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2d
comment Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?
Thanks mixedmath. It must indeed be the root finding algorithm that struggles here. I found the same issue in Mathematica that also only at a precision of $<40$ digits could find the $\rho$. I did indeed find the many other zeros as well, but given some earlier questions I raised on MO (e.g. this one mathoverflow.net/questions/89518/…), I believe these can all be proven to be on the critical line (which appears to be slightly more difficult to prove for the $\rho$s... :-) ).
2d
accepted Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?
Jun
29
comment Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?
Yes, all other root finding works fine up to very high accuracies and also for the exponent $1$ I find all the $\rho$s. However, when I plot the zeros for exponent $2$ at $1/2+s*i$ it can be clearly seen that the expected roots "at the $\rho$s" have a "parabolic dip", whilst the other zeros (that I all find at all accuracies) have a sharp downward "spike". You might be right that it is the shape of the curve at the zeros that confuses Newton's method.
Jun
29
comment Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?
Steven, I am using the RootFinding[Analytic] function in Maple and I believe this is based on Newton's method.
Jun
29
asked Does $\zeta(s)^2 \pm \zeta(1-s)^2$ have roots at the $\rho$s?
Jun
16
comment Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?
Chappers, you are probably right, although this is one of the very few Euler products that I know of that converges for $s=1$ ($\beta(1)$ is a well-known formula). Hence it could be an indication that further convergence exists. Numercal evidence does indeed suggest (very slow) convergence continues in the strip, however certainly not $\le \frac12$.
Jun
16
revised Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?
Added elegant formula just for info.
Jun
16
asked Does the Euler product for the Dirichlet $\beta$-function converge for all $\Re(s)>\frac12$?
Jun
9
revised Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?
removed additional question since it doesn't work.
Jun
9
revised Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?
added bit more text under addition and included a follow up question.
Jun
7
revised Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?
Added a further simplification for the equation inducing zeros on the line $\Re(s)=\frac12$
Jun
6
asked Are all complex zeros of ${\frac {\zeta \left( s+1 \right) }{\zeta \left( s-1 \right) }}\pm\, 2\,\pi\frac{2-s}{s\,(s+1)}$ on the critical line?
May
18
comment Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
Anixx, it actually is defined for all indices in $\mathbb{C}$ of the polygamma. See link here: maplesoft.com/support/help/Maple/view.aspx?path=Psi I tested it in a Maple and it works fine. Was surprised to learn a function exists that links $\zeta(s)$ to its derivative. Through the functional equation you can then also link $\zeta'(s)$ and $\zeta'(1-s)$. However $\Psi^{s}(1)$ is defined in terms of $\zeta(s)$, so we won't gain much.
May
14
revised Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
Added formula linking zeta'(s) and zeta(s) (but there is a snag).
May
11
answered Can the Riemann Zeta derivative be expressed in terms of Riemann Zeta?
May
2
asked Does this 'alternating' series with $\Lambda(n)$ converge for all $\Re(s)>0$?
Apr
16
asked Is there anything known about the value where the Euler and Hadamard products for $\zeta(s)$ are equal?
Apr
9
answered How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Apr
9
revised How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Turned update into an answer.
Apr
9
revised How could I get access to more than the first 2 mln non-trivial zeros of $\zeta(s)$?
Added solution to my problem.