## On the proximity of additive and multiplicative functions.(English)Zbl 1388.11067

Summary: Given an additive function $$f$$ and a multiplicative function $$g$$, let $$E(f,g;x)=\#\{n\leqslant x: f(n)=g(n)\}$$. We study the size of $$E(f,g;x)$$ for functions $$f$$ such that $$f(n)\neq 0$$ for at least one integer $$n>1$$. In particular, we show that for those additive functions $$f$$ whose values $$f(n)$$ are concentrated around their mean value $$\lambda(n)$$, one can find a multiplicative function $$g$$ such that, given any $$\varepsilon>0$$, then $$E(f,g;x)\gg x/\lambda(x)^{1+\varepsilon}$$. We also show that given any additive function satisfying certain regularity conditions, no multiplicative function can coincide with it on a set of positive density. It follows that if $$\omega(n)$$ stands for the number of distinct prime factors of $$n$$, then, given any $$\varepsilon>0$$, there exists a multiplicative function $$g$$ such that $$E(\omega,g;x)\gg x/(\log\log x)^{1+\varepsilon}$$, while for all multiplicative functions $$g$$, we have $$E(\omega,g;x)=o(x)$$ as $$x\rightarrow \infty$$.

### MSC:

 11N25 Distribution of integers with specified multiplicative constraints 11A25 Arithmetic functions; related numbers; inversion formulas
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### References:

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