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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 views

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 …
mike's user avatar
  • 603
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 …
mike's user avatar
  • 603
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 …
mike's user avatar
  • 603
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 …
mike's user avatar
  • 603
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 …
mike's user avatar
  • 603
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/ …
mike's user avatar
  • 603