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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.

362 questions
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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 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 $(a-1/2, 1/2],... 1answer 160 views ### Subspaces of$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 ... 1answer 1k views ### 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)$$\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm dx=... 0answers 137 views ### Estimating integral of product of terms \cos(t\log p) I would like to prove the following proposition from A. Harper's paper "Sharp conditional upper bound for moments of the Riemann Zeta Function" Proposition. Let T be large and let n=p_1^{\... 0answers 142 views ### An integral involving 1/\zeta(s) and the zeros of \zeta(s) In my thesis, i stumbled across the following problem: Define$$F(\sigma, T)=\frac{1}{T}\int_{-T}^{T} \frac{1}{|\zeta(\sigma+it)|^2} \mathrm{d}T$$where \zeta denotes the Riemann zeta function. Is ... 0answers 104 views ### Questions on Riemann's explicit formula If we consider this version of the prime-counting function$$\pi_0(x) = \frac{1}{2} \lim_{h\to 0} (\pi(x+h) + \pi(x-h))$$(with \pi being the normal prime-counting function), then we can write \... 2answers 306 views ### Dynamics of Riemann zeta function Has the dynamics of the Riemann zeta function been studied? By dynamics I mean the limiting behavior of the sequence of iterates s, \zeta (s), \zeta (\zeta (s)), \zeta (\zeta (\zeta (s)))\dots for ... 0answers 70 views ### Supremum of certain modified zeta functions at 1 Let D be an integer number and let \chi be the Dirichlet character defined by$$\chi(m) = 0 \text{ if$m$even, } \chi(m) = (D/m) \text{ if$m$odd,}$$where (D/m) denotes the Jacobi symbol. ... 3answers 5k views ### Why is so much work done on numerical verification of the Riemann Hypothesis? I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ... 2answers 168 views ### On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function Define \theta(x)=\sum_{p\leq x} \log p , where p>1 denotes a prime. Nicolas proved that if the Riemann zeta function \zeta(s) vanishes for some s with \Re(s)\leq 1/2 + b, where b\in(0, 1/... 1answer 125 views ### Bounding the second moment of |\zeta(\sigma+i t)|^2 for 0<\sigma<1 Let$$I(\sigma,T)=\int_0^T |\zeta(\sigma+ i t)|^2 dt.$$Unconditional bounds and asymptotics for I(\sigma,T), 1/2\leq \sigma <1, have been known since Hardy and Littlewood (see Chapter 7 of ... 0answers 146 views ### Riemann Explicit Formula I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula:$$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)... 0answers 68 views ### 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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