On the distribution of primes in short intervals (Q1309200)

Chebyshev's weighted prime counting function \(\psi(x)\) has a classical explicit formula given in terms of a sum over the complex zeros \(\rho\) of the Riemann zeta function. The error term \(E(x,T)\) associated with the formula, taking only such zeros in \(| \rho| \leq T\), satisfies \(E(x,T) \ll x\log^ 2x/T\). Now write \[ J_ 1(N,H) = \int^{2N}_ N\bigl|\psi(x+H) - \psi(x) - H\bigr|^ 2 dx, \] and the purpose of the paper is to give an analysis of the relationship between the two estimates \[ J_ 1(N,H) = o(NH^ 2)\quad\text{and}\quad \int^{2N}_ N| E(x,T)|^ 2 dx = o\biggl({N^ 3\over T^ 2\log N}\biggr), \] for \(H\geq N^ \theta\) and \(T\leq N^{1-\theta}\log N\) respectively, with \(0 < \theta < 1\). The authors rewrite the explicit formula by the introduction of a weighted factor on the sum involving \(\rho\), whereby the ``local problem'' involving primes can be taken into account. This naturally introduces a much more complicated error term, for which mean square estimates are given in terms of \(J_ 1(N,H)\). We omit the statements of the theorems involved, except to say that the results can then be used to deduce that the two estimates above are equivalent.