On mean values of random multiplicative functions

Yuk-Kam Lau (1) , Gérald Tenenbaum (2) , Jie Wu (2)

1 HKU - The University of Hong Kong
2 IECN - Institut Élie Cartan de Nancy

Let P denote the set of primes and {f (p)} p∈P be a sequence of independent Bernoulli random variables taking values ±1 with probability 1/2. Extending f by multiplicativity to a random multiplicative function f supported on the set of squarefree integers, we prove that, for any ε > 0, the estimate nx f (n) √ x (log log x) 3/2+ε holds almost surely—thus qualitatively matching the law of iterated logarithm, valid for independent variables. This improves on corresponding results by Wintner, Erd˝ os and Halász .

Domaines: Théorie des nombres [math.NT]

Yuk-Kam Lau, Gérald Tenenbaum, Jie Wu. On mean values of random multiplicative functions. Proceedings of the American Mathematical Society, 2012, ⟨10.1090/S0002-9939-2012-11332-2⟩. ⟨hal-01278413⟩