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Results tagged with riemann-zeta-function
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user 20038
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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Inverse of the Riemann zeta function
The question is about the value distribution of $\zeta(s)$; it is considered (without speaking of inverse) in some detail in Chapter XI of Titchmarsh's book.
22
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Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?
For the primes in the denominator, there is an amusing heuristic based on the fact that $n \equiv n^{-1}\pmod p$ holds for all $n\geq 1$ (coprime to $p$) only for $p=2$ and $p=3$. So for these primes …