0
votes
0answers
25 views

$g(x) = \sum_{m=1}^{\infty}f(mx)$ if and only if $f(x) = \sum_{m=1}^{\infty}\mu(m)g(mx)$

This is Problem 1.1.10 from book Problems in analytic number theory by Ram Murthy. It says, given the condition $$ \sum_{k=1}^{\infty} d_3(k)|f(kx)| < \infty $$ where $d_3(k)$ denotes the number of ...
0
votes
0answers
53 views

Conjecture related to the Erdős discrepancy problem

Conjecture: If $k \in \mathbb{N}$ and $S$ is an infinite set of primes, then the multiplicative $\pm$-sequence generated by $S$ contains $+^k$ as a substring infinitely often. (If $S$ is allowed to ...
2
votes
2answers
65 views

Orthogonality de Möbius

Does anyone know how prove that $$\sum_{n\leqslant x}\mu(n)\xi(n) =o(x)$$ when $\xi(n)$ is a multiplicative functions? I found one commentary that exist a connection of this problem with the Theory of ...
5
votes
2answers
171 views

Sum of $n \sigma(n)$

What is known about the asymptotic behavior of $$ -\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ? $$ It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.