Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fonctions compl\`etement multiplicatives de somme nulle   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 (Hal\'asz,
Montgomery--Vaughan), or from improved versions of classical results (Ingham,
Ska{\l} ba) about $f(n)$ under assumptions on the $f*1(n)$, * denoting the
multiplicative convolution (theorems 10 and 11).

35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys