# Ramanujan's ternary quadratic form

@article{Ono1997RamanujansTQ, title={Ramanujan's ternary quadratic form}, author={Ken Ono and Kannan Soundararajan}, journal={Inventiones mathematicae}, year={1997}, volume={130}, pages={415-454} }

do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer. Consequently if a genus consists of a single class with representative f(x, y, z), then f represents every eligible integer. In the…

## 20 Citations

Positive-definite ternary quadratic forms with the same representations over ℤ

- MathematicsInternational Journal of Number Theory
- 2020

Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over [Formula: see text], they are equivalent over [Formula: see text] or constant…

Eisenstein Series of 3/2 Weight and Eligible Numbers of Positive Definite Ternary Forms

- Mathematics
- 2001

A general algorithm is given for the number of representations for a positive integer n by the genus of a positive definite ternary quadratic form with form ax2 + by2 + cz2. Using this algorithm, we…

The eligible numbers of positive definite ternary forms

- Mathematics
- 2000

Abstract. Several nontrivial genera of positive ternary forms with small discriminants have been studied in this paper. Especially we prove that there are only finitely many, square-free eligible…

On sums of four pentagonal numbers with coefficients

- MathematicsElectronic Research Archive
- 2020

The pentagonal numbers are the integers given by $p_5(n)=n(3n-1)/2\ (n=0,1,2,\ldots)$. Let $(b,c,d)$ be one of the triples $(1,1,2),(1,2,3),(1,2,6)$ and $(2,3,4)$. We show that each $n=0,1,2,\ldots$…

Steinhaus tiling problem and integral quadratic forms

- Mathematics
- 2006

A lattice L in R n is said to be equivalent to an integral lattice if there exists a real number r such that the dot product of any pair of vectors in rL is an integer. We show that if n ≥ 3 and L is…

Representations of finite number of quadratic forms with same rank

- MathematicsThe Ramanujan Journal
- 2020

Let $m, n$ be positive integers with $m\le n$. Let $\kappa(m,n)$ be the largest integer $k$ such that for any (positive definite and integral) quadratic forms $f_1,\ldots,f_k$ of rank $m$, there…

Shimura lifting of modular forms of weight 3/2

- Mathematics
- 2016

We show that there is the Shimura lifting map of $$\mathbf {M}_{3/2}(N,\chi _{0})$$M3/2(N,χ0) to $$\mathbf {M}_{2}(N/2,\chi _{0}^{2})$$M2(N/2,χ02) for any natural number N divisible by 4 and for any…

Some universal quadratic sums over the integers

- MathematicsElectronic Research Archive
- 2019

Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as…

PRIME-UNIVERSAL QUADRATIC FORMS AND

- MathematicsBulletin of the Australian Mathematical Society
- 2019

A positive-definite diagonal quadratic form $a_{1}x_{1}^{2}+\cdots +a_{n}x_{n}^{2}\;(a_{1},\ldots ,a_{n}\in \mathbb{N})$ is said to be prime-universal if it is not universal and for every prime $p$…

REPRESENTATIONS OF BINARY FORMS BY QUATERNARY FORMS

- Mathematics
- 2007

In this paper we study a family of quaternary forms which represent almost all binary forms of a certain type. The result follows from the representation number by the genus of ternary forms and a…

## References

SHOWING 1-10 OF 31 REFERENCES

Note on a theorem of Cassels

- Mathematics
- 1957

be a quadratic form in n variables (n > 2) with integral coefficients. Cassels (1) has recently proved the interesting result that if the equation / = 0 is properly soluble in integers xlt..., xn,…

A second genus of regular ternary forms

- Mathematics
- 1995

In the paper [2] Hsia noted that the forms x 2 + xy + y 2 +9 z 2 and x 2 +3 y 2 +3 yz +3 z 2 constitute a genus and that both forms are regular; he asked whether there exist any other genera…

History of the Theory of Numbers

- MathematicsNature
- 1924

THE third and concluding volume of Prof. Dickson's great work deals first with the arithmetical. theory of binary quadratic forms. A long chapter on the class-number is contributed by Mr. G. H.…

Bounds for the least solutions of homogeneous quadratic equations

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1956

In this note I obtain bounds for the least integral solutions of the equationin terms ofFor ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1),…

Multiplicative Number Theory

- Mathematics
- 1967

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The…

Hyperbolic distribution problems and half-integral weight Maass forms

- Mathematics
- 1988

(Actually n ~ is replaced by d(n)log ~ 2n where d(n) is the divisor function.) A striking application of(1.2) is to give the uniform distribution of certain lattice points in Z 3 on a sphere centered…

Introduction to Elliptic Curves and Modular Forms

- Mathematics
- 1984

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book…

The Theory of the Riemann Zeta-Function

- Mathematics
- 1987

The Riemann zeta-function embodies both additive and multiplicative structures in a single function, making it our most important tool in the study of prime numbers. This volume studies all aspects…

Modular Forms of Half Integral Weight

- Mathematics
- 1973

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function
$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty…

The first nontrivial genus of positive definite ternary forms

- Mathematics, Biology
- 1995

The first nontrivial genus of positive ternary forms has discriminant 7. The paper presents all known results concerning this genus, including some computations