# Questions tagged [circle-method]

The circle-method tag has no usage guidance.

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### Estimates for the integral over minor arcs of exponential sums with special coefficients

Consider $h(n)$ to be an arithmetical function. 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 $Q_0<q\leq Q$.
Fix $N\geq Q$ ...

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### Approximation in the circle method

I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$.
One of the ...

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### 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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### 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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### 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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### $L^1$ 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=...

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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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### 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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### 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 $L(s,\chi)$ have $\...

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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 ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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
$$\...