On Ramanujan expansions with multiplicative coefficients

Maurizio Laporta ()
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Maurizio Laporta: Università degli Studi di Napoli Dipartimento di Matematica e Applicazioni

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 2, 486-504

Abstract: Abstract The Ramanujan sum $$c_n(a)$$ c n ( a ) is related to the Möbius function $$\mu$$ μ , since $$c_n(a)=\mu (n)$$ c n ( a ) = μ ( n ) whenever $$a,n\in {{\mathbb {N}}}$$ a , n ∈ N are such that $$\gcd (n,a)=1$$ gcd ( n , a ) = 1 . An arithmetic function g admits a Ramanujan expansion in $$A\subseteq {{\mathbb {N}}}$$ A ⊆ N if $$g(a)=\sum {{\widehat{g}}}(n)c_n(a)$$ g ( a ) = ∑ g ^ ( n ) c n ( a ) for any $$a\in A$$ a ∈ A , where $${{\widehat{g}}}$$ g ^ is a suitable arithmetic function. Assuming that $${{\widehat{g}}}$$ g ^ is multiplicative, Coppola has recently proved that the convergence of the subseries $$\sum _{\gcd (n,a)=1} {{\widehat{g}}}(n)\mu (n)$$ ∑ gcd ( n , a ) = 1 g ^ ( n ) μ ( n ) yields that of $$\sum {{\widehat{g}}}(n)c_n(a)$$ ∑ g ^ ( n ) c n ( a ) by providing a factorization of the latter series in terms of the former. Such a factorization allows us to come across some explicit relationships between g and $${{\widehat{g}}}$$ g ^ without the benefit of the absolute convergence of the Ramanujan expansion. Among other things, we establish a recursive formula that relates the values attained at the prime powers by g and $${{\widehat{g}}}$$ g ^ . Finally, we provide with a converse of Coppola’s results and a strengthening of ours in case $${{\widehat{g}}}$$ g ^ is a multiplicative function such that $$|{{\widehat{g}}}(p)|

Keywords: Ramanujan expansion; Multiplicative function; Series identity; Primary 11A25; Secondary 11N37; 11K65 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00068-x

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