Edna Jones
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
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A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but 6 candidates.
An integral quadratic polynomial (with positive definite quadratic part) is called almost universal if it represents all but finitely many positive integers. In this paper, we introduce the conductor of a quadratic polynomial, and give an effective characterization of almost universal ternary quadratic polynomials with odd prime power conductor.
We prove the Oppenheim conjecture for indefinite ternary diagonal forms of the type $x^{2}+y^{2} -αz^{2}$ where $ α$ is an irrational number. Our method is explicit in the sense that we are able to construct a solution to the problem and we obtain an effective bound on the solution. The method is geometrical and is based on continued fractions.
Let k\geq 2 and consider the Diophantine inequality |x_1^k-\alp_2 x_2^k-\alp_3 x_3^k| <\tet. Our goal is to find non-trivial solutions in the variables x_i, 1\leq i\leq 3, all of size about P, assuming that \tet is sufficiently large. We study this problem on average over \alp_3 and generalize previous work of Bourgain on quadratic ternary diagonal forms to general degree k.
Covariants and simultaneous diagonalization of pairs of ternary quadratic forms, and binary quartic forms
本文建立了二元四次型与三元二次型对协变间的对应，并给出了任意特征零域上二次型对的标准对角化方法及有理数域上可对角化的精确判据。
A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$ such that $k\nmid\ell$ there exists an almost $(k,\ell)$-universal diagonal ternary form. We also conjecture that there are only finitely many primes $p$ for which a $(p,\ell)$-universal diagonal ternary form exists (for any $\ell<p$) and we show the results of computer experiments that speak in favor of the conjecture.
In this paper, we proved a theorem that every large enough odd number can be represented as the sum of three almost equal Piatetski-Shapiro primes.
In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.
Arithmetic progressions represented by diagonal ternary quadratic forms
本文基于三元二次型理论，研究并证明了由L. Pehlivan、K. S. Williams和Z.-W. Sun提出的若干对角三元二次型具有表示特定等差数列的$(d,r)$-万有性。
In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.
In the present paper we prove that every sufficiently large odd integer $N$ can be represented in the form \begin{equation*} N=p_1+p_2+p_3\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such that $p_1=x^2 + y^2 +1$, $p_2=[n^c]$.
For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum $aP_5(x)+2^rbP_5(y)+2^scP_5(z)$ represents all but finitely many positive integers.
The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.
We prove that the representations numbers of a ternary definite integral quadratic form defined over F_q[t], where F_q is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to its successive minima. Equivalently, such a quadratic form is determined up to integral isometry by its theta series.
We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal
Almost all primes are not needed in Ternary Goldbach
该研究针对三元哥德巴赫猜想，通过可容许同余扩张系统构造素数集$\u2119$，证明即便剔除几乎所有素数，仅用$\u2119$中的素数仍可满足该猜想，深化了对素数加法性质的认识。
We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.
In this paper we prove the main conjectures of Berkovich and Jagy about weighted averages of representation numbers over an S-genus of ternary lattices (defined below) for any odd squarefree S \in N. We do this by reformulating them in terms of local quantities using the Siegel-Weil and Conway-Sloane formulas, and then proving the necessary local identities. We conclude by conjecturing generalized formulas valid over certain totally real number fields as a direction for future work.
For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of reduced residues mod k.
We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.
An integer $n$ is said to be ternary if it is composed of three distinct odd primes. In this paper, we asymptotically count the number of ternary integers $n \leq x$ with the constituent primes satisfying various constraints. We apply our results to the study of the simplest class of (inverse) cyclotomic polynomials that can have coefficients that are greater than 1 in absolute value, namely to the $n^{th}$ (inverse) cyclotomic polynomials with ternary $n$. We show, for example, that the corrected Sister Beiter conjecture is true for a fraction $\ge 0.925$ of ternary integers.
We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.
A cyclotomic polynomial Φ_n(x) is said to be ternary if n=pqr with p,q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ_{pqr}(x) with p<q<r, p and q fixed and r a free prime.
Hecke modules associated to certain even ternary lattices of half-discriminant $\mathbf{N^2}$
该研究针对半判别式为$N^2$的定偶三元格对应的Hecke模，借助Watson型变换序列，建立其与经典模形式空间的Jacquet-Langlands型显式对应，为相关数论问题提供了新的研究路径。
In this paper, we give an explicit expression for a certain family of ternary cyclotomic polynomials: specifically $Φ_{p_{1}p_{2}p_{3}}$, where $p_{1}<p_{2}<p_{3}$ are odd primes such that $p_{2} \equiv1 \mod p_{1}$ and $p_{3} \equiv1 \mod {p_{1}p_{2}}$. As an application of the explicit expressions, we give an exact formula for the number of nonzero terms in the polynomials in the family, which in turn immediately shows that the density (number of non-zeros terms / degree) is roughly inversely proportional to $p_{2}$, when $p_{1}$ is sufficiently large.
Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three subsets of $\mathcal{P}$. Let $\underlineδ(P_{i})$ $(i=1,2,3)$ denote the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that if $\underlineδ(P_{1})>5/8$, $\underlineδ(P_{2})\geq5/8$, and $\underlineδ(P_{3})\geq5/8$, then for every sufficiently large odd integer n, there exist $p_{i} \in P_{i}$ such that $n=p_{1}+p_{2}+p_{3}$. The condition is the best possible.
A (positive definite and integral) quadratic form is said to be $\textit{prime-universal}$ if it represents all primes. Recently, Doyle and Williams in [2] classified all prime-universal diagonal ternary quadratic forms, and all prime-universal diagonal quaternary quadratic forms under two conjectures proposed by themselves. In this article, we classify all prime-universal diagonal quadratic forms regardless of ranks. Furthermore, we prove, so called, $67$-Theorem for a diagonal quadratic form to be prime-universal.
Hecke modules associated to certain ternary lattices of half-discriminant $p^2$ and $4p^2$
该研究针对半判别式为$p^2$和$4p^2$的定三元格对应的Hecke模，证明其与经典模形式空间的Jacquet-Langlands型对应并给出剩余亏格猜想，通过Watson型变换实现。
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualita…
Density ternary Goldbach for primes in a fixed residue class
本文证明，在同余 1 mod 3 的素数中，相对密度超过 1/2 的子集可表示所有充分大且同余 0 mod 3 的奇数为三元素数和，并证明该密度阈值最优。
We show that for every fixed $A>0$ and $θ>0$ there is a $\vartheta=\vartheta(A,θ)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log n)^θ$. Then for all $q_{3}\leq Q_{3}$, all reduced residues $a_{3}$ mod $q_{3}$, almost all $q_{2}\leq Q_{2}$, all admissible residues $a_{2}$ mod $q_{2}$, almost all $q_{1}\leq Q_{1}$ and all admissible residues $a_{1}$ mod $q_{1}$, there exists a representation $n=p_{1}+p_{2}+p_{3}$ with primes $p_{i}\equiv a_{i} (q_{i})$, $i=1,2,3$.
We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all non-diagonal coefficients in $2\mathcal{O}_K$) and prove that no such forms in three variables are universal (i.e., represent all totally positive elements of $\mathcal{O}_K$). This provides further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable el…
A ternary inclusion-exclusion polynomial is a polynomial of the form \[ Q_{p,q,r}=\frac{(z^{pqr}-1)(z^p-1)(z^q-1)(z^r-1)} {(z^{pq}-1)(z^{qr}-1)(z^{rp}-1)(z-1)}, \] where $p$, $q$, and $r$ are integers $\ge3$ and relatively prime in pairs. This class of polynomials contains, as its principle subclass, the ternary cyclotomic polynomials corresponding to restricting $p$, $q$, and $r$ to be distinct odd prime numbers. Our object here is to continue the investigation of the relationship between the coefficients of $Q_{p,q,r}$ and $Q_{p,q,s}$, with $r\equiv s\pmod{pq}$. More specifically, we conside…
Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.
Let $$γ^*=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$.\\ We prove on assumption of the Generalized Riemann Hypothesis that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where the $p_i$ are of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$, for $i=1,2$ and the decimal expansion of $p_3$ does not contain the digit $a_0$.\\ The proof merges methods of J. Maynard from his paper on the infinitude of primes with restricted digits, results…
The arithmetic of ternary diagonal equation is considered for degree d >1, with the outcome that the set of coefficients for which the equation admits a non-zero integer solution is shown to have density zero.
Let $$γ^*:=\frac{8}{9}+\frac{2}{3}\:\frac{\log(10/9)}{\log 10}\:(\approx 0.919\ldots)\:,\ γ^*<\frac{1}{c_0}\leq 1\:.$$ Let $γ^*<γ_0\leq 1$, $c_0=1/γ_0$ be fixed. Let also $a_0\in\{0,1,\ldots, 9\}$. In [23] we proved on assumption of the Generalized Riemann Hypothesis (GRH), that each sufficiently large odd integer $N_0$ can be represented in the form $$N_0=p_1+p_2+p_3\:,$$ where for $i=2, 3$ the primes $p_i$ are Piatetski-Shapiro primes - primes of the form $p_i=[n_i^{c_0}]$, $n_i\in\mathbb{N}$ - whereas the decimal expansion of $p_1$ does not contain the digit $a_0$. In this paper we replace …
The ternary Goldbach conjecture, or three-primes problem, states that every odd number $n$ greater than $5$ can be written as the sum of three primes. The conjecture, posed in 1742, remained unsolved until now, in spite of great progress in the twentieth century. In 2013 -- following a line of research pioneered and developed by Hardy, Littlewood and Vinogradov, among others -- the author proved the conjecture. In this, as in many other additive problems, what is at issue is really the proper usage of the limited information we possess on the distribution of prime numbers. The problem serves a…
In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended Shimura correspondence, we obtain explicit formulas for the number of representations of $|D|n^2$ by the above type of quadratic forms, where $D$ is either a square-free integer or a fundamental discriminant such that $(-1)^{(\ell-1)/2}D > 0$. We demonstrate our method with many examples, in particular, we obtain all the formulas (when $\ell =5$) obtained in t…
We use recent advances in the local evaluation of Brauer elements to study the role played by {\it odd} torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over {\it arbitrary} number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, so that the odd torsion does not obstruct weak approximation. We also no…
Some mean value theorems in the style of Bombieri-Vinogradov's theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd n\not\equiv 1 (6), Goldbach's ternary problem n=p_1+p_2+p_3 is solvable with primes p_1,p_2 in short intervals p_i \in [X_i,X_i+Y] with X_{i}^{θ_i}=Y, i=1,2, and θ_1,θ_2\geq 0.933 such that (p_1+2)(p_2+2) has at most 9 prime factors.
We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hanke's celebrated…
Small solutions of ternary quadratic congruences with averaging over the moduli
本文研究了三次二次同余式的解的存在性，通过模数平均方法，证明了在特定条件下解的高度上限为Q^{3/8+ε}α₂^{ε}。
We show that many of Ramanujan's modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 3y^2 +3z^2: x,y,zin Z}|, just to mention one among many similar positive results of this type. In particular, we prove the recent conjecture of H. Yesilyurt and the first author stating that for any n in N |{n= x(x+1)/2 + y^2 +z^2 : x,y,z in Z}| >= |{n= x(x+1)/2 + 7y^2 + 7z^2: x,y,z in Z}|. We prove a variety of identities for certain ternary forms with discriminants 1…
Small Solutions of generic ternary quadratic congruences
本文研究一般三元二次同余的小解，证明在大多数系数下解的渐近公式成立，突破指数1/2障碍，并在固定素数幂及Lindelöf假设下改进指数。
Equidistribution of Solutions of Ternary Quadratic Congruences Modulo Prime Powers
本文研究素数幂模三元二次同余方程在固定点附近且 $z$ 与 $p$ 互素的解，通过光滑化方法在 $N\geq q^{1/2+\varepsilon}$ 时建立渐近公式并证明解存在。
The author showed that any homogeneous algebraic Diophantine equation of the second order can be converted to a diagonal form using an integer non-orthogonal transformation maintaining asymptotic behavior of the number of its integer solutions. In this paper, we consider the transformation to the diagonal form of a wider class of algebraic second-order Diophantine equations, and also we consider the conditions for converting higher order algebraic Diophantine equations to this form. The author found an asymptotic estimate for the number of integer solutions of the diagonal Thue equation of odd…
Ternary polynomial relations and periods of cusp forms
该研究在特征零域上证明了Bachmann与van Ittersum的三元多项式关系猜想，确定齐次解结构，关联其与尖点形式空间，并在Lean 4中形式化了核心结果。
Small solutions of generic ternary quadratic congruences to general moduli
本文研究一般奇数模数下三元二次同余方程的小解，利用Burgess和Heath-Brown的特征和估计突破指数1/2障碍，在平均意义下对几乎所有系数得到解的渐近公式，并提出最小解大小的精细猜想。
Kitaoka's Conjecture and sums of squares
本文建立了全实域上三元经典万有二次型存在性与2的所有全正倍数为平方和之间的联系，并证明了Kitaoka猜想对奇判别域成立。
Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori
本文通过研究三次二次型在整数上的取值，验证了伯里-塔布猜想在三维扭体中的适用性，解决了特殊对角型在不同缩放下的分布问题。
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