Abstract
We characterize some non-negative multiplicative functions $f(n)$ such that $\lim_{x\rightarrow +\infty}\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A }} f(n)$ exists and is positive, but there exists a subset $A(f)$ of $N$ of density $1$ such that $\lim_{x\rightarrow +\infty }\frac{1}{x}\sum_{{1\leq n\leq x }\atop {n\in A(f)}} f(n)=0$. An application to the case of the Ramanujan $\tau$-function is provided.
Citation
Jean-Loup Mauclaire. "On some arithmetical multiplicative functions." Funct. Approx. Comment. Math. 35 219 - 233, January 2006. https://doi.org/10.7169/facm/1229442625
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