# Zeroes of $s+\sum\limits_{n=2}^\infty \frac{(-1)^{n+1}}{n^s\ln n}$?

Where are the solutions of the equations

$$s+\sum\limits_{n=2}^\infty \dfrac{1}{n^s\ln n}=0\quad \text{and}\quad s+\sum\limits_{n=2}^\infty \dfrac{(-1)^{n+1}}{n^s\ln n}=0 ?$$

Since the functions on the LHS are the integral of the zeta function and the alternating zeta function, it might make a difference if one assumes RH or not.

I am considering zeroes in the entire complex plane , hence after analytic continuation. Although I am mainly interested in the zeross with $\operatorname{Re} s>-1$.

Is it true that all non real zeroes are in the critical strip $0<\operatorname{Re} s<1$ apart from at most a finite amount ?

I wondered if integral representations are insightful.

It is clear that for $\operatorname{Re} s>\alpha$ there is no solution when the positive real $\alpha$ is sufficiently large.

I assume these integrals of these zeta functions cannot be expressed as an infinite product over primes, is that true ?

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Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta. – Lord_Farin Sep 7 '13 at 11:26
Roots $\approx$ zeroes $\ne$ solutions. – Did Sep 7 '13 at 11:59