# 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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### Analytic continuation and convergence of a Riemann zeta related function

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### Special zeta value and zeroes

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### Analytic continuation of Euler product $\phi(s)=\prod_p(1+p^{-s})^{-1}$

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### Scaled Riemann zeta function with no zero in the critical strip

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### Truncated Euler products, Dirichlet eta function, and convergence issues

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### On integral relating logarithmic of absolute value of Zeta function:

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### Optimality of the Riemann Hypothesis

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### A generating function for non-trivial zeros of Riemann zeta function

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### $\frac{1}{\pi} \int_{0}^\infty \frac{\log|\left(\frac{1}{2}+it\right)\zeta\left(\frac{1}{2}+it\right)|}{\frac{1}{4}+t^2}dt $

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### Is there a conjecture about the bounds (constant or a function) of $\sum_{n \le x} \mu(n)/\sqrt{n}$

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### On infinite sum containing logarithmic derivative of Zeta function and Möbius function:

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### Is $\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$ correct at all?

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### Applications of Jensen's Formula to entire functions of finite order

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### On modified Euler product

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### Deriving an asymptotic for $\pi(x)$ directly from $\log \zeta(s)$?

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### Functional equation and/or growth estimates for a shifted L function

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### Continuing an analytic continuation of the Dirichlet $\eta$-function?

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### Trying to evaluate an integral relating to $\zeta (3)$

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### Mean values of $\zeta(s)$ for $\Re(s)=1/2$ vs $\Re(s)\ne 1/2$

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### Maximal analytic continuation of $\varphi(s)=\sum_{n\ge1} e^{-n^s}$

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### A *natural* polynomial expansion of the Riemann $\xi(s)$ function

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### How can collections of rational zeta series that are equal to $\sum_{n=2}^{\infty} (\zeta(n) - 1)^{p} $ be obtained?

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### Incredibly accurate recursions for the Riemann Zeta function

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### Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers

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### Lower bound of the modulus $|\eta(s)|$ of the Dirichlet Eta function if $0.6 < \Re(s) < 0.9$

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### A Hadamard product representation for Keiper's $\tau$-function?

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### Some properties of special Dirichlet series, connection to Riemann Hypothesis

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### On some property of the zeros of $\zeta(s)$ in the complex plane

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### About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$

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### Riemann's attempts to prove RH

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### More mysteries about the zeros of the Riemann zeta function

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### Analytic continuation of the Euler product of odd primes to $s=1$?

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### Binomial transform of Dirchlet series (2)

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### Is there a Riemann Hypothesis criterion utilizing sum of squares of divisors?

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### Closed form of the sum $\sum_{r\ge2}\frac{\zeta(r)}{r^2}$

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### Proving that the Riemann zeta function is zero free on Re=1 using the prime number theorem

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### Value of $\zeta(3/2)$?

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### Prove that the real part of this limit converges to $\frac{1}{2}$

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### Who formulated the conjecture that the set of real parts of zeros of the Riemann zeta function is dense in $[0,1]$?

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### Is the set of subsequential limits of the normalized gaps between critical zeroes of zeta stable under $s\mapsto 1/s$?

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### Is there a connection between the sequence of a finite number of Stieltjes constants and the integer partitions number?

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### Could analytically deriving the next non-trivial zero of $\zeta(s)$ be made rigorous up to a fixed accuracy?

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### What is known about sums of the form $\sum_{n=2}^{\infty}[\zeta(n)-1]^{p} $?

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### Can $ x \sum_{k=1}^{\infty} \frac{1}{k} \Big{(}- \gamma - \psi \big{(}1-\frac{x}{k} \big{)} \Big{)} $ be simplified?

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### Algebraic independence of shifts of the Riemann zeta function

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### Are there variations of Ramaswami's formula for the analytic continuation of the Riemann zeta function?

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### Generalization of the The Liouville Lambda function

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### Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula

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### On Soundararajan's explicit formula

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