On Jutila's integral in the circle problem

We study a ‘correlation’ function $\mathcal{K}_{P} = \mathcal{K}_{P}(T;H,U)$ of the error term$P(t)$ in the circle problem, that is, the integral of the product$P(t)P(t+U)$ over the interval $(T,T+H]$, $1 ė  U, H ė  T$. The case of small $U$, $1\le U\ll \sqrt{T}$, was in essence studied by Jutila in 1984.It turns out that, for all these $U$ and sufficiently large $H$,$\mathcal{K}_{P}$ attains its maximum possible value. In this paper we study the caseof ‘large’ $U$, $\sqrt{T}\ll U\le T$, when the behaviour of $\mathcal{K}_{P}$ becomes morecomplicated. In particular, we prove that the correlation function may be positive and negativeof maximally large modulus as well as having very small modulus on sets of values of $U$of positive measure.

circle problem, Jutila's conjecture, Jutila's formula, correlation function, simultaneousapproximations