On Pseudosquares and Pseudopowers

TL;DR

This paper investigates the distribution of pseudosquares and pseudopowers, providing new bounds and equidistribution results, some conditional on the Riemann Hypothesis, advancing understanding of their properties and distribution.

Contribution

It establishes bounds on pseudosquares using character sums and proves conditional equidistribution results for pseudopowers, improving previous bounds under the Riemann Hypothesis.

Findings

Pseudosquares are equidistributed in short intervals.

Conditional bounds on pseudopowers are improved under RH.

Provides both unconditional and RH-conditional distribution results.

Abstract

Introduced by Kraitchik and Lehmer, an $x$-pseudosquare is a positive integer $n\equiv1\pmod 8$ that is a quadratic residue for each odd prime $p\le x$, yet is not a square. We use bounds of character sums to prove that pseudosquares are equidistributed in fairly short intervals. An $x$-pseudopower to base $g$ is a positive integer which is not a power of $g$ yet is so modulo $p$ for all primes $p\le x$. It is conjectured by Bach, Lukes, Shallit, and Williams that the least such number is at most $\exp(a_g x/\log x)$ for a suitable constant $a_g$. A bound of $\exp(a_g x\log\log x/\log x)$ is proved conditionally on the Riemann Hypothesis for Dedekind zeta functions, thus improving on a recent conditional exponential bound of Konyagin and the present authors. We also give a GRH-conditional equidistribution result for pseudopowers that is analogous to our unconditional result for pseudosquares.