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3 questions with no upvoted or accepted answers
2
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answers
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Solving a system of differential-like equations for reverse Euler-Maclaurin summation
Aim
A particular instance of a rational zeries that has as of yet not been evaluated is:
\begin{align}
Z:= \sum_{n=1}^{\infty} \frac{\zeta(2n)}{(2n)!}. \label{EM1} \tag{EM1}
\end{align}
This sum ...
- 4,829
1
vote
0
answers
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Prove or disprove that $|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$
$|(1/\zeta)^{(n)}(x)| \leq \frac{n!}{(x-\frac{1}{2})}$ for all real $x>1$.
I had this conjecture for a long time. I tried various methods and techniques but they all failed. It might also be wrong ...
- 178
1
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Integral of $|1/\zeta(\sigma+i T)|$ (or $|(1/\zeta(\sigma+i T))^{(k)}|$) on a horizontal half-line in the left upper quadrant
Let $T_0\geq 20$. Let $L$ be the half-line from $-\infty + i T$ to (say) $-1/2 + i T$. Since $|\zeta(s)|$ is roughly proportional to $(T/2 \pi e)^\sigma$ for $s=\sigma+ i T$ on $L$, it is clear that ...
- 20.2k