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Results tagged with riemann-zeta-function
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user 307675
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.
11
votes
Density of fake zeros of Zeta
To add to GH from MO's answer, Chapter 10 of Iwaniec and Kowalski's "Analytic Number Theory" is another good reference. A few of the classical papers are
Ingham's "On the Estimation of $N(\sigma,T)$, …
- 1,990
7
votes
Accepted
On the nearest integer to $\zeta(1-1/B),B \ge 2$
We can make the error mentioned by Wojowu in his comment explicit by using some results on the Laurent coefficients of the zeta function. There are a few results on this, but I'll just use Theorem 2 o …
- 1,990
9
votes
1
answer
344
views
Large values of $\zeta(1/2+it)$ from sums of short moments
In a now classical paper, Iwaniec proved the following theorem.
Theorem. Let $T \geq 2$, $T^{1/2} < T_0 \leq T$ and $T \leq t_1 < t_2 < \cdots < t_R \leq 2T$, $t_{r+1} - t_r \geq T_0$. Then
$$
\tag{1} …
- 1,990
10
votes
Accepted
Large values of $\zeta(1/2+it)$ from sums of short moments
I wrestled with this question for an annoyingly long time. Now understanding things much better, I'm a bit embarrassed that this took me so long to figure out (such is the process of learning though). …
- 1,990