2
votes
2answers
34 views

Zeta Function $\zeta(1\pm1/n)$ and Euler's constant.

How do I show that $$\lim_{n\to\infty}{\zeta(1+1/n)+\zeta(1-1/n)}=2\gamma$$ and $$\lim_{n\to\infty}{\zeta(1+1/n)-\zeta(1-1/n)}=\infty,$$ where $\gamma$ is the Euler's constant?
11
votes
1answer
548 views

Derivative of Riemann zeta, is this inequality true?

Is the following inequality true? $$\gamma -\frac{\zeta ''(-2\;n)}{2 \zeta '(-2\;n)} > \log (n)-\gamma$$ This for $n$ a positive integer, $n=1,2,3,4,5,...$, and more precisely when $n$ approaches ...
2
votes
1answer
66 views

Pole of Riemann zeta and Riemann zeta zeros, prove this relation.

Prove this relation: $$\displaystyle \lim_{s\to 1} \, \left(\zeta (s)-\frac{\zeta '(s-1+\rho _n)}{\zeta \left(s-1+\rho _n\right)}\right)=\gamma -\frac{\zeta ''(\rho _n)}{2 \zeta '(\rho ...
3
votes
2answers
234 views

Elementary derivation of certian identites related to the Riemannian Zeta function and the Euler-Mascheroni Constant

Is the proof of these identities possible, only using elementary differential and integral calculus? If it is, can anyone direct me to the proofs? ( or give a hint for the solution ) ...
5
votes
1answer
147 views

Define integral for $\gamma,\zeta(i) i\in\mathbb{N}$ and Stirling numbers of the first kind

Consider the integral $$\int\limits_0^{\infty}e^{-x}x^k\ln(x)^n\dfrac{dx}x$$ For $n=3$ we have ...
6
votes
3answers
451 views

Limit of Zeta function

I'm looking for a reference for (or an elementary proof of) $$ \lim_{s \rightarrow 1} \left( \zeta(s) - \frac{1}{s-1} \right) = \gamma$$ Thanks for your help.