arXiv subjects

Amir Akbary

Publications and source records attributed to Amir Akbary.

21 records · Page 2Linked to original sources

arXiv2013

A variant of the Bombieri-Vinogradov theorem with explicit constants and applications

We give an effective version with explicit constants of a mean value theorem of Vaughan related to the values of ψ(y, χ), the twisted summatory function associated to the von Mangoldt function Λand a Dirichlet character χ. As a consequence of this result we prove an effective variant of the Bombieri-Vinogradov theorem with explicit constants. This effective variant has the potential to provide explicit results in many problems. We give examples of such results in several number theoretical problems related to shifted primes.

math.NT↗

arXiv2013

Artin prime producing polynomials

We define an Artin prime for an integer $g$ to be a prime such that $g$ is a primitive root modulo that prime. Let $g\in \mathbb{Z}\setminus\{-1\}$ and not be a perfect square. A conjecture of Artin states that the set of Artin primes for $g$ has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of finding a fixed integer $g$ and a prime producing polynomial $f(x)$ with the property that a long string of consecutive primes produced by $f(x)$ are Artin primes for $g$. By employing some results of Moree, we propose a general method for finding such polynomials $f(x)$ and integers $g$. We then apply this general procedure for linear, quadratic, and cubic polynomials to generate many examples of polynomials with very large Artin prime production length. More specifically, among many other examples, we exhibit linear, quadratic, and cubic (respectively) polynomials with 6355, 37951, and 10011 (respectively) consecutive Artin primes for certain integers $g$.

math.NT↗

arXiv2013

Limiting distributions of the classical error terms of prime number theory

In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we establish limiting distribution results for the error term in the prime number theorem for an automorphic $L$-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.

math.NT↗