Real zeros of real odd Dirichlet -functions

Let be a real odd Dirichlet character of modulus , and let be the associated Dirichlet -function. As a consequence of the work of Low and Purdy, it is known that if and , , , then has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if , then has no positive real zeros.

     

All numbers whose positive divisors have integral harmonic mean up to

A positive integer is said to be harmonic when the harmonic mean of its positive divisors is an integer. Ore proved that every perfect number is harmonic. No nontrivial odd harmonic numbers are known. In this article, the list of all harmonic numbers with is given. In particular, such harmonic numbers are all even except .

     

Elliptic curves with nonsplit mod representations

We calculate explicitly the -invariants of the elliptic curves corresponding to rational points on the modular curve by giving an expression defined over of the -function in terms of the function field generators and of the elliptic curve . As a result we exhibit infinitely many elliptic curves over with nonsplit mod representations.

     

The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if is an abelian extension of relative degree , an odd prime, we prove the -part of the Brumer-Stark conjecture for all odd primes with belonging to a wide class of base fields. In the same setting, we study the -part and -part of Brumer-Stark with no special restriction on and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

     

-estimate for the discrete Plateau Problem

In this paper we prove the convergence rates for a fully discrete finite element procedure for approximating minimal, possibly unstable, surfaces.

Originally this problem was studied by G. Dziuk and J. Hutchinson. First they provided convergence rates in the and norms for the boundary integral method. Subsequently they obtained the convergence estimates using a fully discrete finite element method. We use the latter framework for our investigation.

     

Class numbers of some abelian extensions of rational function fields

Let be a monic irreducible polynomial. In this paper we generalize the determinant formula for of Bae and Kang and the formula for of Jung and Ahn to any subfields of the cyclotomic function field By using these formulas, we calculate the class numbers of all subfields of when and are small.

     

Linear algebra algorithms for divisors on an algebraic curve

We use an embedding of the symmetric th power of any algebraic curve of genus into a Grassmannian space to give algorithms for working with divisors on , using only linear algebra in vector spaces of dimension , and matrices of size . When the base field is finite, or if has a rational point over , these give algorithms for working on the Jacobian of that require field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to deal with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Florian Hess' 1999 Ph.D. thesis, which works with function fields as extensions of . However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor 1987), superelliptic curves (Galbraith, Paulus, and Smart 2002), and curves (Harasawa and Suzuki 2000); in all those cases, one can attain a complexity of .

     

Chebyshev's bias for composite numbers with restricted prime divisors

Let denote the number of primes with . Chebyshev's bias is the phenomenon for which ``more often' \pi(x;d,r)$">, than the other way around, where is a quadratic nonresidue mod and is a quadratic residue mod . If for every up to some large number, then one expects that for every . Here denotes the number of integers such that every prime divisor of satisfies . In this paper we develop some tools to deal with this type of problem and apply them to show that, for example, for every .

In the process we express the so-called second order Landau-Ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants.

     

An estimate for the number of integers without large prime factors

denotes the number of positive integers and free of prime factors y$">. Hildebrand and Tenenbaum provided a good approximation of . However, their method requires the solution to the equation , and therefore it needs a large amount of time for the numerical solution of the above equation for large . Hildebrand also showed approximates for , where and is the unique solution to . Let be defined by for 0$">. We show approximates , and also approximates , where . Using these approximations, we give a simple method which approximates within a factor in the range , where is any positive constant.

     

Bounds for computing the tame kernel

The tame kernel of the of a number field  is the kernel of some explicit map , where the product runs over all finite primes  of  and is the residue class field at . When is a set of primes of , containing the infinite ones, we can consider the -unit group  of . Then has a natural image in . The tame kernel is contained in this image if  contains all finite primes of  up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in  all primes with norm up to  , where  is the discriminant of . Using this bound, one can find explicit generators for the tame kernel, and a ``long enough' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough' means. However, using theorems from Keune, we can show that the tame kernel is computable.

     

Finding -strong pseudoprimes

Let be odd primes and . Put

Then we call the kernel, the triple the signature, and the height of , respectively. We call a -number if it is a Carmichael number with each prime factor . If is a -number and a strong pseudoprime to the bases for , we call a -spsp . Since -numbers have probability of error (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of (the smallest strong pseudoprime to all the first prime bases). If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement.

In this paper, we first describe an algorithm for finding -spsp(2)'s, to a given limit, with heights bounded. There are in total -spsp's with heights . We then give an overview of the 21978 - spsp(2)'s and tabulate of them, which are -spsp's to the first prime bases up to ; three numbers are spsp's to the first 11 prime bases up to 31. No -spsp's to the first prime bases with heights were found. We conjecture that there exist no -spsp's to the first prime bases with heights and so that

which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those -spsp's is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates of -spsp's and their prime factors to Miller's tests, and obtain the desired numbers. At last we speed our algorithm for finding larger -spsp's, say up to , with a given signature to more prime bases. Comparisons of effectiveness with Arnault's and our previous methods for finding -strong pseudoprimes to the first several prime bases are given.

     

Two kinds of strong pseudoprimes up to

Let be an odd composite integer. Write with odd. If either mod or mod for some , then we say that is a strong pseudoprime to base , or spsp() for short. Define to be the smallest strong pseudoprime to all the first prime bases. If we know the exact value of , we will have, for integers , a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the are known for . Conjectured values of were given by us in our previous papers (Math. Comp. 72 (2003), 2085-2097; 74 (2005), 1009-1024).

The main purpose of this paper is to give exact values of for ; to give a lower bound of : ; and to give reasons and numerical evidence of K2- and -spsp's to support the following conjecture: for any , where (resp. ) is the smallest K2- (resp. -) strong pseudoprime to all the first prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: with primes and ; and that a -spsp is an spsp and a Carmichael number of the form: with each prime factor mod .)

     

Asymptotics of recurrence coefficients for orthonormal polynomials on the line--Magnus's method revisited

We use Freud equations to obtain the main term in the asymptotic expansion of the recurrence coefficients associated with orthonormal polynomials for weights on the real line where is an even polynomial of fixed degree with nonnegative coefficients or where . Here for some real -1$">.

     

Computing weight modular forms of level

For a prime we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space of modular forms of weight and level . For we define a special Hecke stable subspace of which contains the space of modular forms with CM by the ring of integers of and we describe the calculation of the corresponding Brandt matrices.

     

On the embedding problem for representations

Let denote the double cover of corresponding to the element in where transpositions lift to elements of order and the product of two disjoint transpositions to elements of order . Given an elliptic curve , let denote its -torsion points. Under some conditions on elements in correspond to Galois extensions of with Galois group (isomorphic to) . In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for having a Galois extension with gives a homomorphism . As a corollary we can prove (if has conductor divisible by few primes and high rank) the existence of -dimensional representations of the absolute Galois group of attached to and use them in some examples to construct modular forms mapping via the Shimura map to (the modular form of weight attached to) .

     

Convergence of nonconforming -cycle and -cycle multigrid algorithms for second order elliptic boundary value problems

The convergence of -cycle and -cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems.

     

Four-dimensional lattice rules generated by skew-circulant matrices

We introduce the class of skew-circulant lattice rules. These are -dimensional lattice rules that may be generated by the rows of an skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree . We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for we have constructed an infinite sequence of rules that has a limit rho index of . This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.

     

Minimality and other properties of the width- nonadjacent form

Let be an integer and let be the set of integers that includes zero and the odd integers with absolute value less than . Every integer can be represented as a finite sum of the form , with , such that of any consecutive 's at most one is nonzero. Such representations are called width- nonadjacent forms (-NAFs). When these representations use the digits and coincide with the well-known nonadjacent forms. Width- nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. We provide some new results on the -NAF. We show that -NAFs have a minimal number of nonzero digits and we also give a new characterization of the -NAF in terms of a (right-to-left) lexicographical ordering. We also generalize a result on -NAFs and show that any base 2 representation of an integer, with digits in , that has a minimal number of nonzero digits is at most one digit longer than its binary representation.

     

On the greatest prime factor of with effective constants

Let denote a prime. In this article we provide the first published lower bounds for the greatest prime factor of exceeding in which the constants are effectively computable. As a result we prove that it is possible to calculate a value such that for every x_0$"> there is a with the greatest prime factor of exceeding . The novelty of our approach is the avoidance of any appeal to Siegel's Theorem on primes in arithmetic progression.

     

Comparison theorems of Kolmogorov type and exact values of -widths on Hardy-Sobolev classes

Let be a strip in complex plane. denotes those -periodic, real-valued functions on which are analytic in the strip and satisfy the condition , . Osipenko and Wilderotter obtained the exact values of the Kolmogorov, linear, Gel'fand, and information -widths of in , , and 2-widths of in , , .

In this paper we continue their work. Firstly, we establish a comparison theorem of Kolmogorov type on , from which we get an inequality of Landau-Kolmogorov type. Secondly, we apply these results to determine the exact values of the Gel'fand -width of in , . Finally, we calculate the exact values of Kolmogorov -width, linear -width, and information -width of in , , .