# Questions tagged [riemann-zeta-function]

The Riemann zeta function is the function of one complex variable $s$ defined by the series $\zeta(s) = \sum_{n \geq 1} \frac{1}{n^s}$ when $\operatorname{Re}(s)>1$. It admits a meromorphic continuation to $\mathbb{C}$ with only a simple pole at $1$. This function satisfies a functional equation relating the values at $s$ and $1-s$. This is the most simple example of an $L$-function and a central object of number theory.

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### On the spacing of the zeros of the Riemann zeta function

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### Is the following recursion formula for $\zeta(2n)$ known?

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### How often does the Mertens function vanish?

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### How large can $|\zeta(\sigma + it)|$ be for $\sigma<1/2$?

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### Is there an analogue of the Balazard-Saias-Yor criterion for the Riemann Hypothesis for finite fields?

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### On a certain representation of the Riemann zeta function

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### An integral involving the argument of the Gamma function and the Riemann Hypothesis

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### Inquiry on the bound for $\int_{0}^{T} \Big|\log|\zeta(1/2 + it)| \Big| \mathrm{d}t$

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### Upper bound of $\zeta$-function on critical strip

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### A computation in Conrey's paper on Riemann zeta function

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### A curious relationship betwen $|\zeta(\sigma+it)|$ and $|\zeta(1-\sigma - it)|$

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### Asymptotics for $\int_{0}^{T} \zeta(\sigma+ it) \mathrm{d}t$

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### On a certain representation for the Riemann zeta function in Montgomery-Vaughan

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### Is there an analogue for the Balazard et al criterion for the Generalised Riemann Hypothesis?

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### Is there a (Riemann) explicit formula for $\sum_{p\le x}\frac{1}{p}$ involving a sum over the non-trivial zeros ρ of the Riemann zeta function?

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### Witten zeta function v.s. Riemann zeta function

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### Enquiry on an equality involving the Riemann zeta function

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### Closed form for an integral involving the Riemann zeta function at the critical line

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### $L_2$ bounds for $\zeta(1/2 + it)$ and a related integral

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### On some inequality involving the Riemann zeta function integral at $\Re(s)=1/2$

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### Does the Riemann Xi function possess the universality property?

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### Remainder term in an integral linked to the Riemann zeta function

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### Hardy-Littlewood vs heuristics on the zeta zeros

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### $\sum_{n=1}^\infty \Lambda(n) e^{-nz}$ and $L(s,\chi)$

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### Strange inequality with $\zeta(5)$ [closed]

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### Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

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### What is the best known upper bound for $( \gamma_{n+1}-\gamma_{n})\max_{\{T\in(\gamma_{n},\gamma_{n+1})\}}(\vert\zeta(1/2+iT)\vert) $?

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### Analogues of the Riemann zeta function that are more computationally tractable?

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### Motivated account of the prime number theorem and related topics

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### Order of magnitude of extremely abundant numbers and RH

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### Partial product of Euler factors

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### Dirichlet eta function and Stirling Permutations

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### The Riemann hypothesis as a problem in analysis

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### A recurrence relation for $\zeta(2n)$ - reference request

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### “Long-standing conjectures in analysis … often turn out to be false”

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### On the Dirichlet series for $1/\zeta(s)$ at $\Re(s)=1/2$

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### An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]

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### On a certain integral representation for Hurwitz zeta functions

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### On the real part of the Riemann zeta function inside the critical strip

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### Confusion about Montgomery's Pair Correlation Conjecture

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### On a certain integral representation for Dirichlet L-functions

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### Is Riemann zeta function injective in some strips $a<\Re(s)<b$, where $0\leq a<b \leq 1$?

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### Why did Euler consider the zeta function?

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### Dirichlet series as rational zeta expressions

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### A mixed of the Dedekind zeta function and the L-function

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### Is the Riemann zeta function surjective?

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### On reasonable asymptotic estimates for some integral involving the logarithm of the Riemann zeta function

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### Computing Mertens' function in time O(sqrt(x)) - in practice

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### Product representations of the Riemann zeta function

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