On multiplicative functions which are small on average

Details On multiplicative functions which are small on average On multiplicative functions which are small on average

2011-11-11

arxiv.org/abs/1111.2659v1

Mathematics

Number Theory

33 pages

Scientific paper

The prime number theorem in arithmetic progressions is equivalent to the statement that certain multiplicative functions of modulus at most 1 are small on average. Motivated by this fact we study the general problem of when such a function f is small on average. Halasz showed that unless f `pretends to be' n^{it} for some fixed t, then f has mean value 0, and gave a quantitative estimate for the rate of decay of the partial sums of f. However, there are functions f for which Halasz's estimate is very far from the truth. We study those functions f which are significantly smaller on average than the bound provided by Halasz. We show that unless f pretends to be \mu(n)n^{it} for some t, then f(p) is small on average too. Our methods yield a new proof of the Siegel-Walfisz theorem.

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