Fonctions complètement multiplicatives de somme nulle
Fonctions complètement multiplicatives de somme nulle
摘要：
和为零的完全乘法函数（$CMO$），本文讨论$CMO$，即完全乘法（$CM$）函数$f$使得$f(1)=1$和$\sum\limits\_1^\infty f(n)=0$ . $CM$是指所有$(a,b)\in \N^{*2}$的$f(ab)=f(a)f(b)$，因此$f$由$f(p)$，$p$素数定义。假设$f$是$CM$，给出$f(p)$的条件，必要条件或充分条件，两者都是可能的，使$f$是$CMO$：这是作者的一般目的. $f$的$CMO$特征标在序列$(f(p))$的轻微修改下不变（定理3）.同样的想法也适用于更一般的情况下（定理4）。一般的陈述后，排序，包括$CMO$的例子（定理5），该文件致力于“小”的功能，即功能的形式$\frac{f(n)}{n}$，其中$f(n)$有界。下面是一个典型的结果：如果对于所有$p$都有$|f(p)|\le 1$和$Re\, f(p)\le0$，则$\big(\frac{f(n)}{n}\big)$是$CMO$的充分必要条件是$\sum \, Re\, f(p)/p=-\infty$（定理8）。在假设$|1+f(p)|\le 1$和$f(2)\not=-2$成立的条件下，给出了另一个充要条件（定理7）。第三个结果只给出了一个充分条件（定理9）。定理7和8需要辅助结果，这些结果要么来自现有文献（Halász，Montgomery-Vaughan），要么来自在$f*1(n)$上的假设下关于$f(n)$的经典结果（Ingham，Skaubba）的改进版本，* 表示乘法卷积（定理10和11）。
英文摘要：
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 $\sum\limits\_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^{*2}$, 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 $\frac{f(n)}{n}$, where the $f(n)$ are bounded. Here is a typical result : if $|f(p)|\le 1$ and $Re\, f(p)\le0$ for all $p$, a necessary and sufficient condition for $\big(\frac{f(n)}{n}\big)$ to be $CMO$ is $\sum \, Re\, f(p)/p=-\infty$ (theorem 8). Another necessary and sufficient condition is given under the assumption that $|1+f(p)|\le 1$ 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).
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