All Questions
8 questions
5
votes
1
answer
594
views
Exponential sum vs. exponential integral via Poisson summation
When we want to estimate an exponential sum
$$
\sum_{M<m\le M'}e(f(m))
\quad\text{with}\quad
1\le M\le M'\le 2M
\quad\text{and}\quad
e(x):=\exp(2\pi ix)
$$
where $e(x):=\exp(2\pi ix)$
and the phase ...
- 363
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
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
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
476
views
Fourier support condition in the paper 'A study guide for the $l^2$ decoupling theorem'
I'm currently reading Bourgain and Demeter's study guide for the $l^2$ decoupling theorem (https://arxiv.org/pdf/1604.06032.pdf). I have some trouble with understanding the proof of Proposition 8.4.
...
- 61
2
votes
0
answers
167
views
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
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
42
votes
7
answers
5k
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 ...
- 7,337