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seen Jul 20 at 22:04

Jul
2
awarded  Curious
Jun
9
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Greg, $G(i)$ (and all $\Re(s)=0$) indeed also diverges when evaluating the series, however Maple does assign the value $-13.23372427475...$ (guess some form of analytic continuation is applied). Not sure how to calculate this number, but is seems to be the average value of an infinite amount of finite odd and finite even series.
Jun
9
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Thanks Greg. I tested convergence with increasingly higher finite series and it also seems to work for the imaginary axis for $\Re(s) \ne 0$, however indeed difficult to prove.
Jun
9
accepted Question about the convergence of an infinite series in all of $\mathbb{C}$.
Jun
8
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
Ethan. Got it and corrected it in the question.
Jun
8
revised Question about the convergence of an infinite series in all of $\mathbb{C}$.
Sharpened the $G(0)=\frac14$ explanation.
Jun
8
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
@ellya. Same here, but started to get used to it somewhat :) You are right though, that I should articulate this much sharper in my question. The series do diverge for $G(0)$ in the usual sense and I was already applying/presuming some analytic continuation. Will correct it.
Jun
8
comment Question about the convergence of an infinite series in all of $\mathbb{C}$.
At $s=0$ the series becomes $\frac12$ * Grandi's series that has a Cesàro sum of $\frac12$, hence $G(0)$ could be assigned a value of $\frac14$.
Jun
8
asked Question about the convergence of an infinite series in all of $\mathbb{C}$.
Apr
13
revised The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?
Fixed error. Forgot the factor s(s-1) on the right hand side in the second equation.
Apr
13
asked The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?
Mar
21
comment Equality between an infinite product and an infinite series. How can I reconcile both?
@Gerry. I see that and believe your statement is indeed true for many cases, however there still is the option that two formulas evaluating to the same number actually do have a connection (with the Euler product as a trivial example).
Mar
21
revised Equality between an infinite product and an infinite series. How can I reconcile both?
Added a more worked out 'Euler product' example to explain the question further.
Mar
12
comment MoebiusMu product
@Fred. You're not talking to yourself but to myself now, right ? :-) Poor joke, I know. However, could you elaborate a bit on how you calculated the numerical convergence? Keen to reproduce it. Note that $5040 = 7!$.
Mar
5
comment Equality between an infinite product and an infinite series. How can I reconcile both?
@Lucian. Thanks. Studied both links, however still can't see a direct connection for the product/series with power 2. Note that the infinite products are 'alternating'. There is however a clear connection with the Wallis product for the power 1 (that I now added to my question).
Mar
5
revised Equality between an infinite product and an infinite series. How can I reconcile both?
Added a similar equation for the power 1 and an additional thought.
Mar
5
asked Equality between an infinite product and an infinite series. How can I reconcile both?
Mar
2
asked Question about the zeros and poles of the PrimeZeta function.
Feb
23
comment Convergence of a modified sum of prime reciprocals for all $s \in \mathbb{C}$?
Many thanks Daniel. Very helpful! Your logic obviously also applies to using integers $n$ instead of primes $p$ i.e. $\displaystyle f(s) := \sum^\infty_{n=2} \left( \frac{1}{n-\frac{1}{n^s}}- \frac{1}{n+\frac{1}{n^s}} \right)$ should converge in the same way. I experimented with $f(s)$ and found closed forms at integer values of $s$: $f(0)=-f(2)=\frac32$, $f(-1)=-f(3)=-\frac34 - i +\gamma + \Psi(i-1)-\frac{i}{2} \pi \coth(\pi)$, $f(-2)=-f(4)=\frac16$. After that the closed forms get complicated. Have not found similar closed forms for the primes.
Feb
23
accepted Convergence of a modified sum of prime reciprocals for all $s \in \mathbb{C}$?