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 ...

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Explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
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The Correlation of the Mobius Function and Dirichlet Characters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$ In other words $$\phi_{\chi}(n)=\sum_{d|...
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Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$ D(n) = \sum_{k=1}^{n}d(k) , $$ where $$ d(n) = \sum_{k|n}^{n}1. $$ One can observe the following pattern in the values of $...
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lower bound for $\Re\zeta(1+it)$

Hi is there any lower bound for $\Re\zeta(1+it)$. I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$. If it is true, is there any reference to prove it. thanks
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Characterizing essential singularities

In the paper Picture of an essential singularity, an analogy is made between the multipolar moments of infinitesimal charge distributions and the lines of constant modulus/argument around an essential ...
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Riemann's $\zeta$ function and the uniform distribution on $[-1,0]$

http://math.stackexchange.com/questions/64566/riemanns-zeta-function-and-the-uniform-distribution-on-1-0 Stackexchange isn't getting really excited about this, so here it is. The $n$th cumulant of ...
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Are the nontrivial zeros of the Riemann zeta simple?

A few years ago, I found on arXiv an article in which the authors (I think they were at least two to write it) claimed to have proven that the non trivial zeros of the Riemann zeta function were all ...
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If the Riemann Hypothesis fails, must it fail infinitely often?

That is must there either be no non-trivial zeros off the critical line or infinitely many? I'm sure that no one believes otherwise, but I've never seen a theorem in the literature addressing this. ...
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Upper bound for real part of Riemann Zeta function zeros

I have been reading about Riemann Zeta function $\zeta(s)$ and have been thinking about it for some time. I did some calculations, and reached a conclusion where $\Re(\rho) \le \log_2(3) - 1$ as $\Im(\...