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Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as $$\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}$$ where $Re(s)>1$. Let $\omega$ be either the trivial character or the sign character i.e. $x\mapsto sign(x)$. Define a partial $\Psi$-function as $$\Psi(s,\omega;N,a):=\sum_{\substack{0\neq n\in\mathbf{Z}\newline n\equiv a\pmod{n}}}\frac{\omega(n)}{|n|^s}a\pmod{N}}}\frac{\omega(n)}{|n|^s},$$ for $Re(s)>1$. Then it is well known that $\zeta(s;N,a)$ and $\Psi(s,\omega;N,a)$ admit a meromorphic continuation to $\mathbf{C}$ with at most of pole of order $1$ at $s=1$. If $\omega$ is the sign character then $$\Psi(s,\omega;N,a)+\Psi(s,1;N,a)=2\zeta(s;N,a)$$ and $$\zeta(s;N,a)-\zeta(s;N,-a)=\Psi(s,\omega;N,a).$$ Note that when $N>1$, the functions $\Psi$ and $\zeta$ do not have an Euler product. So here are 2 natural questions:

Q1: For a fixed $N$ do we know if there exists a constant $C_N>1$ such that if $Re(s)>C_N$ then $\Psi$ and $\zeta$ do not vanish (if the answer is yes then how to prove it)?

Q2: What do we know in general about the nontrivial zeros of $\Psi$ and $\zeta$?

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# non-trivial zeros of partial zeta functions

Let $N,a\in\mathbf{Z}_{\geq 1}$. Define a partial $\zeta$-function as $$\zeta(s;N,a):=\sum_{\substack{n\geq 1\newline n\equiv a\pmod{N}}} \frac{1}{n^s}$$ where $Re(s)>1$. Let $\omega$ be either the trivial character or the sign character i.e. $x\mapsto sign(x)$. Define a partial $\Psi$-function as $$\Psi(s,\omega;N,a):=\sum_{\substack{0\neq n\in\mathbf{Z}\newline n\equiv a\pmod{n}}}\frac{\omega(n)}{|n|^s},$$ for $Re(s)>1$. Then it is well known that $\zeta(s;N,a)$ and $\Psi(s,\omega;N,a)$ admit a meromorphic continuation to $\mathbf{C}$ with at most of pole of order $1$ at $s=1$. If $\omega$ is the sign character then $$\Psi(s,\omega;N,a)+\Psi(s,1;N,a)=2\zeta(s;N,a)$$ and $$\zeta(s;N,a)-\zeta(s;N,-a)=\Psi(s,\omega;N,a).$$ Note that when $N>1$, the functions $\Psi$ and $\zeta$ do not have an Euler product. So here are 2 natural questions:

Q1: For a fixed $N$ do we know if there exists a constant $C_N>1$ such that if $Re(s)>C_N$ then $\Psi$ and $\zeta$ do not vanish (if the answer is yes then how to prove it)?

Q2: What do we know in general about the nontrivial zeros of $\Psi$ and $\zeta$?