A Bound for a Sum of Products of Two Characters and Its Application - Teerapat Srichan (Mathematics, 2023)
Using the exponent pair method, a bound is derived for the sum <semantics><mstyle displaystyle="true"><munder><mstyle mathsize="60%">∑</mstyle>manb≤x</munder></mstyle><msubsup>χ1a</msubsup>(m)<msubsup>χ2b</msubsup>(n)</semantics>, where <semantics>a,<mspace width="4pt"></mspace>b</semantics> are fixed positive integers, <semantics>χ1,<mspace width="4pt"></mspace>χ2</semantics> are primitive Dirichlet characters modulo <semantics>q1</semantics> and <semantics>q2</semantics>, respectively, and <semantics><msubsup>χ1a</msubsup>,<mspace width="0.277778em"></mspace><msubsup>χ2b</msubsup></semantics> are not principal characters. As an application, an estimate for the error term in an asymptotic formula for the number of square-full integers simultaneously belonging to two arithmetic progressions is obtained.