# Questions tagged [riemann-zeta]

Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

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### Is there a reason why the Maclaurin coefficients of the Riemann zeta function are asymptotically close to -1?

If I look at the numerical values of the Maclaurin series of the Riemann zeta function I see that they approach -1 extremely quickly. In fact, if I take $\zeta(x)=\sum_{n=0}^\infty a_n x^n$ then ...
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### A limit involving the $n$th derivative of the reciprocal of the Riemann Zeta function: $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ [closed]

Does the limit $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ exist? If so, how can we determine its precise value, or is there any way to approximate it with high precision? ($(1/\zeta)^{(n)}(2)$ is ...
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### Proof of zeta functional equation in Edwards

I'm trying to understand the first proof of the zeta functional equation given in Riemann's Zeta Function by H M Edwards. Referring to the excerpt from page 13 below, I'm stuck on how he derives the ...
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### is this solution correct $\frac {\partial}{\partial x} \int_0^∞ \frac{\sin((x+it)\arctan(t))}{((1+t^2)^{(x+it)/2} (e^{2\pi t} -1))} dt =0$?

when I was reading about the Riemann zeta function I found out this integral $\ \frac {\partial}{\partial x} \int_0^∞ ​ \frac{\sin((x+iy)\arctan(t))}{((1+t^2)^{(x+iy)/2} (e^{2\pi t} -1))} dt$ and ...
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### what is the value of d/ds (integral(0,infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt) [closed]

I was reading about the Riemann zeta function and found the integral form of it ζ(s)= 1/(s-1) + 1/2 + 2(integral(0, infinity) sin(s arctan t)/((1-t^2)(s/2) (e^(2pi t) -1)) dt). When I was trying to ...
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### Finding the Taylor Series for $f\left(x\right)=\frac{x}{e^{x}-1}$ [closed]

I want to see how one would figure out that $\frac{x}{e^{x}-1}=\sum_{n=0}^{\infty}B_{n}\frac{x^{n}}{n!}$ where $B_{n}$ represents the nth Bernoulli number with the convention that $B_{1}=-\frac{1}{2}$....
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On page 7 Edwards gives Euler’s factorial function: $\displaystyle n! = \int_0^{\infty} e^{-x} x^n dx$ for $n=1,2,3,\ldots$ On the next page he gives the same in Gauss’ notation $\displaystyle \Pi(s) ... • 447 0 votes 0 answers 37 views ### Show$\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$for even n I am working on an exercise with aim to prove$\frac{1}{\pi^n}\zeta(n) \in \mathbb{Q}$using Fourier series. Here's the outline: We first let$f_1(x) = x-\frac{1}{2}$, then we can express it with ... • 51 1 vote 0 answers 34 views ### maximal continuation of$\Pi_2(x)$Consider the functions for$k\in \Bbb N$$$\Pi_k(x) := \sum_{n \in \Bbb N} e^{\frac{\log^k n}{\log x}}$$$\Pi_1(x)$converges for real$1/e<x<1$.$\Pi_1(x)$is a Riemann zeta function i.e.$\...
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The prime zeta function is defined for $\mathfrak{R}(s)>1$ as $P(s)=\sum_{p} \ p^{-s}$, where $p \in \mathbb{P}$. It is well-know this series converges whenever $\mathfrak{R}(s)>1$. Now, ...