In an old diary I read this :
Consider the following functions :
Let $s,t,v$ be complex.
V-zeta function :
$$ \zeta_v(s) = \sum_{n = 1}^{\infty} \frac{1}{n^s + v} $$
T-zeta function :
$$ \zeta_t(s) = \prod_{p = 2}^{\infty} \frac{p^s}{p^s - t} $$
where $p$ denotes taking the product over the primes.
And we are suppose to consider their behaviour, symmetry , analytic continuations , poles and especially their zero's IN PARTICULAR when $v,t$ are real.
(Notice these are NOT the Hurwitz or Barnes zeta functions !)
Now clearly $\zeta_v(s) = \zeta_t(s) = \zeta(s) $ when $v = 0,t = 1$ so this might be intresting !! So it is connected to the Riemann zeta function and the Riemann Hypothesis. I mean if $v,t$ get close to $0,1$ maybe all the nontrivial zero's get very near the critical line ?
As you probably guessed many properties of the riemann zeta function have analogues for the T-zeta function. In fact the T-zeta function seems easier to understand than the V-zeta function. For instance clearly the cases $t = r$ and $ t = -r $ are easily related.
It is clear that the individual terms being summed ( $\frac{1}{n^s + v} $) or multiplied ( $\frac{p^s}{p^s - t}$ ) create poles by themselves and at $s = 0$. But one wonders if the functions have nontrivial poles NOT created by these individual terms nor at integer $s$.
One also wonders if these functions can be extended to a meromorphic function on the entire complex plane. Or maybe we get a natural boundary ? Or a singularity ?
I searched for these functions in the standard literature but found nothing.
However it does remind me of (limits in) perturbation theory.
I have seen perturbation theory used for differential equations, dynamics, calculusn solving polynomials , combinatorics etc but not much for number theory or special functions outside of the above.
But maybe that is just my lack of knowledge.
Have these functions been studied before ?
Is perturbation theory a good idea for these kind of things ?
Plots would be nice to see too.
I assume for $Re(s) > 1,Re(v) > 0,Im(v) = 0$ or $Re(s) > 1,Im(t) = 0$ we get no zero's from the V-zeta or T-zeta function. ( analogue to the prime number theorem proved with the Riemann zeta function ) but maybe I am wrong.
As for perturbation theory I assume these are the 2 most interesting ways to do perturbation on the zeta function , right ??