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Results tagged with riemann-zeta-function
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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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Density of fake zeros of Zeta
I am investigating whether or not there exist $\epsilon > 0$ such that $\zeta(s) \neq 0$ on the strip $1-\epsilon < \Re(s) \leq 1$.
Suppose not. Then given $\delta > 0$ there exists a zero of zeta $\ …
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answers
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What are the implications of a zero of zeta off the critical line
So what happens if there is a non-trivial zero of the Riemann zeta function off the critical line? Has there been any work in the following direction: We know from Landaus theorem that there is a po …
- 3,699
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What are the obstructions to showing that $\zeta$ doesn't vanish on the strip $1- \varepsilo...
Most (if not all) of the proofs of the Prime Number Theorem that I have seen in the
literature rely on the fact that the Riemann zeta function, $\zeta(s)$, does not vanish
on the line ${\rm Re}(s) = 1 …
- 3,699
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How did Riemann calculate the first few non-trivial zeros of the zeta-function?
Does anyone know how Riemann calculated the first few non-trivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z …
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answer
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Riemann's attempts to prove RH
I read somewhere that Riemann believed he could find a representation of the zeta function that would allow him to show that all the non-trivial zeros of the zeta function lie on the critical line. I …
- 3,699
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answer
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Prime number theorem via the explicit formula
Can the prime number theorem be obtained from the explicit formula,
$\psi(x)=x-\sum_{\zeta(\rho)=0}\frac{x^\rho}{\rho}+O(1)$?
Here, $\psi(x)=\sum_{k=1}^\infty\sum_{p^k<x}\log p$
- 3,699
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Heuristic argument for the Riemann Hypothesis
Is there a heuristic argument that supports the validity of the Riemann hypothesis or are we just relying on numerical evidence? Moreover, what is the strongest theorem that supports the validity of …
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