Questions tagged [circle-method]

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7 votes
0 answers
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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 ...
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4 votes
0 answers
66 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} \...
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0 votes
0 answers
75 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)...
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3 votes
1 answer
236 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 ...
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1 vote
2 answers
200 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 ...
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  • 303
1 vote
1 answer
158 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 ...
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12 votes
1 answer
367 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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  • 539
6 votes
3 answers
790 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 ...
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  • 16.7k
1 vote
1 answer
239 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, ...
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8 votes
0 answers
348 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}...
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27 votes
4 answers
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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}(...
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1 vote
1 answer
383 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 ...
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  • 791
6 votes
1 answer
397 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,\...
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  • 725
0 votes
0 answers
402 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 ...
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14 votes
1 answer
814 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,...
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  • 725
2 votes
1 answer
94 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 ...
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3 votes
0 answers
141 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 ...
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8 votes
0 answers
203 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)...
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  • 163
1 vote
0 answers
249 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?
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2 votes
2 answers
557 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, ...
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3 votes
2 answers
505 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 $$\...
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