On correlations of certain multiplicative functions

R. Balasubramanian, Sumit Giri, Priyamvad Srivastav

Published 2015-11-06Version 1

In this paper, the correlation function of the form $\frac{1}{x}\sum_{n \leq x} F(n) G(n-h)$, were $F=f*1$ and $G=g*1$ with $f$ and $g$ small on an average, has been considered. We obtain an estimation formulae for this mean value depending on different order of magnitude of $f$ and $g$. We compute an average of shifted Euler totient function $\varphi(n-h)$ over square-free integers $n$ and in general expressions of the type $\sum_{n \leq x} \mu^2(n) G(n-h)$, where $G$ is close to $1$ on primes. For certain cases where $f$ and $g$ are small, we make some improvements in the error terms of the corresponding estimations.