On primes in special sequences with applications to Carmichael numbers

TL;DR

This paper investigates primes within special sequences, such as Beatty and Piatetski-Shapiro sequences, using exponential sums, leading to improved results on primes and applications to Carmichael numbers.

Contribution

It introduces new methods involving exponential sums to analyze primes in nonhomogeneous sequences, improving previous results on primes and Carmichael numbers.

Findings

Enhanced bounds for primes in Beatty sequences

Improved results for primes in Piatetski-Shapiro sequences

Applications to properties of Carmichael numbers

Abstract

By involving some exponential sums related to $\Lambda(n)$ in arithmetic progression, we can obtain some new results for von Mangoldt function over {\bf nonhomogeneous} Beatty sequences in arithmetic progressions, which improve some recent results of Banks-Yeager unconditionally. On the other hand, we also considered the primes over Piatetski-Shapiro sequences in arithmetic progressions, which gives a continuous improvement of the results of \cite{BBB}. These results can be used to improve some results related to the Carmichael numbers.