Zobrazeno 1 - 10
of 1 045
pro vyhledávání: '"11N05"'
Autor:
Silberger, Donald
Let $n,d$, and $k$ be positive integers where $n$ and $d$ are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval $[kd,\infty)$ of positive integers into a family of finite sets $X$ for which the sum of the reci
Externí odkaz:
http://arxiv.org/abs/2412.02049
Autor:
Silberger, Donald
We treat the functions $\star^k:{\mathbf N}\rightarrow{\mathbf N}$ where $\star:x\mapsto \star x := x(x+1)$. The set $\{\star^k x: \{x,k\}\subset{\mathbf N}\}$ offers a short and direct proof that the set of primes is infinite. We introduce the ``mot
Externí odkaz:
http://arxiv.org/abs/2412.00253
Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude
Externí odkaz:
http://arxiv.org/abs/2411.19762
Autor:
Zhao, Tianyu
For $i\in \{1,2,3\}$, let $E_i(x)$ denote the error term in each of the three theorems of Mertens on the asymptotic distribution of prime numbers. We show that for each $i\in \{1,2\}$, the Riemann hypothesis is equivalent to the condition $\int_2^X E
Externí odkaz:
http://arxiv.org/abs/2411.18903
We prove that $\mathop{\mathbb{E}}_{m \leq M} \mathop{\mathbb{E}}_{n \leq N} \Lambda(n) \Lambda\bigl(n + \lfloor m^c \rfloor\bigr) = 1 + \rm{O}(\log^{2 - Bc} N)$, where $c > 2$ is a non-integer, $B \geq 3/c$, and $M$ is of order $N^{1/c} \log^{-B} N$
Externí odkaz:
http://arxiv.org/abs/2411.17599
Autor:
Johnston, Daniel R.
We demonstrate the impact of a generic zero-free region and zero-density estimate on the error term in the prime number theorem. Consequently, we are able to improve upon previous work of Pintz and provide an essentially optimal error term for some c
Externí odkaz:
http://arxiv.org/abs/2411.13791
Autor:
Suzuki, Masatoshi
We show that the sign constancy for the values of certain weighted summatory functions of the von Mangoldt function implies the Riemann hypothesis or the generalized Riemann hypothesis for Dirichlet $L$-functions. While such sign constancy is challen
Externí odkaz:
http://arxiv.org/abs/2411.07436
Autor:
Chandran, Arya, K, Vishnu Namboothiri
Cohen-Ramanujan sum, denoted by $c_r^s(n)$, is an exponential sum similar to the Ramanujan sum $c_r(n):=\sum\limits_{\substack{h=1\\{(h,r)=1}}}^{r}e^{\frac{2\pi i n h}{r}}$. An arithmetical function $f$ is said to admit a Cohen-Ramanujan expansion $
Externí odkaz:
http://arxiv.org/abs/2411.11890
Autor:
Quesada-Herrera, Emily
We give an exposition of some connections between Fourier optimization problems and problems in number theory. In particular, we present some recent conditional bounds under the generalized Riemann hypothesis, achieved via a Fourier optimization fram
Externí odkaz:
http://arxiv.org/abs/2411.05095
Autor:
Starichkova, Valeriia
In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is due to Hea
Externí odkaz:
http://arxiv.org/abs/2411.01845