Fonctions compl\`etement multiplicatives de somme nulle
Jean-Pierre Kahane, Eric Saias (LPMA)

TL;DR
This paper characterizes completely multiplicative functions with sum zero, providing necessary and sufficient conditions, especially for bounded functions, and explores their invariance and specific cases like Euler's example.
Contribution
It offers new criteria for identifying completely multiplicative functions with zero sum, including invariance properties and detailed conditions for bounded functions.
Findings
Conditions on prime values for CMO functions are established.
Necessary and sufficient criteria for bounded functions are provided.
Invariance of CMO property under small modifications is demonstrated.
Abstract
Completely multiplicative functions whose sum is zero ().The paper deals with , meaning completely multiplicative () functions such that and . means for all , therefore is well defined by the , prime. Assuming that is , give conditions on the , either necessary or sufficient, both is possible, for being : that is the general purpose of the authors.The character of is invariant under slight modifications of the sequence (theorem 3). The same idea applies also in a more general context (theorem 4).After general statements of that sort, including examples of (theorem 5), the paper is devoted to "small" functions, that is, functions of the form , where the are bounded. Here is a typical result : if $|f(p)|\le…
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Theories · Meromorphic and Entire Functions
