Related papers: A note on Dedekind zeta values at 1/2

We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $\zeta(2n+1)$, and we shall also give a generalization of the transformation formula…

We fix a number field $K$ and study statistical properties of the ring $\mathcal{O}_K[\gamma]\cap K$ as $\gamma$ varies over algebraic numbers of a fixed degree $n\geq 2$. Given $k\geq 1$, we explicitly compute the density of $\gamma$ for…

Assuming the Generalized Riemann Hypothesis, Bach has shown that one can calculate the residue of the Dedekind zeta function of a number field K by a clever use of the splitting of primes p < X, with an error asymptotically bounded by 8.33…

In this paper we address the question of non-vanishing of $L'(0,f)$ where $f$ is an algebraic valued periodic function. In 2011, Gun, Murty and Rath studied the nature of special values of the derivatives of even Dirichlet-type functions…

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of…

The Riemann Zeta function $\zeta(s)$ never vanishes in the region : $$ \Re s \ge 1- \frac1{5.70176 \log |\Im s|} \quad \quad (|\Im s| \ge 2). $$

We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized…

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.…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$. Let $F(s)= \sum_{n=1}^\infty f(n)n^{-s}$ be the associated Dirichlet series and $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ be the truncated Dirichlet…

We study the problem of bounding the least prime that does not split completely in a number field. This is a generalization of the classic problem of bounding the least quadratic non-residue. Here, we present two distinct approaches to this…

Given a number field $L\neq \mathbb{Q}$, we obtain new and explicit zero-free regions for Dedekind zeta-functions of $L$, which refine the previous works of Ahn--Kwon, Kadiri, and Lee. In particular, for low-lying zeros, we extend Kadiri's…

Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $\Psi$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical…

We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a unipotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these…

For any periodic function $f:{\mathbb N} \to {\mathbb C}$ with period $q$, we study the Dirichlet series $L(s,f):=\sum_{n\geq 1} f(n)/n^s.$ It is well-known that this admits an analytic continuation to the entire complex plane except at…

Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $\zeta_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$…

We continue our investigation of the distribution of the fractional parts of $a \gamma$, where $a$ is a fixed non-zero real number and $\gamma$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We…

Recently, the author defined multiple Dedekind zeta values [5] associated to a number K field and a cone C. These objects are number theoretic analogues of multiple zeta values. In this paper we prove that every multiple Dedekind zeta value…

By employing the assessment of the asymptotic size of various sums of G\'{a}l studied by La Bret\`eche and Tenenbaum, we provide an improvement on the recent result of A. Bondarenko, P. Darbar, M. V. Hagen, W. Heap, and K. Seip regarding…

A discussion involving the evaluation of the sum $\sum_{0<\gamma\le T} |\zeta(1/2+i\gamma)|^2$ is presented, where $\gamma$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Three theorems involving certain…

Levinson and Montgomery proved that the Riemann zeta-function $\zeta(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $\zeta'(1/2+it)=0$ implies $\zeta(1/2+it)=0$.…