A note on Titchmarsh divisor problem under the generalized Riemann hypothesis

Abstract

Let $\mathrm{\Lambda }\left(n\right)$ be the von Mangoldt function and $\tau \left(n\right)$ the divisor function. We focus on the summation

$$T\left(x\right)=\sum _{1<n\le x}\mathrm{\Lambda }\left(n\right)\tau \left(n-1\right).$$

Under the Riemann hypothesis related to Dirichlet $L$-functions, it is proved that

$$T\left(x\right)=x{P}_1\left(logx\right)+O\left(x^{1-𝜃}\right)$$

holds for $𝜃<\frac{1}{6}\times 10^{-4}$. Here $P_1\left(t\right)$ is a polynomial of degree $1$.