Generalization of the Ford-Zaharescu Theorem

Abstract

We derive an asymptotic formula for the sum $$H=\sum _{0<{\gamma }_k⩽T,1⩽k⩽m}h(a_1{\gamma }_1+a_2{\gamma }_2+\cdots +a_m{\gamma }_m),$$ where $a_1,a_2,\dots ,a_m$ are integers whose sum equals zero, ${\gamma }_1,\dots ,{\gamma }_m$ independently run through the imaginary parts of the non-trivial zeros of the Riemann zeta function, each zero occuring in the sum the number of times of its multiplicity, and the function $h$ belongs to some special class.

32 pages, 1 figure