Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as
$$
D(n) = \sum\_{k=1}^{n}d(k) ,
$$
where
$$
d(n) = \sum\_{k|n}^{n}1.
$$


One can observe the following pattern in the values of $D(n)$,


$$
\lbrace{D(n)\rbrace}=\lbrace \overbrace{1,3,5,\;}^{3,odd}\overbrace{8,10,14,16,20,}^{5,even}\overbrace{23,27,29,35,37,41,45,}^{7,odd}\cdots \rbrace
$$


where groups of odd elements alternate with groups of even elements and where the $n^{th}$ group has $2n-1$ elements, (we can see that the pattern pesists). Now, based on this (but it is not necessary that this pattern of $D(n)$ is verified for all $n$ we can only assume that such a similar pattern exists), and considering that any number can be written as
$$
\begin{align\*}
n=p\_{1}^{\alpha\_{1}}\cdot p\_{2}^{\alpha\_{2}}\cdot p\_{3}^{\alpha\_{3}} \cdots p\_{n}^{\alpha\_{n}}
\end{align\*}
$$
where $p\_{i}$ are prime numbers, one can define the following arithmetical functions
$$
a(n)=
\begin{cases}
1, & \text{if all } D(\alpha\_1), D(\alpha\_2), \ldots, D(\alpha\_n) \text{ are even} ,
\\\\
\\\\
0, & \text{if one or more of the } D(\alpha\_{i}) \text{ is odd},
\end{cases}
$$
and
$$b(n)\;=\;
\begin{cases}
(-1)^{n\_1+n\_2+\cdots+n\_i}, &\text{if } \alpha\_i = n\_i^2, 
\\\\
\\\\
0, & \text{if } \alpha\_i \text{ is not of the form } n\_i^2,
\end{cases}
$$
($b(n)=$[A197774](https://oeis.org/A197774)) then we can define two Dirichlet series $A(s)=\sum\_{k=1}^{\infty}\frac{a(k)}{n^{s}}$ with $a(1)=1$ and $B(s)=\sum\_{k=1}^{\infty}\frac{b(k)}{n^{s}}$ with $b(1)=1$.


Both of the Dirichlet series have, respectively, the following Euler's products
$$
A(s) = \prod\_{p\in \mathbb{P}}\left(1+\frac{1}{p^{4s}}+\cdots+\frac{1}{p^{8s}}+\frac{1}{p^{16s}}+\cdots+\frac{1}{p^{24s}}+\cdots\right)
$$
and
$$
B(s) = \prod\_{p\in \mathbb{P}}\left(1 - \frac{1}{p^{s}}+\frac{1}{p^{4s}}-\frac{1}{p^{9s}}+\frac{1}{p^{16s}}-\frac{1}{p^{25s}}+\frac{1}{p^{36s}}-\cdots\right)
$$
First we can see that $A(s)$ is absolutely convergent for $s>\frac{1}{4}$. Secondly we can observe that $B(s)$ is related to $\vartheta\_{4}(0,x)=1-x+x^{4}-x^{9}+x^{16}\cdots$ (the Jacobi Theta function) by
$$
\begin{equation\*}
B(s) = \prod\_{p\in \mathbb{P}}\left(\frac{1}{2} \vartheta\_{4}(0,p^{-s})+1 \right)
\end{equation\*}
$$
and thirdly
$$
\begin{equation\*}
\zeta(s) = \frac{A(s)}{B(s)}
\end{equation\*}
$$
**We can think of $b(n)$ as a generalization of $\mu(n)$, the Möbius function and we can assume that if $B(s)$ converges for $\Re{s}>\frac{1}{2}$ then $\zeta(s)$ has no zeros on the right of $\frac{1}{2}$, just like the Mertens function**, where
$$
\begin{equation\*}
\frac{1}{\zeta(s)} = s\int\_{1}^{\infty}\frac{M(x)}{x^{s+1}}dx
\end{equation\*}
$$
similarly for $\zeta(s)$ we have
$$
\begin{equation\*}
\zeta(s) = \frac{A(s)}{ s\int\_{1}^{\infty}\frac{B(x)}{x^{s+1}}dx}
\end{equation\*}
$$
where $M(x)$ is the Mertens function
$$
\begin{equation\*}
M(x)=\sum\_{1\leq n \leq x}\mu(x)
\end{equation\*}
$$
and $B(x)$ is
$$
\begin{equation\*}
B(x)=\sum\_{1 \leq n \leq x}b(x)
\end{equation\*}
$$
My question is: Were these Dirichlet series, $A(s)$ and $B(s)$, studied before and related to the $\zeta(s)$-function the way I did? Or... is this something new?


Thanks.



Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function
If $B(s)$ is any Dirichlet series which factors as an Euler product
$\prod\_{p\in\mathbb{P}}B\_p(s)$ satisfying
$$ B\_p(s)=1-\frac{1}{p^s}+O\left(\frac{1}{p^{2\sigma}}\right),\qquad s=\sigma+it, $$
then $B(s)=A(s)/\zeta(s)$, where $A(s)$ is given by an Euler product which is absolutely and locally uniformly convergent for $\Re(s)>1/2$. As a result, the poles of $B(s)$ are precisely the zeros of $\zeta(s)$ in $\Re(s)>1/2$, hence if the Dirichlet series $B(s)$ converges here, then the Riemann Hypothesis follows. 


In fact it is not hard to relate directly the partial sums of the coefficients $b(n)$ of $B(s)$ to those of $\mu(n)$ and vice versa, since $b$ is a convolution $b=a\ast\mu$ where $a(n)$ is a sparse sequence, namely the coefficients of $A(s)$. In short, the exact shape of $b(n)$ is irrelevant as long as the above conditions on $B\_p(s)$ are met, which in turn can be translated to: (1) the coefficients are multiplicative, (2) $b(p)=-1$ for $p$ prime, (3) $b(p^k)$ is not too large for $k>1$. Your example is one of many.


