Showing papers in "International Journal of Number Theory in 2006"
TL;DR: A comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs are given.
Abstract: The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs.
67 citations
TL;DR: In this paper, the fundamental Euler sum identity is shown to be a small mathematical gem, and various generalizations for multiple harmonic (Euler) sums and their connections are discussed, illustrating both the wide variety of techniques used to study such sums and the attraction of their study.
Abstract: We give thirty-two diverse proofs of a small mathematical gem — the fundamental Euler sum identity We also discuss various generalizations for multiple harmonic (Euler) sums and some of their many connections, thereby illustrating both the wide variety of techniques fruitfully used to study such sums and the attraction of their study.
64 citations
TL;DR: The main aim of as discussed by the authors is to characterize the formal power series β(|β| > 1) such that dβ(1) is finite, eventually periodic or automatic.
Abstract: Let β be a fixed element of 𝔽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk . Any formal power series in is expanded in this way into dβ(f) = (ai(X))i≥1, where . The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case).
32 citations
TL;DR: In this paper, the structure of finite subsets of an abelian group satisfying |A + B| ≤ |A| + |B| -1 is described as a dual result of Kemperman's.
Abstract: A well-known result by Kemperman describes the structure of those pairs (A, B) of finite subsets of an abelian group satisfying |A + B| ≤ |A| + |B| -1. We establish a description which is, in a sense, dual to Kemperman's, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne.
30 citations
TL;DR: In this paper, the authors prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that has rank ≥ 2.
Abstract: We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.
29 citations
TL;DR: In this paper, the authors give a simple, direct proof of a theorem involving partitions into distinct parts, where multiples of 7 come in two colours, and show that the theorem is correct.
Abstract: We give a simple, direct proof of a theorem involving partitions into distinct parts, where multiples of 7 come in two colours.
20 citations
TL;DR: In this paper, the authors considered the relative Pellian equations and showed that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solved the system completely for | c| ≥ 1544686.
Abstract: In this paper we consider the family of systems (2c + 1)U2 - 2cV2 = μ and (c - 2)U2 - cZ2 = -2μ of relative Pellian equations, where the parameter c and the root of unity μ are integers in the same imaginary quadratic number field . We show that for |c| ≥ 3 only certain values of μ yield solutions of this system, and solve the system completely for |c| ≥ 1544686. Furthermore we will consider the related relative Thue equation and solve it by the method of Tzanakis under the same assumptions.
20 citations
TL;DR: Self-dual codes over the Galois ring GR(4,2) are investigated, of special interest are quadratic double circulant codes and Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.
Abstract: Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.
19 citations
TL;DR: For a finite set A of positive integers, this paper studied the partition function pA(n), which enumerates the partitions of the positive integer n into parts in A, and gave simple proofs of some known and unknown identities and congruences.
Abstract: For a finite set A of positive integers, we study the partition function pA(n). This function enumerates the partitions of the positive integer n into parts in A. We give simple proofs of some known and unknown identities and congruences for pA(n). For n in a special residue class, pA(n) is a polynomial in n. We examine these polynomials for linear factors, and the results are applied to a restricted m-ary partition function. We extend the domain of pA and prove a reciprocity formula with supplement. In closing we consider an asymptotic formula for pA(n) and its refinement.
17 citations
TL;DR: For positive integers n and k, it is possible to choose primes P1, P2,…, Pk such that Pi | (n + i) for 1 ≤ i ≤ k whenever n + 1, n + 2, etc., n + k are all composites and n ≤ 1.9 × 1010 as mentioned in this paper.
Abstract: For positive integers n and k, it is possible to choose primes P1, P2,…, Pk such that Pi | (n + i) for 1 ≤ i ≤ k whenever n + 1, n + 2,…, n + k are all composites and n ≤ 1.9 × 1010. This provides a numerical verification of Grimm's Conjecture.
15 citations
TL;DR: In this article, a weak form of the reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of a CM number field is proposed.
Abstract: Let p be an odd prime. For any CM number field K containing a primitive pth root of unity, class field theory and Kummer theory put together yield the well known reflection inequality λ+ ≤ λ- between the "plus" and "minus" parts of the λ-invariant of K. Greenberg's classical conjecture predicts the vanishing of λ+. We propose a weak form of this conjecture: λ+ = λ- if and only if λ+ = λ- = 0, and we prove it when K+ is abelian, p is totally split in K+, and certain conditions on the cohomology of circular units are satisfied (e.g. in the semi-simple case).
TL;DR: In this article, the image of the theta series under the Hecke operators T(p)2 and (1 ≤ j ≤ n ≤ 2k) was realized as a sum of thets attached to certain sublattices of a rank 2k ℤ-lattice L equipped with a positive definite quadratic form.
Abstract: We apply the Hecke operators T(p)2 and (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)2.
TL;DR: In this article, it was shown that for some constant K > 0, the lower bound for the distribution of odd values of τk(n) is equivalent to a lower bound on the number of partitions of n. This estimation slightly improves a preceding result of S. Ahlgren who obtained the above lower bounds for K = 0.
Abstract: Let p(n) denote the number of partitions of n, and for i = 0 (resp. 1), Ai(x) denote the number of n ≤ x such that p(n) is even (resp. odd). In this paper, it is proved that for some constant K > 0, holds for x large enough. This estimation slightly improves a preceding result of S. Ahlgren who obtained the above lower bound for K = 0. Let and ; the main tool is a result of J.-P. Serre about the distribution of odd values of τk(n). Effective lower bounds for A0(x) and A1(x) are also given.
TL;DR: In this paper, the authors introduce the notion of caliber, cal(A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B.
Abstract: We introduce the notion of caliber, cal(A, B), of a strictly increasing sequence of natural numbers A with respect to another one B, as the limit inferior of the ratio of the nth term of A to that of B. We further consider the limit superior t(A) of the average order of the number of representations of an integer as a sum of two elements of A. We give some basic properties of each notion and we relate the two together, thus yielding a generalization, of the form t(A) ≤ t(B)/cal(A, B), of a result of Cassels specific to the case where A is an additive basis of the natural numbers and B is the sequence of perfect squares. We also provide some formulas for the computation of t(A) in a large class of cases, and give some examples.
TL;DR: In this article, the authors consider the number of r-tuples of square-free numbers in a short interval and prove that it cannot be much bigger than the expected value and also establish an asymptotic formula if the interval is not very short.
Abstract: We consider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also establish an asymptotic formula if the interval is not very short.
TL;DR: A new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers ℤM is presented.
Abstract: The nonlinear congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present a new type of discrepancy bound for sequences of s-tuples of successive nonlinear congruential pseudorandom numbers over a ring of integers ℤM.
TL;DR: For two particular classes of elliptic curves, Ahlgren and Ono as mentioned in this paper established congruences relating the coefficients of their corresponding modular forms to combinatorial objects, which resemble a super-congruence for the Apery numbers conjectured by Beukers and proved by Ahngren and ono in [1].
Abstract: For two particular classes of elliptic curves, we establish congruences relating the coefficients of their corresponding modular forms to combinatorial objects. These congruences resemble a supercongruence for the Apery numbers conjectured by Beukers and proved by Ahlgren and Ono in [1]. We also consider the trace Tr2k(Γ0(N), n) of the Hecke operator Tn acting on the space of cusp forms S2k(Γ0(N)). We show that for (n, N) = 1, these traces interpolate p-adically in the weight aspect.
TL;DR: In this article, the authors solved the problem of finding pairs of Heron triangles and rectangles, such as (5,5,6) and [2 × 6] with a common area and a common perimeter, by parametrized by a family of elliptic curves.
Abstract: We solve a problem of Bill Sands, to find pairs of Heron triangles and rectangles, such as (5,5,6) & [2 × 6] or (13,20,21) & [6 × 21] which have a common area and a common perimeter. The original question was posed for right-angled triangles, but there are no nondegenerate such. There are infinitely many isosceles triangles and these have been exhibited by Guy. Here we solve the general problem; the triangle-rectangle pairs are parametrized by a family of elliptic curves.
TL;DR: In this article, the authors used Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integer solutions to the quartic Diophantine equation (m2 + m + 1)X4 - (m 2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2+ 4m + 3).
Abstract: Bumby proved that the only positive integer solutions to the quartic Diophantine equation 3X4 - 2Y2 = 1 are (X, Y) = (1, 1),(3, 11). In this paper, we use Thue's hypergeometric method to prove that, for each integer m ≥ 1, the only positive integers solutions to the Diophantine equation (m2 + m + 1)X4 - (m2 + m)Y2 = 1 are (X,Y) = (1, 1),(2m + 1, 4m2 + 4m + 3).
TL;DR: In this article, the authors used the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series.
Abstract: We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q2; q3)∞/(q; q3)∞ and . In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.
TL;DR: In this article, the authors find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four-dimensional cubic hypersurfaces.
Abstract: In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four-dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of hypergeometric series on certain CM points. Our methods are based on the calculation of the Picard–Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.
TL;DR: In this article, it was shown that the zeros of the Eisenstein series Ek in the standard fundamental domain other than i and ρ are transcendental for a general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz.
Abstract: Kohnen showed that the zeros of the Eisenstein series Ek in the standard fundamental domain other than i and ρ are transcendental. In this paper, we obtain similar results for a more general class of modular forms, using the earlier works of Kanou, Kohnen and the recent work of Getz.
TL;DR: In this paper, the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.
Abstract: We state a form of the modular relation in which the functional equation appears in the form of an expression of one Dirichlet series in terms of the other multiplied by the quotient of gamma functions and illustrate it by some concrete examples including the results of Koshlyakov, Berndt and Wigert and Bellman.
TL;DR: In this paper, the authors studied the distribution of densities of a set B of nonnegative integers such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways.
Abstract: If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet–Thue–Morse sequence, {0} ∪ {2n:n ∈ ℕ}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that has an odd number of partitions, then B is the set of pentagonal numbers {n(3n + 1)/2:n ∈ ℤ}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.
TL;DR: In this article, the authors considered an analogue of Artin's primitive root conjecture for units in real quadratic fields and showed that there is at least one unit in each of these fields which satisfies the above version.
Abstract: We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.
TL;DR: In this article, a generalization of Carlitz's results was obtained for Nq when -1 is a power of p modulo dD, where D = lcm[d1,…,dn], $d={\rm gcd}(mj,q -1), 1 ≤ j ≤ n.
Abstract: Let Nq be the number of solutions to the equation \[ (x_1^{m_1}+\cdots+x_n^{m_n})^k=ax_1 \cdots x_n \] over the finite field 𝔽q = 𝔽ps. Carlitz found formulas for Nq when m1 = ⋯ = mn = 1, k = 2, n = 3 or 4, p > 2; and when m1 = ⋯ = mn = 2, k = 1, n = 3 or 4, p > 2. In earlier papers, we obtained some generalizations of Carlitz's results. In this paper, we find formulas for Nq when -1 is a power of p modulo dD, where D = lcm[d1,…,dn], $d={\rm gcd}(\sum_{j=1}^nM/m_j-kM,(q-1)/D)$, M = lcm[m1,…,mn], dj = gcd(mj,q - 1), 1 ≤ j ≤ n.
TL;DR: The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given in this article.
Abstract: The complete description of the discrete part of the Lagrange and Markov spectra of the imaginary quadratic fields with discriminants -20 and -24 are given. Farey polygons associated with the extended Bianchi groups Bd, d = 5, 6, are used to reduce the problem of finding the discrete part of the Markov spectrum for the group Bd to the corresponding problem for one of its maximal Fuchsian subgroup. Hermitian points in the Markov spectrum of Bd are introduced for any d. Let H3 be the upper half-space model of the three-dimensional hyperbolic space. If ν is a hermitian point in the spectrum, then there is a set of extremal geodesics in H3 with diameter 1/ν, which depends on one continuous parameter. This phenomenon does not take place in the hyperbolic plane.
TL;DR: In this article, it was shown that the residue difference sets of power n are not known to exist for n = 2,4,6, as do similar sets that include the zero element.
Abstract: Qualified residue difference sets of power n are known to exist for n = 2,4,6, as do similar sets that include the zero element. Both classes of sets are proved non-existent for n = 8.
TL;DR: In this paper, the authors generalize this result to the context of holomorphic cusp forms on the upper half space and derive a Dirichlet L-function at s = 2.
Abstract: A classical theorem of Ramanujan relates an integral of Dedekind eta-function to a special value of a Dirichlet L-function at s = 2. Ahlgren, Berndt, Yee and Zaharescu have generalized this result [1]. In this paper, we generalize this result to the context of holomorphic cusp forms on the upper half space.
TL;DR: In this article, a geometric lower bound for the height of a hypersurface defined and irreducible over the rationals is given, which is an analogue of a result of F. Amoroso and S. David.
Abstract: Here we are concerned on Bogomolov's problem for hypersurfaces; we give a geometric lower bound for the height of a hypersurface of (i.e. without condition on the field of definition of the hypersurface) which is not a translate of an algebraic subgroup of . This is an analogue of a result of F. Amoroso and S. David who give a lower bound for the height of non-torsion hypersurfaces defined and irreducible over the rationals.