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# 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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### Solution of $s\zeta'(s)-\zeta(s)=0$ for some infinite strip $\subset\{0<\Re s<1\}$

I did the following calculation, first I take the $k$-th derivative with respect $s$ of the Mellin transform of the fractional part $\{\frac{1}{t}\}$ defined for $0<\Re s<1$ as it is showed in ...
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### Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral

The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
2answers
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### Explicit formula: explicit work with general smoothing?

The following is a literature question, in the sense that I already know how to do what I am asking about, and in fact have already done it; now I'd like to write a brief historical overview as an ...
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### On some analytic property of the Riemann zeta function

Denote by $\zeta$ the Riemann zeta function. For $\Re(s)=\sigma>0$, it is well known that $$\sum_{n\leq x} n^{-s} = \zeta(s) + \frac{x^{1-s}}{1-s}+ O(x^{-\sigma}).$$ But do there exist infinitely ...
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Denote by $\zeta$ the Riemann zeta function. Define $$F_{y}(x)= \int_{-\infty}^{\infty} \frac{x^{iu}\zeta(1/2 + it + iu)}{u^2 + y^2} \mathrm{d}u.$$ For some fixed real number $t$, is there any $y&... 1answer 308 views ### Conditional bound on RH for$\Re\left(\sum_{p\leq\sqrt{x}}\frac{(1/2)}{p^{1+2it}}\right)$I would like to prove that Assume RH. Let$T$large,$2\leq x \leq T^2$and$T\leq t \leq 2T$, then $$\log|\zeta(1/2+it))|\leq \Re\left(\sum_{p\leq x}\frac{1}{p^{1/2+1/\log x+it}}\frac{\log(x/p)}... 0answers 246 views ### Alternative approaches to Zudilin's proof of Apéry's theorem The article Wadim Zudilin, Apéry's theorem. Thirty years after, Int. J. Math. Comput. Sci. 4 (2009), no. 1 pp 9–19. (arXiv:math/0202159, with the title An elementary proof of Apéry's theorem) ... 1answer 131 views ### Complex integral of logarithmic derivative of \zeta I want to prove that for any x\geq 2 we have$$ \begin{split} -\frac{\zeta^{\prime}}{\zeta}(s)&=\sum_{n\leq x}\frac{\Lambda(n)}{n^s}\frac{\log(x/n)}{\log x}+\frac{1}{\log x}\left(\frac{\zeta^{\... 0answers 101 views ### How to prove a bound for derivative of Zeta function For some c > 0.$\xi''(s) = O(\log^3(t))$for$|t| \geq 2$and$\sigma > 1 - \frac{c}{\log(|t|)}$. 1answer 184 views ### Zeros of polynomial approximations of the Riemann$\zeta$function I know next to nothing about analytic number theory, or the theory of the Riemann$\zeta$function in particular, so the following might be too elementary to deserve more than derision; even so it ... 0answers 179 views ### What is easier to find, the next prime number or next zero of zeta function? I mean at a fairly large height. At what height does the difficulty, change sign? Let us give the number of the prime numbers, with 5 decimals accurate. (When we use the zeros of zeta function ... 0answers 121 views ### How to estimate the integral of Riemann zeta function with error term trends to zero as T trends to infinity?$\zeta(s)$denotes the Riemann zeta function. For$T>0$large enough $$\int_{T}^{2T}|\zeta(\frac{1}{3} + it) \ \zeta(\frac{2}{3} + it)|dt = \ ?$$ 1answer 327 views ### Tight error terms for partial sums$\sum_{n\leq x} 1/n^s$(a) Let$s>1$,$x>0$be real. Then it is not hard to see that $$\sum_{n\leq x} \frac{1}{n^s} \leq \zeta(s) - \frac{1}{(s-1) x^{s-1}} + \frac{1}{2 x^s},$$ basically because$x\mapsto 1/x^s$is ... 2answers 225 views ### Robin's inequality and the zeros of the Riemann zeta function Robin showed that if$a\in(1/2, 1]$is the supremum of the real parts of the zeros of the Riemann zeta function$\zeta(s)$, then$f(x)=\Omega_{\pm} (x^{-b})$, where$b$is some number on$(a-1/2, 1/2],...
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### Subspaces of $L^2(0,1)$ dense on every truncation $L^2(c,1)$

It may be better to move this to a separate question. Let me call a linear subspace $V \subset L^2(0,1)$ to be tame if, for every linear subspace $W \subset V$, either $W$ is dense in $L^2(0,1)$, or ...

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