On primes in special sequences with applications to Carmichael numbers
Wei Zhang
TL;DR
The paper addresses the distribution of primes and the von Mangoldt function in special sequences (Beatty and Piatetski-Shapiro) within arithmetic progressions, with implications for Carmichael numbers. It develops exponential-sum techniques for $\Lambda(n)$ in APs, using Fourier/Vaaler expansions to handle Beatty indicators and Dirichlet approximation for finite-type irrationals $\alpha$, achieving an unconditional improvement over Banks–Yeager bounds. For Piatetski-Shapiro primes, Vaughan-type identities and refined Type I/II sum estimates yield an explicit error bound $O\left(x^{11/14+\gamma/7+\varepsilon}\right)$ for $1<c<12/11$, improving prior work and enabling stronger Carmichael-number results in this context. Overall, the work provides sharper asymptotics for primes in these structured sequences and strengthens the connections to Carmichael-number constructions, via improved exponential-sum bounds and Diophantine-approximation techniques.
Abstract
By involving some exponential sums related to $Λ(n)$ 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.