All Questions
Tagged with analytic-number-theory ca.classical-analysis-and-odes
63 questions
25
votes
3
answers
3k
views
Understanding zeta function regularization
I attended a talk this morning on Ray-Singer torsion, in which Rafael Siejakowski introduced zeta function regularization in a compelling way. The goal is to define the determinant of a positive self-...
- 22.1k
3
votes
2
answers
597
views
lower bound for $\Re\zeta(1+it)$
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it.
thanks
- 163
16
votes
1
answer
2k
views
Why is the functional equation of the Riemann zeta function equivalent to the Poisson summation formula?
We can derive from the Poisson summation formula the modularity of the Theta function, which results in the functional equation. In his book on the Riemann Zeta function, Patterson mentions also that ...
- 11.2k
2
votes
1
answer
1k
views
What is the Stirling formula for x(x+1)(x+2)...(x+n-1)?
Let x be a complex number.
What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?
- 3,804
9
votes
2
answers
1k
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On rational functions with rational power series
Let $f(z)=\sum_{n\geq 0}a_n z^n$ be a Taylor series with rational coefficients with infinitely non-zero $a_n$ which converges
in a small neighboorhood around $0$. Furthermore, assume that
\begin{...
- 7,609
24
votes
4
answers
8k
views
What does log convexity mean?
The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ as the unique function $f(x)$ on the positive reals such that $f(1)=1$, $f(x+1)=xf(x)$, and $f$ is logarithmically convex, i.e. $\...
- 11.1k
14
votes
6
answers
1k
views
Consequence of equidistribution or not?
Let $\theta \not\in \mathbb{Q}$. We know that $(n\theta)_{n \geq 1}$ is equidistributed modulo 1.
Let $\epsilon_n = \mathrm{sign}\bigl(\sin(n\pi \theta)\bigr)$ and $S_N= \sum_{n=1}^N \epsilon_n$.
I'...
- 2,829
4
votes
2
answers
1k
views
Product over the primes
I'm trying to estimate the product
$$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$
where $p,q,r,s$ are primes.
This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea ...
- 9,114
7
votes
2
answers
948
views
Uniform variant of Stirling's approximation
Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where
$E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute ...
- 4,671
1
vote
1
answer
613
views
What is the value of the regularized incomplete beta function at x=0.5?
What is $I_{0.5}(a,b)$ where I is the regularized incomplete beta function?
- 598
7
votes
1
answer
446
views
at which rational points does the Hypergeometric function take rational values
A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...
- 605
22
votes
9
answers
3k
views
When does the zeta function take on integer values?
Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...
- 7,013
18
votes
2
answers
3k
views
Zeta-function regularization of determinants and traces
The short answer to my question may be a pointer to the right text. I will give all the background I know, and then ask my questions in list form.
Let A be an operator (on an infinite-dimensional ...
- 54.6k