All Questions
Tagged with quadratic-forms analytic-number-theory
32 questions
6
votes
0
answers
380
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
2
votes
0
answers
59
views
Equidistribution of lattice points on quadratic forms without certain values
I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, ...
- 313
3
votes
1
answer
230
views
Duke and Schulze-Pillot condition for equidistribution
When regarding a ternary quadratic form $Q(x,y,z)$, is a classic question to consider which integers $n$ can be represented by $Q$. It is also classic to wonder how "well distributed" are ...
- 313
1
vote
0
answers
129
views
Siegel's formula for generalized theta series with characteristics?
Siegel's formula(Siegel-Weil) directly relates the weighted sum of theta functions to Eisenstein series. (Or equivalently, the weighted sum of the cusp form is zero). I wonder if there is a ...
- 11
7
votes
0
answers
224
views
the gaps between values of a positive binary quadratic form at integer points
Suppose that $s$ is a positive irrational number. Consider all possible values of the sum $x^2+sy^2$ where $x$ and $y$ are integers. These values form a set $W=\{0=w_1<w_2<\ldots\}$. Can the ...
4
votes
1
answer
197
views
Conics and triples of binary quadratic forms
Let $C \subset \mathbb{P}^2$ be a planar conic curve, defined by a ternary quadratic form $Q(x_1, x_2, x_3)$ say. Suppose that $C(\mathbb{Q}) \ne \emptyset$, or equivalently, that $C$ is everywhere ...
- 26.9k
3
votes
1
answer
370
views
Close integer solutions to $ab-cd=1$?
I am looking for infinite set of Diophantine solutions.
Suppose we require
$$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$
$$a,b,c,d\in\mathbb Z$$
then can we still find ...
- 1,826
0
votes
0
answers
113
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 $$N(F,w,m)...
- 429
3
votes
1
answer
378
views
On quadratic forms in four variables
Let $F$ be a non-singular integral quadratic form in four variables. Then a result of Heath-Brown from the 90's states for $m \to \infty$, $$|\{ x \in \mathbb{Z}^4 \,:\, F(x) = m \}| = C_F\sigma(F,m)m ...
- 429
2
votes
0
answers
58
views
Number of representations by the norm in a division algebra corresponding to endomorphism rings of elliptic curves
Let $E$ be a supersingular curve over a field of characteristic $p$ with endomorphism ring $\mathcal O_D$ which is a maximal order in a division ring $D$ over $Q$ ramified at $p$ and $\infty$.
The ...
- 7,746
6
votes
0
answers
211
views
some problems on sum of two squares
During my experiments with "Mathematica" I arrived to the following observations. My question is that are they interesting, known, solved or not. If they are known could you please give me a reference....
- 841
5
votes
1
answer
208
views
applications of finding least quadratic nonresidue mod $p$?
I saw some papers from famous mathematicians (assuming GRH or without it) which are devoted to finding bound for least quadratic nonresidues modulo prime number $p$.
My question is that why it is so ...
- 841
8
votes
1
answer
720
views
Are lattice points in thin spherical shells uniformly distributed?
Consider the spherical shell (annulus)
$$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq |
x|\leq R+r \}.$$ Think of the limit $R \to \infty$.
Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are ...
0
votes
1
answer
125
views
Polyhedral conditions for quadratic inequalities in fixed dimension
Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy
$$0<T_1,T_2,T_3,T_4$$
conditions?
Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
- 13.9k
0
votes
1
answer
104
views
Elementary description to count of perfect squares - II
What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+...
- 13.9k
0
votes
1
answer
133
views
Elementary description to count of perfect squares - I
Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
- 13.9k
17
votes
2
answers
4k
views
On Siegel mass formula
I have asked this question exactly here. The question is as follows:
I am interested deeply in the following problem:
Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be an ...
12
votes
1
answer
499
views
A diophantine equation in $\mathbb{N}$
While I was working on a paper on graph theory, I encountered a problem which I think is a number-theory-problem. I don't know if there are any tools to answer the question.
Find all natural numbers $...
- 351
5
votes
2
answers
231
views
Bounded version of linear and quadratic Hasse--Minkowski theorem
The Hasse-Minkowski theorem states that if
$$Q(x_1,\ldots,x_n) = \sum_{i,j=1}^n a_{ij} x_ix_j$$
is a quadratic form with $a_{ij} \in \mathbb Z$ and $\det (a_{ij}) \neq 0$, then the equation
$$Q(x_1,\...
- 13.9k
8
votes
3
answers
870
views
Asymptotic formula for sums of four squares?
Does there exist asymptotic formula for ways to write n as sum of four squares? Or can this be proved impossible? I can only find reference for sums of five squares.
- 83
8
votes
0
answers
264
views
Number of representations of an integer as the dot product of integer vectors
Let $r_k(n)$ denote the number of solutions in positive integers to the equation: $$n = a_1 b_1 + a_2 b_2 + \ldots + a_k b_k.$$
What estimates and/or asymptotics are available for
(1) $...
- 13k
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}(...
- 20.4k
24
votes
1
answer
694
views
two's and three's survive in gcd of Lagrange
Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+...
- 43.2k
2
votes
1
answer
223
views
Uniformity in the error term of counting ideals up to a certain norm
There is a beautiful formula to count the number of ideals $I$ in the ring of integers $\mathcal{O}_K$ of a number field $K$, given by
$$\sum_{n \leq X} a_n \sim C_K X,$$
where $a_n$ is the number ...
- 26.9k
3
votes
1
answer
188
views
Are those $2$ quadratic forms congruent over $\mathbb{Z}[1/q]$
Let $q$ be a natural number (the first cases of interest being $q = 10,12$ or $15$), and let $n = q^2+q+1$. Also, let $I_n$ be the $n\times n$ identity matrix, and let $A_n$ be the $n\times n$ ...
- 1,017
2
votes
0
answers
110
views
Bounds on the number of zeros of a quadratic form
Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and $Q(...
- 6,214
6
votes
1
answer
462
views
The number of integral solutions to $x^2+y^2-az^2=0$
I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ x^2+y^...
- 6,214
18
votes
2
answers
3k
views
Many representations as a sum of three squares
Let $r_3(n) = \left|\{(a,b,c)\in {\mathbb Z}^3 :\, a^2+b^2+c^2=n \}\right|$. I am looking for the maximum asymptotic size of $r_3(n)$. That is, the maximum number of representations that a number can ...
- 1,072
7
votes
0
answers
673
views
Mock modular forms and (indefinite) quadratic forms
Define the function
$$f(q,z,y) = \sum_{n \ge 0,m,l} c(n,m,l) q^n z^m y^l$$
where $c(n,m,l)$ is defined by
$$ c(n,m,l) =
\begin{cases}
(-1)^{s+l} & \text{if } 4n - m^2 + l^2 = 2s(s+1)\\
0 & \...
- 1,298
2
votes
1
answer
376
views
Primes as the first coefficient of a reduced indefinite quadratic form
Given a discriminant d>0 (make it fundamental if that is easier), when can a prime p be the the $x^2$ coefficient of a reduced indefinite quadratic form?
That is, for what p is there a reduced form $...
- 809
14
votes
3
answers
985
views
Achieving consecutive integers as norms from a quadratic field
This question is inspired by my inability to make any progress on Will Jagy's question.
Giving a positive answer to this question should be strictly easier than proving Jagy's conjectures.
Suppose ...
- 156k
2
votes
2
answers
506
views
Lower bounds for split primes in Real quadratic fields
Snippet portion:
From Iwaniec and Kowalski's Analytic Number Theory:
If the class number $h=h(D)$ is small, then there are only few
prime ideals $\bf{p}$ of degree one with small norm. Indeed, if
$p=...
- 809