Let $s$ be a complex number with a strictly positive real part ($Re(s)>0$).
Let $f(s)=\sum_{a^2+b^2>0} \dfrac{(-1)^{a^2+b^2}}{(a^2+b^2)^s}$
where the sum runs over all positive integers $a,b$ such that $a^2+b^2>0$ and not allowing multiplicity (as e.g. $x^2+y^2=q^2+s^2$ is counted only once).
(see David Speyers comment and below)
Notice $a^2+b^2$ is a norm.
Also notice this has a strongly related euler form : $f_2(s)=\prod_{p=a^2+b^2=1 mod 4} (1+1/p^s)$ for odd primes $p$.
($p=a^2+b^2=1 mod 4$ for odd primes $p$ follows from Fermat's Christmas theorem )
Is it true that :
1) $f(s)$ is meromorphic for $Re(s)>0$
2) $\dfrac{f'(s)}{f(s)}$ has a simple pole at $s=1$
3) $\dfrac{f'(s)}{f(s)}$ has no other poles than at $s=1$
4) If $Re(s)>0$ and $f(s)=0$ then $Re(s)=\frac{1}{2}$ or at least that is believed by most mathematicians.
This is an analogue to the Riemann Hypothesis.
Is this a part of the Extended Riemann Hypothesis ?? Or did I misunderstood the Extended Riemann Hypothesis ?
Does all this lead to a kind of PNT for primes of the form 1 mod 4 ? An explicit formula for primes of the form $a^2 + b^2$ based on the zero's of $f(s)$ ?
Analogue questions for $G(s)=\sum_{a^3+b^3+c^3>0} \dfrac{(-1)^{a^3+b^3+c^3}}{(a^3+b^3+c^3)^s}$.
Notice $a^3+b^3+c^3$ is also a norm of a ring. (I think a UFD ring )
Update : Sorry $a^3+b^3+c^3$ is not the norm of a ring apparantly. However the question remains and I note that there about - I think - $O(\frac{x}{\ln^\frac{2}{3}(x)})$ integer numbers between $2$ and $x$ of the form $a^3+b^3+c^3$.
I also updated the OP based upon the helpfull comments of David Speyer. (and my reconsideration of "things")
Update 2 : It seems my estimate for the amount of numbers of the form $a^3+b^3+c^3$ is either wrong or unproven, see my link in my comment.