Multiplicative Functions on Arithmetic Progressions. VII: Large Moduli

A complex valued function g, defined on the positive integers,is multiplicative if it satisfies g(ab) = g(a)g(b) wheneverthe integers a and b are mutually prime. THEOREM 1. Let D be an integer, 2 D x, > 0. Let g be amultiplicative function with values in the complex unit disc. There is a character _{1(mod D)}, real if g is real, such thatwhen 0 < < 1, uniformly for (a, D) = 1, D y, x y x, the implied constantdepending at most upon , .