Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a pair of zeta functions   Let $m$ be a positive integer, and define
$$\zeta_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\omega(n)}}{n^s}\ \ \ \
\text{and} \ \ \ \ \zeta^*_m(s)=\sum_{n=1}^\infty\frac{(-e^{2\pi
i/m})^{\Omega(n)}}{n^s},$$ for $\Re(s)>1$, where $\omega(n)$ denotes the number
of distinct prime factors of $n$, and $\Omega(n)$ represents the total number
of prime factors of $n$ (counted with multiplicity). In this paper we study
these two zeta functions and related arithmetical functions. We show that
$$\sum^\infty_{n=1\atop n\ \text{is squarefree}}\frac{(-e^{2\pi
i/m})^{\omega(n)}}n=0\quad\text{if}\ \ m>4,$$ which is similar to the known
identity $\sum_{n=1}^\infty\mu(n)/n=0$ equivalent to the Prime Number Theorem.
For $m>4$, we prove that $$\zeta_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi
i/m})^{\omega(n)}}n=0 \ \ \ \ \text{and}\ \ \ \
\zeta^*_m(1):=\sum_{n=1}^\infty\frac{(-e^{2\pi i/m})^{\Omega(n)}}n=0.$$ We also
raise a hypothesis on the parities of $\Omega(n)-n$ which implies the Riemann
Hypothesis.

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The categories of this paper are: cond-mat
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The categories of this paper are: solv-int
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
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The categories of this paper are: stat
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The categories of this paper are: physics