On some property of the zeros of $\zeta(s)$ in the complex plane

This property is rather elementary, and not at all specific to $$\zeta$$, so I am wondering if it has any value in studying the zeros of the Riemann zeta function in the critical strip. It is a well known result? I can provide a proof sketch if you are interested, and it has been checked numerically.

If $$\zeta(s)=0$$, with $$s=\sigma +it$$ and $$0<\sigma<1$$ then for all real $$\theta$$, we have

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos(\theta+t\log n)}{n^\sigma}=0.$$

This would be true even if by chance, one of the zeroes is outside the critical line $$\sigma=\frac{1}{2}$$. In particular let $$t_0$$ be the imaginary part of a zero of $$\zeta(s)$$. Then in order to find all $$\sigma$$'s such that $$\zeta(\sigma +it_0)=0$$, we only need to focus on the $$\sigma$$'s that satisfy the following equation for every $$\theta$$:

$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\cos(\theta+t_0\log n)}{n^\sigma}=0.$$

Of course the Riemann Hypothesis (RH) implies that $$\sigma$$ must be equal to $$\frac{1}{2}$$, but my assertion is true even if RH is not true. So if you prove that my equality can only happen if $$\sigma=\frac{1}{2}$$, then you would have proved RH. I have several strong reasons to believe that my equality does not lead to a proof of RH, yet I am wondering if it has any value.

• How do you show convergence of this sum?
– user130903
Jan 7, 2021 at 7:07
• For convergence of the sum, there is a result on Dirichlet series that if the series conditionally converges can at some complex $s_0$, then it also converges on the half-plane $\Re(s-s_0)>0$. The alternating zeta function converges at some positive $s$, hence converges inside the critical strip. The function in question is the real part of $exp(i\theta)$ times the alternating zeta, hence also converges. Jan 7, 2021 at 10:16
• Isn't mathoverflow supposed to be for research level questions ? Jan 7, 2021 at 13:49
• why use $\cos \theta, \sin \theta$ beyond the fact that they have some formulas; if you fix any $f,g$ real functions of any variable(s) independent of $t, \sigma$ and any real functions $A,B(\sigma, t)$ the same follows, namely $f(\theta)A(\sigma, t)+g(\theta)B(\sigma, t)=0$ whenever $A(\sigma, t)+iB(\sigma, t)=0$; not sure what this has to do with anything Jan 7, 2021 at 15:34

That looks like just one of the two components of a "rotated" Dirichlet Eta function (sometimes called Alternate Zeta function): $$e^{i\theta} \; \eta (s)= e^{i\theta} \; \sum _{n=1}^{\infty}{\frac {(-1)^{n+1}}{n^{s}}}$$ It cannot hence help, as it is always possible to find a rotation angle $$\theta$$ that will bring to zero one of the two components. True that the non trivial zeros of $$\eta$$ coincide with those of the Riemann Zeta function $$\zeta$$. But the difficulty is of course that zeros of both said functions require by definition that both components be zero.

Sure enough, if $$\eta(s) = 0$$ then it would be so also under any rotation $$\theta$$.

Conversely, if any one of $$\eta(s)$$ two components remains $$0$$ under any rotation $$\theta$$, then $$\eta(s)=0$$.

• Sure, this series converges for real part >1. How do you show convergence for real part >0?
– user130903
Jan 7, 2021 at 9:50
• @Zero If you are referring to my answer, concerning the series for the $\eta$ function most books on the subject would lay out such a proof. See for example Prof. Stoppel's "A Primer of Analytic Number Theory".
– Luca
Jan 7, 2021 at 9:56
• I will add a proof sketch, but obviously the status of convergence of that series is a tough nut to crack. Jan 7, 2021 at 11:09
• @Vincent Do you really need to do that? I mean, it is a well known fact that the series for the $\eta$ function converges also inside the Critical Strip, and so it must also any of its two components. Whatever the rotation applied to the sum of the series.
– Luca
Jan 7, 2021 at 11:16