All Questions
57 questions
2
votes
0
answers
106
views
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
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
0
votes
1
answer
117
views
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 ...
2
votes
1
answer
202
views
Exponential sums involving smooth truncated divisor functions
Let $p$ be a prime, $a \neq 0$ an integer, let $M,N \gg 1$ and let $\psi,\eta$ be some fixed Schwartz functions. Would you know of any references in the literature where upper bounds for sums such as
$...
- 253
2
votes
2
answers
329
views
$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
- 178
4
votes
0
answers
128
views
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
1
vote
1
answer
244
views
Large sieve type inequality
Let $S_x(t)=\sum_{n\le x} a_n e(nt)$, where $e(x)=e^{2\pi i x}$. Then, the large sieve inequality tells us that
$$
\sum_{q\le Q} \sum_{\substack{0\lt a \lt q \\ (a,q)=1}}|S_x(a/q)|^2 \le (Q^2+4\pi x)\...
- 178
1
vote
0
answers
127
views
an eigenvalue problem for Jacobi Forms
Assume $G(q,z)$ is a Jacobi form of a certain index k. It is known that $G$ can be expanded in a Taylor series with coefficients in the ring of quasi-modular forms (generators $E_2, E_4$ and $E_6$).
$\...
- 43.2k
4
votes
1
answer
248
views
Fourier coefficients of Selberg polynomials
In Montgomery's "Ten Lectures on the Interface Between Number Theory and Harmonic Analysis" a bound for the Fourier coefficients of the Selberg polynomial $S^+_K$ is obtained by using what ...
- 3,625
5
votes
0
answers
246
views
Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
- 20.2k
4
votes
0
answers
103
views
L_infinity norm of signed sums of Fourier characters and discrepancy of Fourier matrices
Consider signed sums $\displaystyle A_f(x) =\sum_{\chi} (-1)^{f(\chi)} \chi(x)$ for some set $S$ of characters of an abelian group $G$, and signing $f$ of the characters. For a fixed set $S$ what is ...
9
votes
2
answers
483
views
Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?
(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...
- 20.2k
6
votes
1
answer
679
views
Fourier optimization problem related to the Prime Number Theorem
Let $\kappa>0$ be given. What is the function
$f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that
$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
- 20.2k
10
votes
1
answer
474
views
A basic estimate of exponential sums
Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:
\begin{equation}
\sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
- 351
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
5
votes
2
answers
484
views
Optimizing a smoothing function with the Prime Number Theorem in mind
Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
- 20.2k
3
votes
2
answers
450
views
Bounds for PNT from Wiener-Ikehara?
What sort of bounds on the error term in the Prime Number Theorem can one obtain through a Wiener-Ikehara approach?
Same question, but for the Mertens function $M(x)=\sum_{n\leq x} \mu(n)$.
- 20.2k
2
votes
0
answers
79
views
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
5
votes
2
answers
245
views
An optimization problem: $\Phi(0)$, $\widehat{\Phi}(0)$, $\Phi$ a majorant
(This is a problem that arose from my own answer to Mean value theorem for Dirichlet series - optimize? )
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such ...
- 20.2k
5
votes
0
answers
207
views
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
2
votes
0
answers
152
views
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
13
votes
1
answer
777
views
Large sieve inequality for sparse trigonometric polynomials
Let $S(\alpha) = \sum_{n\leq N}f(n) e^{2\pi i \alpha n}$ for some arithmetic function $f$. Suppose $\alpha_1, \ldots, \alpha_R$ are real numbers that are $\delta$-spaced modulo $1$, for some $0 < \...
- 253
16
votes
1
answer
845
views
Fourier Series with Mobius coefficients
I assume this question has been considered before, but I can't find an literature on it. Let $\mu(n)$ denote the usual Mobius function and define:
$F(x) : = \sum_{n=1}^{\infty} \frac{\mu(n)}{n}e(nx)$
...
- 13k
9
votes
1
answer
475
views
Error term in Davenport's sum $\sum_{n \leq x } \mu(n) \exp(2 \pi i \alpha n ) $
Reference request:
Davenport proved that for every fixed $N>1 $ one has $$ \sup_{\alpha \in \mathbb R } \left | \sum_{1\leq n \leq x } \mu(n) \exp(2 \pi i \alpha n )\right | = O_N\left( \frac{x}{(\...
- 3,062
0
votes
0
answers
258
views
Lower bound of exponential sum
This question is a close cousin of the following:
Lower bound on exponential sums
Let $\phi:[0,1]\to \mathbb R$ be a smooth function with $\frac 1 {10}<\phi'< 10, \frac 1 {10}<\phi''< 10$. ...
- 751
8
votes
2
answers
1k
views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
- 751
4
votes
1
answer
208
views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
- 12.3k
8
votes
0
answers
139
views
Fourier transform of $I_Y$, $Y=\{\text{numbers with many prime factors}\}$
Let $Y$ be the set of integers $N<n\leq 2 N$ with more than $D \log \log N$ prime factors. We may consider, say, $D = (\log \log N)^{1-\epsilon}$.
We do have rather precise approximations for the ...
- 20.2k
2
votes
2
answers
475
views
Questions concerning the Fourier analysis of $ nx\ \%\ m$
Let $x\ \%\ m$ be the residue of $x$ modulo $m$, i.e.
$$x \equiv x\ \%\ m\pmod{m}$$
The plots of the functions $f_{nm}(x) = nx\ \%\ m$ exhibit characteristic patterns, especially periods of length $...
- 9,720
6
votes
0
answers
214
views
Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
- 1,235
11
votes
1
answer
747
views
Poisson summation formula for number fields
Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...
- 415
1
vote
0
answers
163
views
Is this averaged exponential sum over primes small infinitely often?
Do there exist infinitely many positive integers $N$ such that
$$\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
7
votes
1
answer
488
views
Examples of the large sieve inequality where a constant larger than 1 is needed
Let $S(x) = \sum_{n=0}^{N-1} a_n e^{2 \pi i n x}$ be a trigonometric polynomial of length $N$. The analytic/harmonic large sieve inequality in its sharpest form states that
$$ \sum_{r=1}^R |S(x_r)|^2 ...
- 13k
8
votes
0
answers
398
views
$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual.
Assume that $$\int_0^1\prod_{j=1}...
- 148k
4
votes
1
answer
532
views
An Exponential Sum Restricted to Primes
Let $a,q,N$ be integers such that $N/2 \leq q \leq N$ and $a/q \notin \mathbb{Z}$.
Is the following estimate true, and, if so, how can it be proved?
\[\left|\sum_{1 \leq p \leq N} \exp(2\pi i p a/q) \...
- 217
4
votes
1
answer
383
views
A "nice" trigonometric polynomial approximation of a characteristic function
Let $\delta > 0$ be small and $\chi_{[-\delta, \delta]}(t)$ be a characteristic function on the interval $[-\delta, \delta]$. I am interested in a trigonometric polynomial $S$ such that
$$
\chi_{[-\...
- 3,625
4
votes
0
answers
119
views
Variability of finite Fourier coefficient with length
This is a restricted question related to the one here.
Consider the unnormalized Fourier coefficients of subsets $D_g$ of $\mathbb Z/n \mathbb Z$, denoted by
$$
\hat1_{D_g}(m,n)=\sum_{d \in {D_g}} e\...
- 10.4k
8
votes
0
answers
396
views
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 $J_0$:
$$\sum_{n=0}^\infty r_2(n) \int_0^\infty \pi J_0(2\pi\sqrt{nx}) f(x) \, dx = \sum_{n=...
- 363
6
votes
0
answers
486
views
On Fourier coefficients of nonnegative function
Let $N$ be nonnegative integer, $F(x)$ be a nonnegative real
Lebesgue integrable function defined on $[0,1]$. Suppose that all Fourier coefficients $c(\lambda)=\int_0^1F(x)e^{-2\pi i \lambda ...
- 12.3k
12
votes
2
answers
1k
views
Fourier transform of the critical line of zeta?
This was asked on MSE and got a lot of upvotes but no answers, so I'm posting it here.
Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along ...
- 4,897
4
votes
0
answers
562
views
Best known bounds on certain exponential sums
What are the best bounds currently known for the following exponential sum:
$$\sum_{x < p \le 2x} e(\alpha p^k)$$
for values of $\alpha$ far from a rational with small denominator. ($p$ refers ...
- 1,890
1
vote
0
answers
200
views
Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$
I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is striclty greater than 1(...
- 11
5
votes
0
answers
326
views
Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \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
2
votes
0
answers
239
views
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
285
views
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
3
votes
2
answers
522
views
Exponential Sum Bound
In
http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
$$\...
- 1,890
7
votes
1
answer
759
views
Major arcs in the proof that every odd number is the sum of at most 5 primes
In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
- 1,890
49
votes
2
answers
11k
views
Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
$$\Lambda(n)=...
10
votes
0
answers
2k
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 ...
- 163