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

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
$$ \sum_{P<d\le P+Q}\frac{h(-d)}{\sqrt d}\quad{\text and}\quad\sum_{P<n\le P+Q}\frac{r_3(n)}{\sqrt n}, $$
where $h(-d),r_3(n)$ is the class number of the imaginary quadratic field of discriminant $-d<0$ and $r_3(n)$ is the number of representations of n by the sum of three squares. Bibliography: 7 titles.