A note on Carmichael numbers in residue classes

TL;DR

This paper improves lower bounds on the count of Carmichael numbers within specific residue classes, showing they grow faster than previously established for large values of X depending on the modulus.

Contribution

It advances the understanding of Carmichael numbers in residue classes by establishing a new lower bound that surpasses earlier results.

Findings

Number of Carmichael numbers in a residue class exceeds X^{1/(6 log log log X)} for large X.

The result depends on the modulus of the residue class.

Provides improved asymptotic lower bounds for Carmichael numbers.

Abstract

Improving on some recent results of Matom\"aki and of Wright, we show that the number of Carmichael numbers to $X$ in a coprime residue class exceeds $X^{1/(6\log\log\log X)}$ for all sufficiently large $X$ depending on the modulus of the residue class.