# Tagged Questions

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

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### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$.

We consider an analogue of the Dirichlet $L$-function in $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, ...
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### Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get "...
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### Integer values of the Riemann Zeta function

When $s>1$ is real, the Riemann zeta function $\zeta(s)$ takes all finite positive value $> 1$. I am studying the values of $s$ for which $\zeta(s)$ is a positive integer. I have the following ...
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### On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
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### One-to-one correspondance between zeta zeros and the prime powers?

I have noticed an interesting property related to the Gibbs phenomenon for the Fourier transform of the zeta zeros in Riemann's explicit formula, namely that the rate at which $r\rightarrow 2$ in the ...
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### Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}.$$ I am looking for an inversion formula for the zeta ...
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### Can this approximate closed form of Apery's constant $\zeta(3)$ be improved?

I know that an approximate closed form is not really a solution. However, I would like to present a method that gives a closed form of $\zeta(3)$ that is accurate to the 5th decimal, hoping that it ...
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### Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right)$$ I know that without the power one could ...
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### An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
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### English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung Ã¼ber die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
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### Asymptotic expansion of $\zeta(s)$

It is well known that $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}, \quad \Re[s] > 1, \tag{1}$$ but, if $p \leq N$ denotes the primes less than or ...
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### Can you prove that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$, assuming LeClaire's approximation?

Can you prove using double series reversion that the $13$ first zeros of $(r\zeta(r+\Im i))^2$ have real part $r=\frac{1}{2}$ (as their convergent), with initial guess for the real part $r$ to be ...
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### Zeta zeros standard normal distribution

Below is a partially scaled plot of $\vartheta (\gamma_n) - \pi (n - 3/2) ,$ where $\gamma_n$ is the imaginary part of the $n$th zeta zero, and $\vartheta$ is the Riemann-Siegel theta function, for ...
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### How can I integrate this zeta function expression?

Can you integrate this function: $$f(k)=\exp\left(-\Re\left(\sum\limits_{n=1}^{n=scale} \frac{1}{n} \zeta(1/2+i \cdot k)\sum\limits_{d|n} \frac{\mu(d)}{d^{(1/2+i \cdot k-1)}}\right)\right)$$ with ...
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### How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: \displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&...
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### Is this formula for $\zeta(15)$ true?

Apery gave, \begin{aligned} \zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned} J. Borwein and D. Bradley found this can be generalized to $\zeta(4n+3)$...
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### Order of growth of real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$

On a lark, I decided to calculate (via Newton's method and using mpmath) the real $x_{n}$ such that $\zeta(x_{n}) = 1 + 1/2^{n}$ for as many $n\in\mathbb{N}$ as I could. What sort of surprised me is ...
If the Riemann hypothesis is false, then there has to be a first counterexample for $\zeta(z)=0$ in the critical strip with $\Re(z) \ne \frac{1}{2}$. For such a counterexample, how large would \$T=|\...