Result on the Mobius Function over Shifted Primes

TL;DR

This paper establishes new asymptotic bounds for the summatory Mobius and Liouville functions over shifted primes, improving previous estimates and analyzing autocorrelation functions with both conditional and unconditional proofs.

Contribution

It provides novel asymptotic results for the Mobius and Liouville functions over shifted primes, including bounds and autocorrelation analyses with new proofs.

Findings

New asymptotic bounds for summatory Mobius and Liouville functions

Improved estimates over shifted primes compared to previous results

Conditional and unconditional proofs for autocorrelation functions

Abstract

This article provides new asymptotic results for the summatory Mobius function $\sum_{p \leq x} \mu(p+a) =O \left (x(\log x)^{-c} \right )$ and the summatory Liouville function $\sum_{p \leq x} \lambda(p+a) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne0$ is a fixed parameter, and $c>1$ is an arbitrary constant. These results improve the current estimates $\sum_{p \leq x} \mu(p+a)=(1-\delta)\pi(x)$, and $\sum_{p \leq x} \lambda(p+a)=(1-\delta)\pi(x)$ for $\delta>0$, respectively. Furthermore, a conditional proof for the autocorrelation function $\sum_{p \leq x} \mu(p+a)\mu(p+b) =O \left (x(\log x)^{-c} \right )$, and an unconditional proof for the autocorrelation function $\sum_{p \leq x} \lambda(p+a)\lambda(p+b) =O \left (x(\log x)^{-c} \right )$ over the shifted primes, where $a\ne b$, are also included.