Combinatorical and Algebraic Structures Seminar

Session details

Date: 25.2.2020
Speaker: Mikuláš Zindulka, MFF, Univerzita Karlova
Title: Hardy-Littlewood conjecture for primes having a prescribed primitive root
Abstract: (spoluautoři: Magda Tinková, Ezra Waxman.)
The twin prime conjecture has received a lot of attention in the recent years. Its quantitative form is called the Hardy-Littlewood conjecture. I will attempt to combine it with the famous Artin's primitive root conjecture to address the following problem: Let $g$ be an integer, $|g|>1$. How to estimate the number of prime pairs $p, p+d$ with $p\leq x$, where $g$ is a primitive root both modulo $p$ and modulo $p+d$? Our approach is based on a probabilistic model proposed by Moree. I will present a general method which allows to derive an asymptotic formula for the number of such pairs when $g$ and $d$ are given. The conjectured estimate is of the form $\sim C\frac{x}{(\log x)^2}$, where $C$ is expressed as an explicit infinite product over primes. It is supported by extensive numerical evidence.

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