Jacob's ladders, heterogeneous quadrature formulae, big asymmetry and related formulae for the Riemann zeta-function

Abstract

In this paper we obtain as our main result new class of formulae expressing correlation integrals of the third-order in $Z$ on disconnected sets $\mathring{G}_1(x),\mathring{G}_2(y)$ by means of an autocorrelative sum of the second order in $Z$. Moreover, the distance of the sets $\mathring{G}_1(x),\mathring{G}_2(y)$ from the set of arguments of autocorrelative sum is extremely big, namely $\sim A\pi(T),\ T\to\infty$, where $\pi(T)$ is the prime-counting function.