Related papers: A note on $\theta_2$

The article presents a generalization of the classical Hardy-Littlewood conjecture concerning the density of prime tuples to the case of tuples consisting of almost-prime numbers (numbers with a specified quantity of prime divisors). The…

Let $\omega^*(n)$ be the number of primes $p$ such that $p-1$ divides $n$. Assuming the Elliott--Halberstam Conjecture, we prove a conjecture posted by M. R. Murty and V. K. Murty in 2021 which states that $$\sum_{n\leqslant…

We apply Selberg's trace formula to solve problems in hyperbolic band theory, a recently developed extension of Bloch theory to model band structures on experimentally realized hyperbolic lattices. For this purpose we incorporate the…

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real…

It is shown that if every odd integer $n > 5$ is the sum of three primes, then every even integer $n > 2$ is the sum of two primes. A conditional proof of Goldbach's conjecture, based on Cram\'er's conjecture, is presented. Theoretical and…

The main result of this paper is that for any $1/2 \leq s < 2 - \sqrt{2} \approx 0.5858$, there is a number $\sigma = \sigma(s) < s$ with the following property. Let $\delta > 0$ be small, assume that $A \subset [0,1]$ is a…

For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary…

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these…

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions in…

Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert…

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. Here we completely determine the theta cycle of a weight $k <…

We obtain asymptotic results on the average numbers of Goldbach representations of an interger as the sum of two primes in different arithmetic progressions. We also prove an omega-result showing that the asymptotic result is essentially…

Let $0<\gamma_1\leq \gamma_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\Theta, \vartheta>0$ such that for…

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $\chi_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $\chi$ be a primitive Dirichlet character modulus $q$ with $\chi\neq\chi_r$. In…

According to the similarity theorem on the distributions of the effective prime factors and by using two-part method, Goldbach theorem and, consequently, Goldbach conjecture was proved.

Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random multiplicative function supported on the squarefree integers such that $(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution…

A modified totient function ($\phi_2$) is seen to play a significant role in the study of the twin prime distribution. The function is defined as $\phi_2(n):=\#\{a\le n ~\vert ~\textrm{$a(a+2)$ is coprime to $n$}\}$ and is shown here to…

Based on the Goldbach conjecture and arithmetic fundamental theorem, the Goldbach conjecture was extended to more general situations, i.e., any positive integer can be written as summation of some specific prime numbers, which depends on…

The convergence properties of cycle expanded periodic orbit expressions for the spectra of classical and semiclassical time evolution operators have been studied for the open three disk billiard. We present evidence that both the classical…

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part…