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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.
37
votes
Motivated account of the prime number theorem and related topics
To a certain extent, I think that analytic number theory really is magical, and there's a limit to how natural and motivated it can be. Of the accounts I have seen, the one in Donald Newman's book An …
- 82.6k
28
votes
Accepted
Riemann's attempts to prove RH
The short answer is no. If anyone were aware of such a record, it would surely have been Carl Siegel, who undertook a careful study of Riemann’s unpublished notes. However, Siegel wrote:
Approaches …
- 82.6k
26
votes
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationali...
Regarding your second question, Apéry's amazing formula
$$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$
has inspired the search for analogous formulas for other zeta func …
- 82.6k
4
votes
What is easier to find, the next prime number or next zero of zeta function?
Alex Peter's answer basically answers your question, but perhaps in a somewhat confusing way. The short answer is that it's always easier to find the next prime than the next zeta zero.
Lists of small …
- 82.6k