On Euler's example of a completely multiplicative function with sum 0

2016/03/15 by Jean-Pierre Kahane, Kahane, Jean-Pierre, Eric Saïas +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1603.04832

in French, Comptes Rendus de l'Academie des Sciences. Serie 1, Mathematique, Elsevier, 2016

arxiv created 2016/03/15 · arxiv updated 2016/03/16

Abstract

Euler wrote a formula expressing that l(n)/n is a completely multiplicative function with sum 0 (a CMO function) , where l(n) is the completely multiplicative function equal to -1 on the prime numbers (the Liouville function). We extend this formula to Beurling generalized numbers, using a result of Diamond. As an application we show how to construct a CMO function carried by a set of integers whose counting function is of the form D x^a (1+o(1) ) for any a between 0 and 1.

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