Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A complete Riemann zeta distribution and the Riemann hypothesis   Let $\sigma,t\in{\mathbb{R}}$, $s=\sigma+\mathrm{{i}}t$, $\Gamma (s)$ be the
Gamma function, $\zeta(s)$ be the Riemann zeta function and $\xi(s):=s(s-1)\pi
^{-s/2}\Gamma(s/2)\zeta(s)$ be the complete Riemann zeta function. We show that
$\Xi_{\sigma}(t):=\xi (\sigma-\mathrm{{i}}t)/\xi(\sigma)$ is a characteristic
function for any $\sigma\in{\mathbb{R}}$ by giving the probability density
function. Next we prove that the Riemann hypothesis is true if and only if each
$\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function,
which is defined in this paper, for each $1/2<\sigma<1$. Moreover, we show that
$\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function
when $\sigma=1$. Finally we prove that the characteristic function
$\Xi_{\sigma}(t)$ is not infinitely divisible but quasi-infinitely divisible
for any $\sigma>1$.

37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
18
The categories of this paper are: econ