Density theorems and the mean value of arithmetical functions in short intervals

V. A. Bykovskii

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 12, Tome 212 (1994), pp. 56-70

Résumé

Let $\Gamma=SL_2(\mathbb Z)$ and let $Z_\Gamma(s)$ be the Selberg zeta function. Set $$ \pi_\Gamma(P)=\sum_{N(\mathcal P)\le P}1, $$ where $\mathcal P$ is a primitive hyperbolic class of conjugate elements in $\Gamma$ and $N(\mathcal P)$ is the norm of P. It is shown that for $\mathcal P$. It is shown that for $P^{1/2+\theta}=Q$, $0\le\theta\le1/2$ we have $$ \pi_\Gamma(P+Q)-\pi_\Gamma=\int_P^{P+Q}\frac{du}{\log u}+O_\varepsilon(QP^{-\sigma(\theta)+\varepsilon}), $$ where $$ \sigma(\theta)=\frac{\theta^2}2+O(\theta^3),\qquad\theta\to0. $$ Thus, a conjecture of Iwaniec (1984) is proved. Similar asymptotic formulas are obtained for the sums $$ \sum_{P\le P+Q}\frac{h(-d)}{\sqrt d}\quad{\text and}\quad\sum_{P\le P+Q}\frac{r_3(n)}{\sqrt n}, $$ where $h(-d),r_3(n)$ is the class number of the imaginary quadratic field of discriminant $-d0$ and $r_3(n)$ is the number of representations of n by the sum of three squares. Bibliography: 7 titles.