A note on multiplicative functions resembling the Möbius function

Multiplicative functions resembling the Möbius funciton

A multiplicative function $f$ is said to be resembling the Möbius function if $f$ is supported on the square-free integers, and $f(p)=\pm 1$ for each prime $p$. We prove $O$- and $Ω$-results for the summatory function $\sum_{n\leq x} f(n)$ for a class of these $f$ studied by Aymone, and the point is that these $O$-results demonstrate cancellations better than the square-root saving. It is proved in particular that the summatory function is $O(x^{1/3+\varepsilon})$ under the Riemann Hypothesis. On the other hand it is proved to be $Ω(x^{1/4})$ unconditionally. It is interesting to compare these…

Multiplicative functions supported on the $k$-free integers with small partial sums

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann Hypothesis, then we can improve the exponent $1/k$: There are examples such that the partial sums up to $x$ are $o(x^{1/(k+\frac{1}{2})+ε})$, for all $ε>0$. This generalizes to the $k$-free integers the results of Aymone, `` A note on multiplicative functions resembling the {M}öbius function'', J. Number Theory, 212 (2020), pp. 113--121.

振荡乘性函数的脆弱平均值

Friable averages of oscillating multiplicative functions

本文研究了狄利克雷级数接近黎曼ζ函数负幂的算术函数的脆弱平均值，获得了类似Selberg-Delange方法的渐近展开，并应用于截断求和。