-to-Neumann (D-t-N) operator with partial data,…

We present a ``method'' for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a bijective construction…

We present outlines of a general method to reach certain kinds of $q$-multiple sum identities. Throughout our exposition, we shall give generalizations to the results given by Dilcher, Prodinger, Fu and Lascoux, Zeng, and Guo and Zhang…

We present various identities involving the classical Bernoulli and Euler polynomials. Among others, we prove that $$ \sum_{k=0}^{[n/4]}(-1)^k {n\choose 4k}\frac{B_{n-4k}(z) }{2^{6k}} =\frac{1}{2^{n+1}}\sum_{k=0}^{n} (-1)^k…

In this article we study analytic properties of the multiple Dirichlet series associated to additive and Dirichlet characters. For the multiple Dirichlet series associated to additive characters, the meromorphic continuation is established…

We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a…

The purpose of this paper is to generalize our earlier work on the logarithm of the Riemann zeta-function to linear combinations of logarithms of primitive Dirichlet $L$-functions with constant real coefficients. Under the assumption of…

We present quantitative versions of Bohr's theorem on general Dirichlet series $D=\sum a_{n} e^{-\lambda_{n}s}$ assuming different assumptions on the frequency $\lambda:=(\lambda_{n})$, including the conditions introduced by Bohr and…

An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…

In this note, we study the integrodifferential operator $(I-\Delta)^{\log}$ corresponding to the logarithmic symbol $\log(1+|\xi|^2)$, which is a singular integral operator given by $$ (I-\Delta)^{\log}…

In this paper, we investigate large values of Dirichlet character sums with multiplicative coefficients $\sum_{n\le N}f(n)\chi(n)$. We prove a new Omega result in the region $\exp((\log q)^{\frac12+\delta})\le N\le\sqrt q$, where $q$ is the…

In this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers $(a,b,c)$ with $a+b+c=n-1$, $n\ge 4$. These identities are deduced while translating into homotopical terms the gauge action on the…

For a function $f\colon \mathbb{N}\to\mathbb{N}$, let $$ N^+_f(x)=\{n\leq x: n=k+f(k) \mbox{ for some } k\}. $$ Let $\tau(n)=\sum_{d|n}1$ be the divisor function, $\omega(n)=\sum_{p|n}1$ be the prime divisor function, and…

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 obtain some Bailey pairs associated with indefinite quadratic forms with the $\beta_n$ connected to a finite sum. A new general identity is given, which provides identities for $q$-hypergeometric series, including mock theta functions.

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

Let \sigma(n) = \sum_{d \mid n}d be the usual sum-of-divisors function. In 1933, Davenport showed that that n/\sigma(n) possesses a continuous distribution function. In other words, the limit D(u):= \lim_{x\to\infty} \frac{1}{x}\sum_{n \leq…