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it's known that under GRH the exceptional set in Goldbach is essentially $\ll \sqrt x$ (here and throughout I don't care for $\log $'s and $\epsilon $'s). Ultimately this comes from $$\psi _\chi (\...
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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/...
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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 \...
tomos's user avatar
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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 ...
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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\...
Alexander's user avatar
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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 ...
bbb's user avatar
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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$ ...
Bill Quan Yue's user avatar
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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 \...
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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: ...
Ashvin Swaminathan's user avatar
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Are there some results give the $X$ power saving of the error term in Ternary Goldbach problem, not just the $\log$ power saving?
Adiel Hsueh's user avatar
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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 ...
Aadi Deepchand's user avatar
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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&...
ketsi's user avatar
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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 ...
user50139's user avatar
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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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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 $$N(F,w,m)...
Constantin K's user avatar
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Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m ...
Constantin K's user avatar
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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 ...
fernando's user avatar
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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 ...
The Number Theorist's user avatar
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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 ...
K Hughes's user avatar
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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 ...
H A Helfgott's user avatar
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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, ...
Jan-Christoph Schlage-Puchta's user avatar
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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 $$\int_0^1\prod_{j=1}...
Will Sawin's user avatar
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Let $Q$ be a four-variable positive-definite quadratic form with integer coefficients and let $r_{Q}(n)$ be the number of representations of $n$ by $Q$. The theory of modular forms explains how $r_{Q}(...
Jeremy Rouse's user avatar
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Fermat two square: An odd prime p is expressible as ${\displaystyle p=x^{2}+y^{2},\,}$ with $x, y$ integers, if and only if ${\displaystyle p\equiv 1{\pmod {4}}.}$ Lagrange four square: Every ...
asad's user avatar
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In Hardy-Littlewood's 1923 paper "Some problems of 'Partitio Numerorum' III" it is proven, assuming a weak version of GRH (namely that there is $\varepsilon>0$ s.t. all zeroes of $L(s,\...
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Let $$ f(x,N) = \sum_{0 \leq n\leq N} e(x n^2).$$ Weyl's inequality gives an estimate for $f(x,N)$ when $x$ is near a rational with small denominator. My question is: What estimates are ...
Mark Lewko's user avatar
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For any set $B\subseteq \mathbb{N}$ one can associate the formal series $$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,...
Alufat's user avatar
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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 ...
Mayank Pandey's user avatar
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4 votes
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154 views

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 ...
Mayank Pandey's user avatar
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8 votes
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292 views

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)...
leithian's user avatar
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Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
Mayank Pandey's user avatar
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2 votes
2 answers
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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, ...
Mayank Pandey's user avatar
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3 votes
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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 $$\...
Mayank Pandey's user avatar
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