We study the analogue of the Bombieri–Vinogradov theorem for
$\operatorname{SL}_{m}(\mathbb{Z})$
Hecke–Maass form
$F(z)$
. In particular, for
$\operatorname{SL}_{2}(\mathbb{Z})$
holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on
$\operatorname{SL}_{2}(\mathbb{Z})$
, and
$\operatorname{SL}_{3}(\mathbb{Z})$
Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to
$1/2,$
which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for
$a\neq 0,$
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$
where
$\unicode[STIX]{x1D70C}(n)$
are Fourier coefficients
$\unicode[STIX]{x1D706}_{f}(n)$
of a holomorphic Hecke eigenform
$f$
for
$\operatorname{SL}_{2}(\mathbb{Z})$
or Fourier coefficients
$A_{F}(n,1)$
of its symmetric-square lift
$F$
. Further, as a consequence, we get an asymptotic formula
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$
where
$E_{1}(a)$
is a constant depending on
$a$
. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function
$\unicode[STIX]{x1D70C}(n)d(n-a)$
.