A higher order Levin-Feinleib theorem

DOI10.1007/S12044-022-00721-3arXiv2201.08076WikidataQ122715071 ScholiaQ122715071MaRDI QIDQ6388728

Alisa Sedunova, Ritika Sharma, Olivier Ramaré

Publication date: 20 January 2022

Abstract: When restricted to some non-negative multiplicative function, say f, bounded on primes and that vanishes on non square-free integers, our result provides us with an asymptotic for

s u m_{n l e X} f (n) / n

with error term

O ((l o g X)^{k a p p a - h - 1 + v a r e p s i l o n})

(for any positive

v a r e p s i l o n > 0

) as soon as we have

s u m_{p l e Q} f (p) (l o g p) / p = k a p p a l o g Q + e t a + O (1 / (l o g 2 Q)^{h})

for a non-negative

k a p p a

and some non-negative integer

h

. The method generalizes the 1967-approach of Levin and Fainleib and uses a differential equation.

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37)

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