# Tagged Questions

149 views

208 views

### Are all complex zeros of $\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$ on the critical line for all $z \lt 1$?

Numerical evidence suggests that all complex zeros residing in the critical strip $0 < \Re(s) < 1$ of: $$\frac{\Gamma(s)}{z}Li_s(z) \, \pm \, \frac{\Gamma(1-s)}{z}Li_{1-s}(z)$$ are on the ...
109 views

892 views

### How good is “almost all” when it comes to the Riemann Hypothesis?

Let $N(T)$ be the number of zeroes of the Riemann zeta function $\zeta$ having imaginary part strictly between $0$ and $T$, and let $N_0(T)$ be the number of those zeroes that also have real part ...
570 views

### On extended Riemann Hypothesis and coefficients of Selberg Class L-functions

There is the conjecture that Selberg Class L-functions satisfy RH. So that an L-function needs to have its coefficient multiplicatives (plus other conditions: functional equation,...) in order to ...
81 views

### Closed forms for paired factors of a 4-factor infinite product. Is their shape fixed?

As a follow up on this question, I have now composed the following model of closed forms for infinite products of pairs of factors. At the heart is a 4-factor infinite product shown in red at the ...
311 views

### On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
1k views

566 views

301 views

1k views

### Are the 'semi' trivial zeros of $\zeta(s) \pm \zeta(1-s)$ all on the critical line?

The proof that $\Gamma(z)\pm \Gamma(1-z)$ only has zeros for $z \in \mathbb{R}$ or $z= \frac12 +i \mathbb{R}$ has been given here: Are all zeros of $\Gamma(s) \pm \Gamma(1-s)$ on a line with real ...
552 views

### Is there information about the $\rho$'s hidden in the zeros of $\Re(\chi(s))$ ?

Take the symmetrical form of the completed Zeta-function: $\displaystyle \chi(s) \zeta(s) = \chi(1-s) \zeta(1-s)$ with $\chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2})$. For $s=\sigma + ti$, I ...
It has been proven that: 1) if $s$ is a non trivial zero $\rho$ of $\zeta(s)$ then so is $1−s$. 2) $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$ 3) \$ 0 < \Re(\rho) ...