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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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        Subgroup of the symmetry group of $Zer(\zeta)$ preserving multiplicity

        Let $Zer(\zeta)$ denote the multiset of the non trivial zeros of the Riemann zeta function counted with multiplicity and $G$ the group of isometries of the complex plane preserving this multiset ...
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        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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        On Robin's inequality and the zeros of the Riemann zeta function

        Let $\zeta$ denote the Riemann zeta function. By an argument of Robin http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $\zeta(\rho)=0$ for some $\rho$ with $\Re(\rho) \in (1/2, 1/2 + \beta]$,...
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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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        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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        157 views

        Local phase statistics of the nontrivial Riemann zeros

        (The question is inspired by Owen Maresh's post) The local phase of a nontrivial zero $s$ of the Riemann $\zeta$ is the argument of $\zeta'(s)$. Numerical results on the first 10000 zeros suggest ...
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        Jensen Polynomials for the Riemann Zeta Function

        In the paper by Griffin, Ono, Rolen and Zagier which appeared on the arXiv today, the abstract includes In the case of the Riemann zeta function, this proves the GUE random matrix model ...
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        Do we believe that the distribution of spacings of successive critical zeros of zeta is log-symmetric?

        Let $\gamma^{+}(T)$ be the imaginary part of the critical zero of $\zeta$ closest to $1/2+iT$ with $\gamma^{+}(T)\ge T$ and define similarly $\gamma^{-}(T)$ with a reversed inequality. Let $g(T)$ be ...
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        On an approach on the Hilbert-Polya Conjecture suggested by Schumayer and Hutchison

        In their expository paper, ''Physics of the Riemann Hypothesis arxiv.org/abs/1101.3116v1'', Hutchison and Schumayer suggested the following approach on the Hilbert Polya conjecture, via quantisation ...
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        Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

        $\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
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        Where can I find this result of Ingham?

        Sometime ago, I read somewhere (should be in Titschmarsh) that, if $N(\sigma, T)$ denotes the number of zeros of the Riemann zeta function $\zeta(s)$ with $\Re(s)\geq \sigma>1/2$ up to height $T>...
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        Connection between Fourier analysis and Galois theory

        Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e. $$x \equiv x\ \%\ m\pmod{m}$$ Let $\mu^n_m(x)$ denote multiplication by $n$ modulo $m$, i.e. $$\mu^n_m(x) = nx\ \%\ m$$ Consider the Fourier ...
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        On the spacing of the zeros of the Riemann zeta function

        Suppose the Riemann zeta function has infinitely many zeros $\rho$ with $\Re(\rho)=\sigma$. Does it follow that for every large $T>0$, there exists some $t$ such that $T<t<3T$, where $t=\Im(\...

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