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Published 2015 | Submitted
Book Section - Chapter Open

A Note on Helson's Conjecture on Moments of Random Multiplicative Functions


In this note we are interested in cancellations in sums of multiplicative functions. It is well known that M(x) := Σ n≤x µ(n) = 0(x^(1/2+ε) is equivalent to the Riemann Hypothesis. On the other hand, it is also a classical result that M(x) > x^(1/2+ε) for a sequence of arbitrarily large x. It is in fact conjectured that lim x→ ∞ M(x)/ √x(log log log x)^(5/4) = ±B for some constant B > 0 (see [21]).

Additional Information

© 2015 Springer International Publishing Switzerland. We are grateful to the referee for a careful reading of the paper and for asking several questions which led to Theorem 1.4 and stronger results in Theorem 1.3. The first author is supported by a research fellowship at Jesus College, Cambridge.

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