# Tagged Questions

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

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### speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
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### When $\frac{\pi ^{x}}{\zeta (x)}$ is rational?

When $n$ is a positive integer, we know $$\zeta (2n)=\frac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$$ Now let's say $x>1$ is a real number. Can we say if $\frac{\pi ^{x}}{\zeta (x)}$ is a rational ...
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### Help to solve $\displaystyle \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} dz$

I need help in evaluating the following contour integral: $$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\zeta(2s)\zeta(s-1)}{\zeta(2s-2)} \frac{x^{s}}{s} ds$$ It looks like a complicated ...
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### Growth rate of $\zeta(n)^{-1}$

What is the asymptotic growth rate of $\frac1{\zeta(n)}$? Is it polynomial in $n$?
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### Is the absolute value of the intersection of two functions related to the nontrivial zeros always equal to $\sqrt{2}$?

With $\displaystyle \chi(s)=\pi^{-\frac{s}{2}}\,\Gamma\left(\frac{s}{2}\right)$ and $K(s)=\Psi\left(\frac{s}{2}\right)-\ln(\pi)$, with $\Psi\left(s\right)$ the digamma function, then the Riemann $\xi$-...
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### Root of the $\zeta(s) = s$

What is the root $s_0$ of the equation $\zeta(s) = s$, where $\zeta(s)$ is Euler zeta function? This point $s_0$ has obvious property: the segment $(1,s_0]$ to the left of it is mapping on the half-...
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### Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$

Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$. What happens to the zeta function at these points? For example $\sum_{n=1}^\infty \frac1{n^s}$ is defined for $\Re(s)>1$ and for $\Re(s)>0$ ...
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### How to prove $\lim_{s \rightarrow \infty} \zeta(s) = 1$?

$\lim_{s \rightarrow \infty} \zeta(s) = 1$ I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome. I ...
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### Did Hardy prove that there are countably, or uncountably many zeros on the line Re$(s)=1/2$ of $\zeta(s)$?

It's known that Hardy proved that there are infinitely many zeros of $\zeta(s)$ on the line Re$(s)=\frac{1}{2}$, but did he prove it's countably infinite? Or uncountable?
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### Convexity of the Riemann-zeta function without derivative

Proving that the $\zeta$ function is convex on $(1,+\infty)$ is pretty simple if we use the derivative, but is there a proof without using derivative? I'm allowed to use just the definition of the ...
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