Related papers: On large values of $L(\sigma,\chi)$

Consider the approximation $\tilde{Z}_N(s) = \sum_{n=1}^N n^{-s} + \chi(s) \sum_{n=1}^N n^{1-s}$ of the Riemann zeta function $\zeta(s)$, where $\chi(s)$ is the ratio of the gamma functions. This arise from the approximate functional…

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

Lindel{\"o}f's hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann's zeta function $\zeta(1/2+it)$ is of order $O(t^{\varepsilon})$ for any $\varepsilon>0$ . It is well known…

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the…

In this paper we consider the distribution of large central values of Dirichlet L-functions over cosets of the group of characters modulo q via Soundararajan's resonator method.

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…

We prove an $\Omega$-result for the quadratic Dirichlet $L$-function $|L(1/2, \chi_P)|$ over irreducible polynomials $P$ associated with the hyperelliptic curve of genus $g$ over a fixed finite field $\mathbb{F}_q$ in the large genus limit.…

We establish an asymptotic formula for the first moment and derive an upper bound for the second moment of L-functions associated with the complete family of primitive cubic Dirichlet characters defined over the Eisenstein field. Our…

Let $q\ge3$ be an integer, $\chi$ be a Dirichlet character modulo $q$, and $L(s,\chi)$ denote the Dirichlet $L$-functions corresponding to $\chi$. In this paper, we show some special function series, and give some new identities for the…

Let $0 < a \le 1/2$ and define the quadrilateral zeta function by $2Q(s,a) := \zeta (s,a) + \zeta (s,1-a) + {\rm{Li}}_s (e^{2\pi ia}) + {\rm{Li}}_s(e^{2\pi i(1-a)})$, where $\zeta (s,a)$ is the Hurwitz zeta function and ${\rm{Li}}_s…

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…

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the…

We study lower bounds for the Riemann zeta function $\zeta(s)$ along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the…

In this paper, we prove a lower bound for $\underset{\chi \neq \chi_0}{\max}\bigg|\sum_{n\leq x} \chi(n)\bigg|$, when $x= \frac{q}{(\log q)^B}$. This improves on a result of Granville and Soundararajan for large character sums when the…

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$…

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $\chi_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta…

The Riemann hypothesis is equivalent to the Li criterion governing a sequence of real constants, that are certain logarithmic derivatives of the Riemann xi function evaluated at unity. We investigate a related set of constants c_n, n =…

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial…

We consider the problem of $\Omega$ bounds for the partial sums of a modified character, \textit{i.e.}, a completely multiplicative function $f$ such that $f(p)=\chi(p)$ for all but a finite number of primes $p$, where $\chi$ is a primitive…