17
votes
1answer
625 views

Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?

Mathematica knows that: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$ The von Mangoldt function should then be: ...
5
votes
1answer
192 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
3
votes
3answers
385 views

Modular form on $\Gamma_0(N)$

I recently asked this question on Math.StackExchange with no answer so far. So I thought maybe I can find an answer here. Let $M(k,\Gamma_0(N))$ be a space of modular forms of weight $k$ on ...
6
votes
0answers
697 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 ...
5
votes
0answers
262 views

converse of Weyl criterion

Let $f∈L^1([0,1))$,suppose for all equi-distributed sequence $\{a_n\}^∞_{n=1}$ in $[0,1)$,we have $$\lim_{N\rightarrow \infty} \frac{1}{N}\sum_{k=1}^Nf(a_k)=\int_0^1f.$$ Do we have that $f$ is Riemann ...
3
votes
2answers
498 views

Extension of Poisson Summation formula

Under the condition f continuous, integrable and: $|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0) we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
0
votes
0answers
108 views

Motivation behind the appearance of Bessel functions in partial trace formulas

Bessel functions occur naturally on the Kloosterman side (or geometric side) of Petersson's formula and Kuznetsov's formula. Is there an intuitive explanation for their appearance? For instance, is ...
14
votes
1answer
417 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
4
votes
1answer
701 views

Riemann hypothesis and Kakeya needle problem

The question may be a bit vague. I noticed an analogy that both Riemann hypothesis and Kakeya needle problem has been proved in finite fields. Can somebody shed light on why finite field analogues are ...
2
votes
0answers
192 views

computing a certain contour integral [closed]

I want to compute an integral along a vertical line segment. The function I'm integrating involves the zeta-function, and usually the way such integrals are done treats the line segment as one side ...
39
votes
5answers
3k views

Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line: $$ \zeta(\frac{1}{2} + i k_j) = 0 $$ Let $f$ be the Fourier transform of the sum ...
12
votes
1answer
587 views

On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO. Let $S$ be a finite set of integers. For $P$ a subset of ...
6
votes
4answers
845 views

fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...
3
votes
1answer
397 views

Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that (a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$, (b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...
2
votes
0answers
328 views

How well can you approximate a function by a band-restricted function?

Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$. Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) ...
27
votes
7answers
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 ...
11
votes
4answers
665 views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
6
votes
2answers
544 views

Number of integers coprime to l

A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for $$ \sum_{n \leq x, (n, \ell) = 1} 1 $$ Of course, this is easy to estimate with a trivial error term of ...
4
votes
1answer
450 views

Max of Fourier transform?

Let f be a real-valued function (or distribution) on $\mathbb{R}$. (You can assume it is nice in one way or another.) What would be some practical ways to bound $\max_{\alpha \in \mathbb{R}} ...
7
votes
2answers
614 views

Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...
19
votes
6answers
3k views

explicit formula for Riemann zeros counting function

I've often seen it stated (in vague terms) that there's a Fourier duality between the set of prime numbers and the set of nontrivial Riemann zeta zeros. Because there are various explicit formulae ...
15
votes
3answers
1k views

What is the shortest route to Roth's theorem?

Roth first proved that any subset of the integers with positive density contains a three term arithmetic progression in 1953. Since then, many other proofs have emerged (I can think of eight off the ...
7
votes
3answers
913 views

A Question concerning the Fourier Transform of $\mathbb{R}$

Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$. Consider the subspace ...
6
votes
0answers
174 views

How big is the Fourier transform of the log of a polynomial over the p-adic numbers

Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...
15
votes
4answers
2k views

FFTs over finite fields?

I'm trying to understand how to compute a fast Fourier transform over a finite field. This question arose in the analysis of some BCH codes. Consider the finite field $F$ with $2^n$ elements. It is ...
6
votes
2answers
330 views

How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question how large ( and small) can be the measure of a set where a polynomial takes small values ? This, and other ...
3
votes
1answer
425 views

A sum involving sines

Consider a scaled sine function, $\sin(2\pi x/2^n)$, for some positive integer $n$. For this, I have the following linear combination. $$ \sum_{x=1}^{2^{n-2}} c_x \sin(2\pi x/2^n).$$ (The upper limit ...
18
votes
2answers
1k views

What's the relationship between Gauss sums and the normal distribution?

Let $p$ be an odd prime and $\left( \frac{a}{p} \right)$ the Legendre symbol. The Gauss sum $\displaystyle g_p(a) = \sum_{k=0}^{p-1} \left( \frac{k}{p} \right) \zeta^{ak},$ where $\zeta_p = e^{ ...
2
votes
6answers
1k views

Computing zeta(k), for k odd, using Fourier coefficients

I'm not really sure what topics exactly this falls under, so I apologize if I've misclassified this question. There is a neat way of computing $\sum_{n=1}^{\infty}\frac{1}{n^{2}}$ using Fourier ...
29
votes
1answer
2k views

Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...
1
vote
1answer
932 views

The large sieve for primes

Let Lambda(n) be the von Mangoldt function, i.e., Lambda(n) = (log p) for n a prime power p^k and Lambda(n) = 0 for all n that not prime powers. Let S(alpha) = \sum_{n<=N} \Lambda(n) e(\alpha n). ...