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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
4
votes
1
answer
445
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The sign of an interesting sum involving a Dirichlet character
Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ).
For example for $q=5$ we have
\begin{equation}
\begin{aligned}
\chi_{5,1}&=(1, 1, 1, 1, 0),\\
\ch …
0
votes
Did André Weil predict that the Riemann Hypothesis would be settled by prime number theory r...
I would like to bring your attention to the paper by Andre Weil titled "Two lectures on number theory, past and present". This is based on a talk he gave At Columbia University in 1972.
His term "nu …
1
vote
0
answers
223
views
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 …
6
votes
2
answers
774
views
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 …
2
votes
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:
$$\int_0^1 (1-u)^{r^2-1}f(u) \int_0^1 \frac{\sin(\pi c t u)}{\pi t u} f(u(1-t))u \ dt \ d …
9
votes
2
answers
2k
views
References on Taylor series expansion of Riemann xi function
I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/ …