Upper bounds for $L^q$ norms of Dirichlet polynomials with small $q$

TL;DR

This paper improves upper bounds for the $L^q$ norms of Dirichlet polynomial partial sums, especially for the 1-norm, providing tighter estimates involving logarithmic factors.

Contribution

It establishes new upper bounds for the $q$th norm of Riemann zeta function partial sums when $0<q extless 1$, refining previous results.

Findings

1-norm bounded by $(\log N)^{1/4}(\log\log N)^{1/4}$

Enhanced understanding of the size of Dirichlet polynomial sums

Improved bounds for the Riemann zeta function on the half line

Abstract

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0<q\leqslant 1$. In particular, we show that the 1-norm is bounded above by $(\log N)^{1/4}(\log\log N)^{1/4}$.