A note on multiplicative functions resembling the Möbius function

Abstract

We provide examples of multiplicative functions f supported on the square free integers, such that on primes f (p) = ±1 and such that M f (x) := n≤x f (n) = o( √ x). Further, by assuming the Riemann hypothesis (RH) we can go beyond √ xcancellation.

The Erdős discrepancy problem over the squarefree and cubefree integers

“…If 𝑓 has bounded partial sums, then Λ(𝐻) = 𝑂 (1), and as we show below, the structure of 𝑢 implies that this is impossible in the squarefree case.…”

The Erdős discrepancy problem over the squarefree and cubefree integers

“…In [1], the author addressed the question of how small can we make the partial sums of f = µ 2 g, where g : N → {−1, 1} is a completely multiplicative function. It has been proved that if f is strongly χ-pretentious for some real and non-principal Dirichlet character…”

“…Indeed, we can get an example of this by adjusting, in a standard way, any real non-principal Dirichlet character χ to a completely multiplicative function g : N → {−1, 1}, and then restrict g to the cubefree integers. With an extra effort, one could, perhaps, remove the + from the exponent, by following the same line of reasoning of [1]. Nevertheless, in the cubefree case, we have:…”

“…Clearly f 1 and f 2 are χpretentious. In [1], it has been proved that if χ is real and non-principal, and if we assume RH for L(s, χ), then the statement n≤x f 1 (n)…”

“…Theorem 1.3. Let f 1 and f 2 be as above and let Λ(H) be as in (1). Then, in the squarefree case, Λ(H) H 1/2 ; In the cubefree case Λ(H)…”