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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-...
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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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        On an integral involving $\zeta(1/2 + i\tau)$

        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&...
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        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)}...
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        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) ...
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        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^{\...
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        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 $\sigma > 1 - \frac{c}{\log(|t|)}$.
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        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 ...
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        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 ...
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        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 $$\int_{T}^{2T}|\zeta(\frac{1}{3} + it) \ \zeta(\frac{2}{3} + it)|dt = \ ? $$
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        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 ...
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        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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