Multiplicative Functions on Arithmetic Progressions. VII: Large Moduli

Abstract

A complex valued function g, defined on the positive integers, is multiplicative if it satisfies g(ab) = g(a)g(b) whenever the integers a and b are mutually prime. THEOREM 1. Let D be an integer, 2 ⩽ D ⩽ x, ε > 0. Let g be a multiplicative function with values in the complex unit disc. There is a character χ1(mod D), real if g is real, such that when 0 < γ < 1, ∑ n ⩽ y n ≡ a ( mod D ) g ( n ) − 1 ϕ ( D ) ∑ n ⩽ y ( n , D ) = 1 g ( n ) − χ 1 ( a ) ϕ ( D ) *¯ ∑ n ⩽ y g ( n ) χ 1 ( n ) ≪ y ϕ ( D ) ( log D log y ) 1 / 4 − ɛ uniformly for (a, D) = 1, D ⩽ y, xγ ⩽ y ⩽ x, the implied constant depending at most upon ε, γ.