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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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        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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        Why these surprising proportionalities of integrals involving odd zeta values?

        Inspired by the well known $$\int_0^1\frac{\ln(1-x)\ln x}x\mathrm dx=\zeta(3)$$ and the integral given here (writing $\zeta_r:=\zeta(r)$ for easier reading)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=...
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        Estimating integral of product of terms $\cos(t\log p)$

        I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function" Proposition. Let $T$ be large and let $n=p_1^{\...
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        An integral involving $1/\zeta(s)$ and the zeros of $\zeta(s)$

        In my thesis, i stumbled across the following problem: Define $$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$ where $\zeta$ denotes the Riemann zeta function. Is ...
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        Questions on Riemann's explicit formula

        If we consider this version of the prime-counting function $$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$ (with $\pi$ being the normal prime-counting function), then we can write $\...
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        Dynamics of Riemann zeta function

        Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates $s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots $ for ...
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        Supremum of certain modified zeta functions at 1

        Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
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        Why is so much work done on numerical verification of the Riemann Hypothesis?

        I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ...
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        2answers
        168 views

        On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function

        Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime. Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
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        Bounding the second moment of $|\zeta(\sigma+i t)|^2$ for $0<\sigma<1$

        Let $$I(\sigma,T)=\int_0^T |\zeta(\sigma+ i t)|^2 dt.$$ Unconditional bounds and asymptotics for $I(\sigma,T)$, $1/2\leq \sigma <1$, have been known since Hardy and Littlewood (see Chapter 7 of ...
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        Riemann Explicit Formula

        I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula: $$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...
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        Enquiry on bounds for $\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1.$

        Let $\zeta$ be the Riemann zeta function and $n$ be a positive integer. What are the known (conditional and unconditional) bounds for $f(n)=\frac{1}{(n-1)!}\frac{d^n}{ds^n}((s-1)\zeta(s))$ at $s=1$ ? ...
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        1answer
        106 views

        Cipolla's Prime numbers function: Computing the coefficients of the polynomial

        In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited: [1]: M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti ...

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