# Tagged Questions

690 views

### Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ...
973 views

### Provable zero-free region for any entire function that analytically is similar to zeta(s)

Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$: $f(z)$ is bounded when $\Re z>1+\delta$ $f(z)$ is unbounded when $\Re z=1$ $f(z)$ grows ...
2k views

### How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
750 views

### Request for the proof of a result from Ramanujan's letter to Hardy.

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting : If ...
2k 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 ...
937 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 ...
382 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?
893 views

### 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 ...
2k views

### What does log convexity mean?

The Bohrâ€“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. ...
546 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 $S_N= \sum_{n=1}^N \epsilon_n$. ...
689 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 ...
658 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 ...
451 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?
290 views

### at which rational points does the Hypergeometric function take rational values

A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{6}{5};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric ...
2k 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 ...
2k 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 ...