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Results tagged with riemann-zeta-function
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user 101078
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.
19
votes
Accepted
Are there any papers about this observation of the distribution of the zeros of the zeta fun...
This is called Landau's formula. More precisely, if we extend the von Mangoldt function $\Lambda(n)$ to the function $\Lambda:\mathbb R_+\to \mathbb R$ by $\Lambda(x)=0$ for non-integer $x$, then
$$
\ …
- 7,193
2
votes
On the nearest integer to $\zeta^{(k)}(1-1/B),B \ge 2$
Both conjectures are false. Suppose your conjectured identity holds for all $k\geq 1$ and some $B\geq 2$. It is known that
$$
f(s)=\zeta(s)-\frac{1}{s-1}
$$
is an entire function. Your identity for fi …
- 7,193
4
votes
Accepted
Robin's inequality and the zeros of the Riemann zeta function
It is well-known that the Riemann hypothesis implies
$$
\theta(x)=x+O(\sqrt{x}\ln^2 x).
$$
Therefore, under the Riemann hypothesis we have
$$
\ln\theta(x)=\ln x+O\left(\frac{\ln^2 x}{\sqrt{x}}\right). …
- 7,193
14
votes
Non trivial zeros of Riemann zeta function
Authors of the paper you linked actually define $f(z)$ differently. They have
$$
f(z)=\left(\frac{1}{1-z}-1\right)\zeta\left(\frac{1}{1-z}\right),
$$
so your $f(z)$ is their $f(-z^2)$ and every $\alph …
- 7,193