NONNEGATIVE MULTIPLICATIVE FUNCTIONS ON SIFTED SETS, AND THE SQUARE ROOTS OF −1 MODULO SHIFTED PRIMES

Abstract

AbstractAn oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set. As an application, we show that the normalized square roots of −1 (mod m) are equidistributed (mod 1) as m runs through the shifted primes q − 1.