CMI-IMsc Number Theory Seminar
Date: Tuesday, 22 October 2024
Time: 03:30 - 04:30 PM
Venue: Seminar Hall
Trigonometric sums over primes
Dwaipayan Mazumder
Chennai Mathematical Institute.
22-10-24

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Abstract

Weyl (1916) showed that for any polynomial of the form $f(x)=\alpha_kx^k+\alpha_(k-1)X^(k-1)+...+\alpha_1x$ the fractional parts ${f(x)}$ form a uniformly distributed sequence, provided at least one of $\alpha_1,...,\alpha_k$ is irrational. One can also ask small values of $f(n)$ modulo 1 and in particular, whether there are infinitely many n for which $||f(n)-\lambda||0$. In this lecture I will talk about a work of Harman in that direction where he proved that there are infinitely many primes $p$ such that $||f(p)||