• 1 Tata Institute for Fundamental Research

This paper gives results on four diverse topics. The first result is that the error term for the number of integers $2^u3^v \le n$ is $O((\log n)^{1-\delta})$ with $\delta=(2^{40}(\log3))^{-1}$, using a theorem of A. Baker and G. Wüstholz. The second result is an averaged explicit formula \[ \psi(x) = x-\frac{1}{T} \int_{T}^{2T} \left( \sum \limits_{|\gamma| \le \tau} \frac{x^{\rho}}{\rho} \right) \ d\tau + O \left( \frac{\log x}{\log \frac{x}{T}}\cdot \frac{x}{T} \right) \] for $x \gg T \gg 1$. It then follows, by the Riemann hypothesis, that $\psi (x+h)-\psi (x)= h+ O \left ( h \lambda^{1/2} \right )$ if $h=\lambda x^{1/2} \log x$. The third theme tightens the $\log$ powers in the zero density bounds of Ingham and Huxley, and gives corollaries for the mean-value of $\psi (x+h)-\psi (x)-h$. The fourth remark concerns a hypothetical improvement in the constant 2 in the Brun-Titchmarsh theorem, averaged over congruence classes, and its consequence for $L \left ( 1,\chi \right )$.