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Results tagged with riemann-zeta-function
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user 151669
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.
2
votes
Riemann–Von Mangoldt formula
In addition to 2734364041's answer, this paper of Tim Trudgian may be useful: in particular, Trudgian shows that for all $T\geq e$,
$$\left|N(T)-\frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)-\frac{7 …
- 465
3
votes
Bounds of zeta function near $\Re(s)=1$
In addition to GH from MO's answer, if one wishes to keep the $(\log|\Im(s)|)^{2/3}$ factor, then there is a very recent improvement due to Bellotti [1]. In particular, Bellotti proved that
$$
\zeta(s …
- 465
5
votes
Accepted
Conjectured error term when counting square-free integers
Your guess is correct! It is indeed conjectured that $a=1/4$. A good recent reference is [1]. In particular, it is known that
$$E(x)=\Omega(x^{1/4})$$
and computations have shown
$$|E(x)|<1.12543x^{1/ …
- 465