Results 1  10
of
22
The distribution of values of L(1, χd
 Geom. Funct. Anal
"... Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus d. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with d  ≤ x. Our main concern is to compare the distribution ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
Throughout this paper d will denote a fundamental discriminant, and χd the associated primitive real character to the modulus d. We investigate here the distribution of values of L(1, χd) as d varies over all fundamental discriminants with d  ≤ x. Our main concern is to compare the distribution of values of L(1, χd) with the distribution of “random Euler
LandauSiegel zeroes and black hole entropy,” arXiv:hepth/9903267
"... There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions. ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
There has been some speculation about relations of Dbrane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet Lfunctions.
The efficiency and security of a real quadratic field based key exchange protocol
 DE GRUYTER
, 2001
"... Most cryptographic key exchange protocols make use of the presumed difficulty of solving the discrete logarithm problem (DLP) in a certain finite group as the basis of their security. Recently, real quadratic number fields have been proposed for use in the development of such protocols. Breaking suc ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
Most cryptographic key exchange protocols make use of the presumed difficulty of solving the discrete logarithm problem (DLP) in a certain finite group as the basis of their security. Recently, real quadratic number fields have been proposed for use in the development of such protocols. Breaking such schemes is known to be at least as difficult a problem as integer factorization; furthermore, these are the first discrete logarithm based systems to utilize a structure which is not a group, specifically the collection of reduced ideals which belong to the principal class of the number field. For this structure the DLP is essentially that of determining a generator of a given principal ideal. Unfortunately, there are a few implementationrelated disadvantages to these schemes, such as the need for high precision floating point arithmetic and an ambiguity problem that requires a short, second round of communication. In this paper we describe work that has led to the resolution of some of these difficulties. Furthermore, we discuss the security of the system, concentrating on the most recent techniques for solving the DLP in a real quadratic number field.
Large character sums: Pretentious characters and the PolyaVinogradov theorem
 J. Amer. Math. Soc
"... The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
The best bound known for character sums was given independently by G. Pólya and I.M. Vinogradov in 1918 (see [4], p.135137). For any nonprincipal Dirichlet character χ (mod q) we let M(χ): = max
Archimedes' Cattle Problem
 American Mathematical Monthly
, 1998
"... this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of the problem is
CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
, 2003
"... The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and GrossZagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work for other small class numbers, up to 100, say.” However, more than ten years later, after doing N =5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the GoldfeldOesterlé work, which used an elliptic curve Lfunction with an order 3 zero at the central critical point, to instead consider Dirichlet Lfunctions with lowheight zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of MontgomeryWeinberger. Our method is still quite computerintensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large “exceptional modulus” of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.
Experimental results on class groups of real quadratic fields (Extended Abstract)
 ALGORITHMIC NUMBER THEORY  ANTSIII, LECTURE NOTES IN COMPUTER SCIENCE 1423
, 1998
"... In an effort to expand the body of numerical data for real quadratic fields, we have computed the class groups and regulators of all real quadratic fields with discriminant ∆<10 9. We implemented a variation of the group structure algorithm for general finite Abelian groups described in [2] in the C ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In an effort to expand the body of numerical data for real quadratic fields, we have computed the class groups and regulators of all real quadratic fields with discriminant ∆<10 9. We implemented a variation of the group structure algorithm for general finite Abelian groups described in [2] in the C++ programming language using builtin types together with a few routines from the LiDIA system [12]. This algorithm will be described in more detail in a forthcoming paper. The class groups and regulators of all 303963581 real quadratic fields were computed on 20 workstations (SPARCclassics, SPARC4’s, and SPARCultra’s) by executing the computation for discriminants in intervals of length 10 5 on single machines and distributing the overall computation using PVM [8]. The entire computation took just under 246 days of CPU time (approximately 3 months real time), an average of 0.07 seconds per field. In this contribution, we present the results of this experiment, including data supporting the truth of Littlewood’s bounds on the function L (1,χ∆) [13]and Bach’s bound on the maximum norm of the prime ideals required to generate the class group [1]. Data supporting several of the CohenLenstra heuristics [6,7] is presented, including results on the percentage of noncyclic odd parts of class groups, percentages of odd parts of class numbers equal to small odd integers, and percentages of class numbers divisible by small primes p. We also give new examples of irregular class groups, including examples for primes p ≤ 23 and one example of a rank 3 5Sylow subgroup (3 noncyclic factors), the first example of a real quadratic class group which has a pSylow subgroup with rank greater than 2 and p>3. 1 The L (1,χ¡) Function Much interest has been shown in extreme values of the L (1,χ∆) function [3,14,10,4]. A result of Littlewood [13] and Shanks [14] shows that under the
DISTRIBUTION OF VALUES OF LFUNCTIONS AT THE EDGE OF THE CRITICAL STRIP
"... Abstract. We prove several results on the distribution of values of Lfunctions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power Lfunctions of holomorphic cusp forms in the level aspe ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
Abstract. We prove several results on the distribution of values of Lfunctions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power Lfunctions of holomorphic cusp forms in the level aspect (assuming the automorphy of these Lfunctions) at s = 1, functions in the Selberg class (in the height aspect), and quadratic twists of a fixed GL(m)/Qautomorphic cusp form at s = 1. Introduction and statement of results Values of Lfunctions at the edge of the critical strip are interesting objects that encode deep arithmetic information. For example, the nonvanishing of the Riemann zeta function ζ(s) on the line Re(s) = 1 implies the prime number theorem proved by Hadamard and de
Cryptography Based on Number Fields with Large Regulator
, 2000
"... We explain a variant of the FiatShamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We explain a variant of the FiatShamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also
Rabinowitsch Revisited
"... this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet Lfunction which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n