All Questions
Tagged with analytic-number-theory fourier-analysis
32 questions with no upvoted or accepted answers
10
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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 ...
- 163
9
votes
0
answers
321
views
Best smoothing for the Prime Number Theorem?
There are plenty of proofs of the Prime Number Theorem with explicit error terms - it actually looks like a rather competitive field (see Remark 1.4 in https://arxiv.org/pdf/2204.02588.pdf). Several ...
- 20.2k
8
votes
0
answers
139
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Fourier transform of $I_Y$, $Y=\{\text{numbers with many prime factors}\}$
Let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \log \log N$ prime factors. We may consider, say, $D = (\log \log N)^{1-\epsilon}$.
We do have rather precise approximations for the ...
- 20.2k
8
votes
0
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398
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$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual.
Assume that $$\int_0^1\prod_{j=1}...
- 148k
8
votes
0
answers
396
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Voronoi summation and functional equation, from the point of view of distributions
Consider the Voronoi summation formula for the sum of squares function $r_2$, in terms of Bessel function $J_0$:
$$\sum_{n=0}^\infty r_2(n) \int_0^\infty \pi J_0(2\pi\sqrt{nx}) f(x) \, dx = \sum_{n=...
- 363
6
votes
0
answers
214
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Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
- 1,235
6
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0
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486
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On Fourier coefficients of nonnegative function
Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real
Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda ...
- 12.3k
5
votes
0
answers
246
views
Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
- 20.2k
5
votes
0
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207
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Majorizing $|\{\alpha\}-1/2|$ by trigonometric polynomials
Let $f(\alpha) = |\{\alpha\}-1/2|$. What is the trigonometric polynomial $F_N$ of degree $N$ (i.e., a linear combination $\sum_{n=-N}^N a_n e(\alpha n)$, $a_n\in \mathbb{C}$, where $e(r)= e^{2\pi i r}$...
- 20.2k
5
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0
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326
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Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
- 1,199
4
votes
0
answers
128
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Looking for a generalization of fast Fourier transform form for Gauss sums
I want to compute quickly compute a sum of the form
$$\sum_{k=0}^{N}\sum_{l=0}^{M} e(g^{a^k*b^l})$$
Assume $a^N = b^M = 1$ modulo $q-1$.
Where $e(x) = e^{2\pi ix /q}$. This is very similar to the ...
- 591
4
votes
0
answers
103
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L_infinity norm of signed sums of Fourier characters and discrepancy of Fourier matrices
Consider signed sums $\displaystyle A_f(x) =\sum_{\chi} (-1)^{f(\chi)} \chi(x)$ for some set $S$ of characters of an abelian group $G$, and signing $f$ of the characters. For a fixed set $S$ what is ...
4
votes
0
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119
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Variability of finite Fourier coefficient with length
This is a restricted question related to the one here.
Consider the unnormalized Fourier coefficients of subsets $D_g$ of $\mathbb Z/n \mathbb Z$, denoted by
$$
\hat1_{D_g}(m,n)=\sum_{d \in {D_g}} e\...
- 10.4k
4
votes
0
answers
562
views
Best known bounds on certain exponential sums
What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
- 1,890
3
votes
0
answers
192
views
What smoothing to use for PNT-like results?
Consider a Dirichlet series $\sum_n a_n n^{-s}$ with desirable analytic properties (e.g., analytic extension to $\Re s>0$); one example would be $a_n=\mu(n)$. Say we want to estimate $\sum_{n\leq x}...
- 20.2k
3
votes
0
answers
144
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Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
- 20.2k
3
votes
0
answers
164
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Green-Tao's "Polylogarithmic bound for $r_4(N)$"
On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of $\psi\big(\frac{k}{N}\big)$ for all $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
- 53
2
votes
0
answers
95
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Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
- 20.2k
2
votes
0
answers
187
views
Three optimization problems of uncertainty principle/Paley-Wiener type
Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...
- 20.2k
2
votes
0
answers
79
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For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
- 20.2k
2
votes
0
answers
152
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Non-commutative harmonic analysis on the discrete Heisenberg group
Question: Is there a linear map $\mathcal F$ from the Hilbert space of $\ell^2$ functions on the discrete Heisenberg group to some Hilbert space of functions $ L^2(\bigcup \{\Omega_n\}) $, such that:...
- 9,501
2
votes
0
answers
101
views
Generalised theta series for fixed-rank sublattices
The theta series for a lattice $\Lambda$ is defined by
$$\displaystyle \Theta_\Lambda(q) = \sum_{x \in \Lambda} q^{x \cdot x}.$$
Setting $q=e^{-\pi\tau}$ yields the (maybe more usual) related theta ...
- 313
2
votes
0
answers
167
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Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function
Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: https://math.stackexchange.com/questions/1519724/integrating-a-...
- 761
2
votes
0
answers
239
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Distribution of Fourier coefficients of Maass forms
In the sense of Maass an automorphic function $\phi$ with Laplace-Beltrami eigenvalue $\frac{(d-1)^2}{4}+t^2$ on $d$-dimensional hyperbolic space which can be thought as $\mathbb{R}^{d-1}\times\mathbb{...
- 1,692
1
vote
0
answers
264
views
Fourier transform of fat Cantor set
Let $C_n$ be the set obtained in the $n$-th iteration of the construction of the Smith-Volterra Cantor set, obtained by removing at the $n$-th step $2^{n-1}$ middle intervals of amplitude $1/4^n$. ...
- 213
1
vote
0
answers
127
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an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
1
vote
0
answers
163
views
Is this averaged exponential sum over primes small infinitely often?
Do there exist infinitely many positive integers $N$ such that
$$\sum_{\substack{N/2 \leq q \leq N \\ a/q \notin \mathbb{Z}}} \left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \right|\leq |a|^{o(1)} N^...
- 217
1
vote
0
answers
200
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Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than 1(...
- 11
1
vote
0
answers
285
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Davenport's proof that almost all integers are the sum of 4 cubes
Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
- 1,890
0
votes
1
answer
117
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Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
0
votes
0
answers
258
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Lower bound of exponential sum
This question is a close cousin of the following:
Lower bound on exponential sums
Let $\phi:[0,1]\to \mathbb R$ be a smooth function with $\frac 1 {10}<\phi'< 10, \frac 1 {10}<\phi''< 10$. ...
- 751
0
votes
0
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405
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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 ...
- 1