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).