Questions tagged [circle-method]
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23
questions
3
votes
0
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115
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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
259
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&...
- 125
7
votes
0
answers
192
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 ...
- 473
4
votes
0
answers
79
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
$$
\sum_{n\leq x} \...
- 1,708
0
votes
0
answers
101
views
On question on quadratic forms in four variables
Let $F$ be a non-singular quadratic form in four variables and let $w: \mathbb{R}^4 \to \mathbb{R}$ be a non-negative compactly supported function satisfying certain suitable conditions. Set $$N(F,w,m)...
- 419
3
votes
1
answer
329
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 ...
- 419
1
vote
2
answers
243
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
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1
answer
204
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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
421
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 ...
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6
votes
3
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841
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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 ...
- 19k
1
vote
1
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264
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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
378
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}...
- 131k
27
votes
4
answers
2k
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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}(...
- 19.9k
1
vote
1
answer
463
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
6
votes
1
answer
443
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,\...
- 825
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votes
0
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471
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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 ...
- 11.7k
14
votes
1
answer
865
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,...
- 825
2
votes
1
answer
97
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 ...
- 1,821
4
votes
0
answers
151
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 ...
- 1,821
8
votes
0
answers
221
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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
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0
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270
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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?
- 1,821
2
votes
2
answers
575
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, ...
- 1,821
3
votes
2
answers
514
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
$$\...
- 1,821