Ramanujan's lattice point problem, prime number theory and other remarks.

International audience 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. W\"ustholz. 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 $L \left ( 1,\chi \right )$.