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Results tagged with riemann-zeta-function
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user 33672
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.
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References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/ …
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Is the difference of these two real-rooted functions real-rooted?
During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: $W_{n}(z …
CommunityBot
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2
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Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier tran...
I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.
So I re-post it below.
Riemann $\Xi(z)$ fu …
CommunityBot
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Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function
I am sorry that this is long post. But it might be of interest to you.
This post is related to zeros of partial sum of Taylor series of $e^x-1$.
Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can …
CommunityBot
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Optimization problem arising from the study of zeta zeros
You may set $v=t u$. When $v$ goes from 0 to $u$, $t$ goes from 0 to 1. So that your double integral becomes:
$$\int_0^1 (1-u)^{r^2-1}f(u) \int_0^1 \frac{\sin(\pi c t u)}{\pi t u} f(u(1-t))u \ dt \ d …
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