Year: 2022
Type: preprint
Abstract: This article provides new asymptotic results for the summatory Mobius function $\sum_{p \leq x} μ(p+a) =O \left (x(\log x)^{-c} \right )$ and the summatory Liouville function $\sum_{p \leq x} λ(p+a) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne0$ is a fixed parameter, and $c>1$ is an arbitrary constant. These results improve the current estimates $\sum_{p \leq x} μ(p+a)=(1-δ)π(x)$, and $\sum_{p \leq x} λ(p+a)=(1-δ)π(x)$ for $δ>0$, respectively. Furthermore, a conditional proof for the autocorrelation function $\sum_{p \leq x} μ(p+a)μ(p+b) =O \left (x(\log x)^{-c} \right )$, and an unconditional proof for the autocorrelation function $\sum_{p \leq x} λ(p+a)λ(p+b) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne b$, are also included.
Source: arXiv (Cornell University)
Author: Carella, N. A.
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4283687946