@Article{LanguascoPerelliZaccagnini2010,
  Author    = "A. Languasco and A. Perelli and A. Zaccagnini",
  Title     = "On the {Montgomery}-{Hooley} theorem in short intervals",
  Journal   = "Mathematika",
  Volume    = 56,
  Issue     = 2,
  Pages     = "231--243",
  Year      = 2010,
  Abstract  =
"We study a short-interval version of a result due to Montgomery and
Hooley.
Write
$$
  S(x,h,Q)
  =
  \sum_{q \le Q}
    \sum_{\substack{a = 1 \\ (a, q) = 1}}^q
      \left\vert \psi(x + h; q, a) - \psi(x; q, a)
                 -
                 \frac h {\varphi(q)}
      \right\vert^2
$$
and $\kappa = 1 + \gamma + \log 2\pi + \sum_p (\log p) / p (p - 1)$.
Denote the expected main term by
$M(x, h, Q) = h Q \log (x Q / h) + (x + h) Q \log(1 + h / x) - \kappa h Q$.
Let $\epsilon$, $A > 0$ be arbitrary, $x^{7/12+\epsilon} \le h \le x$
and $Q \le h$.
There exists a positive constant $c_1$ such that
$$
  S(x, h, Q)
  -
  M(X, h, Q)
  \ll
  h^{1/2} Q^{3/2}
  \exp \left(-c_1\frac{(\log 2h/Q)^{3/5}}{(\log\log 3h/Q)^{1/5}} \right)
  +
  h^2 \log^{-A} x.
$$
Now assume \emph{GRH} and let $\epsilon > 0$,
$x^{1/2+\epsilon} \le h \le x$ and $Q \le h$.
There exists a positive constant $c_2$ such that
$$
  S(x, h, Q)
  -
  M(x, h, Q)
  \ll
  \Bigl( \frac hQ \Bigr)^{1/4+\epsilon} Q^2
  +
  h x^{1/2} \log^{c_2} x.
$$"
}