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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.
20
votes
Algebraic independence of shifts of the Riemann zeta function
Hmm, it was more difficult than I expected to leverage universality to establish the claim. But one can proceed by probabilistic reasoning instead, basically exploiting the phase transition in the li …
- 114k
10
votes
The Riemann hypothesis as a problem in analysis
Andrew Booker is of the opinion that it is the nonvanishing of zeta (or an L-function) to the right of the critical strip which is more fundamental than the Euler product. See Slide 10 of his recent …
- 114k
17
votes
Accepted
Better trigonometrical inequalities for $\zeta(s)$?
Assuming the $b_i$ are all distinct (or at least non-zero for $i \neq 0$), this is not possible. (Otherwise there are trivial examples, e.g. $1 + 2 \cos(0 \theta)+ \cos(0 \theta) \geq 0$ or $1 + 4 \c …
- 114k
13
votes
Bound on $L^2$ norm of $1/\zeta(1+i t)$?
This expression is $\asymp T$. I will sketch a proof of the upper bound of $O(T)$; the lower bound is proven similarly.
It is convenient to use the Gonek-Hughes-Keating explicit factorisation
$$ \zet …
- 114k
28
votes
$\psi(x)-x$ on average
In Theorem 1 of
Brent, Richard P.; Platt, David J.; Trudgian, Timothy S., The mean square of the error term in the prime number theorem, ZBL07569752.
it is shown that for sufficiently large $x$ one h …
- 114k
15
votes
Accepted
what would be the consequences on the distribution of primes of $\Lambda=\infty$?
By itself, not very much: the assertion $\Lambda = \infty$ only requires the existence of an arbitrarily sparse sequence of pairs of adjacent zeroes whose normalised spacing goes to infinity arbitrari …
- 114k
16
votes
What is the best known upper bound for $\frac{1}{\zeta'(\rho)}$ assuming the SZC but not the...
We have an exact formula
\begin{align*}
\frac{1}{\zeta'(\rho)} &= \lim_{s \to \rho} \frac{s-\rho}{\zeta(s)} \\
&= \lim_{s \to \rho} \frac{(s-\rho) (s-1) \Gamma(1+s/2) \pi^{-s/2}}{\xi(s)} \\
&= (\rho- …
- 114k
13
votes
What is Ricardo Pérez-Marco's eñe product? Does it explain his statistical results on differ...
My student, Brad Rodgers, has just posted a paper on the arXiv at http://arxiv.org/abs/1203.3275 which proves a partial result towards the repulsion effect (that differences of two imaginary parts of …
- 114k
21
votes
Good uses of Siegel zeros?
With the usual definition of a Siegel zero (involving an unspecified constant $C_\varepsilon$ for each $\varepsilon>0$), it is not easy to talk about a "single" Siegel zero unless one decides to fix e …
- 114k