Result on the Mobius Function over Shifted Primes

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