A note on the two variable Artin's conjecture

Pub Date : 2024-04-23 DOI:10.1016/j.jnt.2024.03.008

S.G. Hazra , M. Ram Murty , J. Sivaraman

引用次数: 0

Abstract

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set ${p ⩽ x : p prime, a \mod p \in 〈 b 〉 \mod p}$ has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs $(a, b)$ $# {p ⩽ x : p prime, a \mod p \in 〈 b 〉 \mod p} ≫ \frac{x}{\log^{2} x} .$ In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers $S = {m_{1}, m_{2}, m_{3}}$ such that $m_{1}, m_{2}, m_{3}, - 3 m_{1} m_{2}, - 3 m_{2} m_{3}, - 3 m_{1} m_{3}, m_{1} m_{2} m_{3}$ are not squares, there exists a pair of elements $a, b \in S$ such that $# {p ⩽ x : p prime, a \mod p \in 〈 b 〉 \mod p} ≫ \frac{x \log \log x}{\log^{2} x} .$ Further, under the assumption of a level of distribution greater than $x^{\frac{2}{3}}$ in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers $S = {m_{1}, m_{2}}$ such that $m_{1}, m_{2}, - 3 m_{1} m_{2}$ are not squares, there exists a pair of elements $a, b \in {m_{1}, m_{2}, - 3 m_{1} m_{2}}$ such that $# {p ⩽ x : p prime, a \mod p \in 〈 b 〉 \mod p} ≫ \frac{x \log \log x}{\log^{2} x} .$

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关于两变量阿尔丁猜想的说明。

1927 年，阿尔丁猜想，对于素数 p 的正密度，任何不是-1 或完全平方的整数 a 都是一个原始根。2000 年，莫雷和斯蒂文哈根提出了所谓的两变量阿尔丁猜想，并证明了对于任何乘法独立的有理数 a 和 b，集合{p⩽x:p 素数，mamodp∈〈b〉modp} 在广义黎曼假设下对于某些戴德金 zeta 函数具有正密度。虽然这类素数的无穷大是已知的，但上述集合大小的唯一无条件下限是拉姆-穆蒂、塞金和斯图尔特在 2019 年提出的，他们证明了对于无穷多的对 (a,b)#{p⩽x:p 素数，mamodp∈〈b〉modp}≫xlog2x。在本文中，我们改进了这一下界。我们特别证明，给定任意三个乘法独立整数 S={m1,m2,m3}，使得m1,m2,m3,-3m1m2,-3m2m3,-3m1m3,m1m2m3 不是正方形，存在一对元素 a,b∈S，使得#{p⩽x：p质，mamodp∈〈b〉modp}≫xloglogxlog2x。此外，根据邦贝里、弗里德兰德和伊瓦尼茨定理（经希斯-布朗修改）中关于分布水平大于 x23 的假设，我们证明了以下条件结果。给定任意两个乘法独立整数 S={m1,m2}，使得m1,m2,-3m1m2 不是正方形，存在一对元素 a,b∈{m1,m2,-3m1m2} 使得#{p⩽x:p 质数，mamodp∈〈b〉modp}≫xloglogxlog2x。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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