Decay of Mean-values of Multiplicative Functions

p 1−f(p) p diverges then the limit in (1.1) exists, and equals 0 = Θ(f,∞). Wirsing’s result settled an old conjecture of P. Erdős and Wintner that every multiplicative function f with −1 ≤ f(n) ≤ 1 had a mean-value. The situation for complex valued multiplicative functions is more delicate. For example, the function f(n) = n (0 6= α ∈ R) does not have a mean-value because 1 x ∑ n≤x n iα ∼ x 1+iα . Note that here ∑ p(1 − Re p)/p diverges but x−1 ∑ n≤x n iα does not tend to 0. G. Halász [5, 6] realized that for complex valued multiplicative functions, the analogue of Wirsing’s result requires the divergence of ∑ p(1−Re f(p)p−iα)/p for all real numbers α. If this holds, then he showed that 1 x ∑ n≤x f(n) → 0, and he quantified how rapidly the limit is attained.