Questions tagged [circle-method]
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32 questions
1 vote
1 answer
160 views
Major arc approximations - partial summation and Gallagher's lemma
The motivation for my question is that I think I need to use Gallagher's lemma for exponential sums (``A large sieve density estimate near $\sigma =1$", Lemma 1, https://link.springer.com/article/...
- 1,536
0 votes
0 answers
83 views
Mean square / Plancherel with Farey dissection
Suppose $S(\alpha )$ is the exponential sum up to $x$ of a sequence with mean square $x$. Trivially $\int _0^1|S(\alpha )|^2d\alpha \ll x$. Should it be true that $\sum _{q,a}\cdot \frac {q}{a}\cdot \...
- 1,536
1 vote
1 answer
238 views
Bourgain and Rudnick circle argument for lattice points in spherical caps
Currently, I'm reading the appendix of Bourgain and Rudnick's paper that considers bounds for eigenfunctions of the Laplacian on the flat torus. The proof breaks down for $d > 3$, but in the ...
- 619
4 votes
1 answer
348 views
Secondary term in asymptotic formula for number of integral solution to quadratic form $F(x)=0$
In Heath-Brown's paper " A new form of the circle method and applications to Quadratic forms", Theorem 7 states that the number of weighted integral solutions in an expanding region $P\...
- 375
2 votes
0 answers
158 views
Additive combinatorics and the Waring-Goldbach problem
I have been reading this article https://arxiv.org/pdf/math/0412220 about the Waring-Goldbach problem. It's really nicely written. They discuss the Waring-Goldbach problem as well as in the end ...
- 21
4 votes
1 answer
299 views
Vinogradov's method for sum of more than three primes
Background: Vinogradov's method for sums of more than three primes Q&A. Can someone confirm that "This formula has been rigorously proven to be asymptotically valid for $k\geqslant 3$ ...
2 votes
1 answer
213 views
Waring's problem with congruence restrictions
I'm interested in a variant of Waring's problem where each variable is restricted to lie in a specified congruence class modulo a fixed integer $p$ (which may be assumed prime). Let $k \geq 2$ and $0 \...
- 21
1 vote
0 answers
101 views
Number of symmetric matrices in a box of bounded determinant
Let $B$ and $T$ be positive real numbers. I'm interested in the following problem, which is about counting $2\times2$ symmetric matrices with bounded determinant and entries lying in a box: Problem: ...
3 votes
0 answers
130 views
The error term of ternary-Goldbach problem
Are there some results give the $X$ power saving of the error term in Ternary Goldbach problem, not just the $\log$ power saving?
- 31
3 votes
0 answers
145 views
How to find the right path of integration to get the asymptotic partition formula
I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers. I am ...
2 votes
1 answer
407 views
Overall idea of estimating major arcs in Waring's problem
This is copied from math.SE after a kind comment's suggestion as I am sure people here are very well knowledged in this method :) I am currently reading Vaughan's "The Hardy-Littlewood Method&...
- 137
7 votes
0 answers
259 views
Birch's theorem for quartic forms in many variables
I would like to understand the application of the Hardy-Littlewood Circle Method to Birch's celebrated theorem on forms in many variables, namely the statement that the point counting function for a ...
- 585
4 votes
0 answers
104 views
Joint mean values of arithmetic functions in sequences and families of sequences
This is a bit of a follow up question to this question I asked a couple days ago. The main content of that post can be phrased as asking for a nontrivial lower bound on the sum $$ \sum_{n\leq x} \...
- 2,142
0 votes
0 answers
149 views
On question on quadratic forms in four variables
Let $F$ be a non-singular quadratic form in four variables and let $w: \mathbb{R}^4 \to \mathbb{R}$ be a non-negative compactly supported function satisfying certain suitable conditions. Set $$N(F,w,m)...
- 429
3 votes
1 answer
458 views
On quadratic forms in four variables
Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m ...
- 429
1 vote
2 answers
342 views
Rademacher expansions for weight 1/2
In a famous paper Rademacher used the circle theorem to give a formula for the fourier coefficients of the partition function $1/f(q)$ where $f(q) = \prod_{n=1}(1-q^n)$, and in another paper he gave ...
- 303
1 vote
1 answer
263 views
Upper bound for the integral over minor arcs of the exponential sum with prime omega function coefficients
Define $\mathfrak{m}$ as the union of the minor arcs of the form $|\alpha-\frac{a}{q}|\leq 1/qQ$, with $(a,q)=1$ and $Q_0<q\leq Q$, with $Q_0\geq N/Q$, for a certain $N\geq Q$ large. Is it ...
12 votes
1 answer
530 views
Is there a physical/geometric proof for L^2 boundedness of Bourgain's maximal function along the squares?
One problem that has bugged me for some time (though I only seriously thought about it for a month several years ago) is to give a physical proof of the L^2 boundedness of Bourgain maximal function ...
- 679
6 votes
3 answers
980 views
Decoupling, efficient congruencing and Vinogradov's main theorem
It seems to be word in the generic corridor that decoupling (as in Bourgain-Demeter-Guth) and efficient congruencing (Wooley) are deeply related, and even that they are deep down the same thing - with ...
- 21.7k
1 vote
1 answer
356 views
Reference Request: Waring's problem for different polynomials
I am looking for a reference for the following statement: For every integer $d$, there exist an integer $k$, such that for all polynomials $P_1, \ldots, P_k$ of degree $d$ there exist integers $N, q, ...
8 votes
0 answers
434 views
$L^1$ norm of Fourier transform of subset sums
Let $n_1,\dots,n_k$ be a set of $k$ natural numbers less than $N$, with $k = (1- \delta) \log_2 N$ for $\delta$ relatively small. Let $e(x) = e^{ 2\pi i x}$, as usual. Assume that $$\int_0^1\prod_{j=1}...
- 161k
27 votes
4 answers
2k views
Which quaternary quadratic form represents $n$ the greatest number of times?
Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
- 21.2k
1 vote
1 answer
569 views
Fermat two square and Lagrange four square via Hardy-Littlewood circle method [closed]
Fermat two square: An odd prime p is expressible as ${\displaystyle p=x^{2}+y^{2},\,}$ with $x, y$ integers, if and only if ${\displaystyle p\equiv 1{\pmod {4}}.}$ Lagrange four square: Every ...
- 841
7 votes
1 answer
665 views
Vinogradov's method for sums of more than three primes
In Hardy-Littlewood's 1923 paper "Some problems of 'Partitio Numerorum' III" it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\...
- 962
0 votes
0 answers
543 views
Weyl sums with polynomial coefficients
Let $$ f(x,N) = \sum_{0 \leq n\leq N} e(x n^2).$$ Weyl's inequality gives an estimate for $f(x,N)$ when $x$ is near a rational with small denominator. My question is: What estimates are ...
- 13.8k
14 votes
1 answer
1k views
Erdös-Turán via Hardy-Littlewood circle method?
For any set $B\subseteq \mathbb{N}$ one can associate the formal series $$f_B(z) = \sum_{b\in B}z^b$$ and obtain $$f_B(z)^k = \sum_{n\geqslant 0} r_{B,k}(n)z^n,$$ where $r_{B,k}(n) = |\{(x_1,\cdots,...
- 962
2 votes
1 answer
117 views
Elaboration of a certain section of a paper by Thanigasalam
In section 11 of this paper by Thanigasalam, it says "... we get $G(10)\le 105$, and this implies that $H(10) \le 107$". However, it is very unclear how this follows. Why is it the case that $G(10)\le ...
- 1,909
4 votes
0 answers
154 views
Goldbach's problem in algebraic number fields [duplicate]
Are there any results on the representation of numbers in algebraic number fields as the sum of primes in the ring of integers in that field? There are some results for Waring's problem in other ...
- 1,909
8 votes
0 answers
287 views
Approximation to a certain Weyl-sum
Let $ S(\alpha)=\sum_{x\leq X}\sum_{y\leq Y}e \left(\alpha x y^3 \right)$, for some $X,Y \geq 1$ and write $\alpha = a/q + \beta$ for $(a,q)=1$, as usual. For the 'classical' cubic Weyl-sum $f(\alpha)...
- 163
1 vote
0 answers
310 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?
- 1,909
2 votes
2 answers
657 views
Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers
How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$ I think that one could use the circle method, ...
- 1,909
3 votes
2 answers
564 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 $$\...
- 1,909