Related papers: On a certain divisor function in Number fields

We investigate an arithmetic function representing a generalization of the gcd-sum function, considered by Kurokawa and Ochiai in 2009 in connection with the multivariable global Igusa zeta function for a finite cyclic group. We show that…

We refine a previous work of K. Matsumoto and H. Ishikawa, obtaining an asymptotic formula for the mean square of the product of the Riemann zeta-function and a Dirichlet polynomial in the critical strip (1/4<$\sigma$<1/2), by obtaining an…

Let $d^{(k)}(n)$ be the $k$-free divisor function for integer $k\ge2$. Let $a$ be a nonzero integer. In this paper, we establish an asymptotic formula \begin{equation*} \sum_{p\leq x} d^{(k)}(p-a) =b_k \cdot x+O\left(\frac{x}{\log x}\right)…

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the…

We study sums of the shape $\sum_{n \leqslant x} f \left( \lfloor x/n \rfloor \right)$ where $f$ is either the von Mangoldt function or the Dirichlet-Piltz divisor functions. We improve previous estimates when $f = \Lambda$ and $f = \tau$,…

We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the…

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and…

Let $d_1 = 1 < d_2 < d_3 < \cdots < d_{\tau(n)} = n$ denote the increasing sequence of the divisors of a positive integer $n$. In this paper, for real or complex values of $\alpha$, we define and study some properties of two new divisor…

Let $\sigma_a^{(N)}(n)=\sum_{d^{N}|n}d^a$. An explicit transformation is obtained for the generalized Lambert series $\sum_{n=1}^{\infty}\sigma_{a}^{(N)}(n)e^{-ny}$ for Re$(a)>-1$ using the recently established Vorono\"i summation formula…

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437].…

In [arXiv:2107.01437], the authors studied the mean-square of certain sums of the divisor function $d_k(f)$ over the function field $\mathbb{F}_q[T]$ in the limit as $q \to \infty$ and related these sums to integrals over the ensemble of…

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…

This paper studies a zeta function of two complex variables (w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov…

We prove that for $s=\sigma+it$ with $\sigma\ge0$ and $0<t\le x$, we have \[\zeta(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+\Theta\frac{29}{14} x^{-\sigma},\qquad \frac{29}{14}=2.07142\dots\] where $\Theta$ is a complex number with…

By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems…

We derive Vorono\"{\dotlessi} summation formulas for the Liouville function $\lambda(n)$, the M\"{o}bius function $\mu(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $\lambda(n)$ requires explicit evaluation…

Here, we study both analytically and numerically, an integral $Z(\sigma,r)$ related to the mean value of a generalized moment of Riemann's zeta function. Analytically, we predict finite, but discontinuous values and verify the prediction…

The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make…

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a…

We begin by defining functions $\sigma_{t,k}$, which are generalized divisor functions with restricted domains. For each positive integer $k$, we show that, for $r>1$, the range of $\sigma_{-r,k}$ is a subset of the interval…