2015/07/17 by Jean-Pierre Kahane, Kahane, Jean-Pierre, Eric Saias +2
Mathematics · #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1507.04858
in French
arxiv created 2015/07/17 · openalex publication_date 2015/07/17 · arxiv updated 2015/07/20 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
Completely multiplicative functions whose sum is zero (CMO).The paper deals with CMO, meaning completely multiplicative (CM) functions f such that f(1)=1 and ∑_1^∞ f(n)=0. CM means f(ab)=f(a)f(b) for all (a,b)∈ \N*2, therefore f is well defined by the f(p), p prime. Assuming that f is CM, give conditions on the f(p), either necessary or sufficient, both is possible, for f being CMO : that is the general purpose of the authors.The CMO character of f is invariant under slight modifications of the sequence (f(p)) (theorem 3). The same idea applies also in a more general context (theorem 4).After general statements of that sort, including examples of CMO (theorem 5), the paper is devoted to "small" functions, that is, functions of the form (f(n))/(n), where the f(n) are bounded. Here is a typical result : if |f(p)|≤ 1 and Re f(p)≤0 for all p, a necessary and sufficient condition for ((f(n))/(n)) to be CMO is ∑ Re f(p)/p=-∞ (theorem 8). Another necessary and sufficient condition is given under the assumption that |1+f(p)|≤ 1 and f(2)\not=-2 (theorem 7). A third result gives only a sufficient condition (theorem 9). The three results apply to the particular case f(p)=-1, the historical example of Euler.Theorems 7 and 8 need auxiliary results, coming either from the existing literature (Halász, Montgomery--Vaughan), or from improved versions of classical results (Ingham, Skał ba) about f(n) under assumptions on the f*1(n), * denoting the multiplicative convolution (theorems 10 and 11).