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 ) 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 χ 1(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 ) = 1g ( n ) −χ 1 ( a )ϕ ( D )*¯ ∑ n ⩽ yg ( n ) χ 1 ( n ) ≪ y ϕ ( D )(log D log y)1 / 4 − ɛuniformly for ( a , D ) = 1, D ⩽ y , x γ ⩽ y ⩽ x , the implied constant depending at most upon ε, γ.