RIEMANN HYPOTHESIS, MÖBIUS’ FUNCTION AND RANDOM
MULTIPLICATIVE FUNCTIONS
(Joint work with Y.-K. Lau & G. Tenenbaum)
Abstract. 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 squarefree integers, we prove that, for any ε > 0, the estimate
X
√
f (n) x (log log x)3/2+ε
n6x
holds almost surely—thus qualitatively matching the law of iterated logarithm,
valid for independent variables. This inequality improves significantly corresponding result of Wintner, Erdős and Halász.
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