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.

     

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.

     

Some new kinds of pseudoprimes

We define some new kinds of pseudoprimes to several bases, which generalize strong pseudoprimes. We call them Sylow -pseudoprimes and elementary Abelian -pseudoprimes. It turns out that every which is a strong pseudoprime to bases 2, 3 and 5, is not a Sylow -pseudoprime to two of these bases for an appropriate prime

We also give examples of strong pseudoprimes to many bases which are not Sylow -pseudoprimes to two bases only, where or

     

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.

     

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.

     

High rank elliptic curves with torsion group

We develop an algorithm for bounding the rank of elliptic curves in the family , all of them with torsion group and modular invariant . We use it to look for curves of high rank in this family and present four such curves of rank  and of rank .

     

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 .

     

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$">.

     

Deciding the nilpotency of the Galois group by computing elements in the centre

We present a new algorithm for computing the centre of the Galois group of a given polynomial along with its action on the set of roots of , without previously computing the group. We show that every element in the centre is representable by a family of polynomials in . For computing such polynomials, we use quadratic Newton-lifting and truncated expressions of the roots of over a -adic number field. As an application we give a method for deciding the nilpotency of the Galois group. If is irreducible with nilpotent Galois group, an algorithm for computing it is proposed.

     

Optimal quadrature for Haar wavelet spaces

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, . The asymptotic orders of the errors are derived for the case of the scrambled -nets and -sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands .

     

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.

     

Random Fibonacci sequences and the number

For the familiar Fibonacci sequence (defined by , and for ), increases exponentially with at a rate given by the golden ratio . But for a simple modification with both additions and subtractions - the random Fibonacci sequences defined by , and for , , where each sign is independent and either or - with probability - it is not even obvious if should increase with . Our main result is that

with probability . Finding the number involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.

     

On equivariant global epsilon constants for certain dihedral extensions

We consider a conjecture of Bley and Burns which relates the epsilon constant of the equivariant Artin -function of a Galois extension of number fields to certain natural algebraic invariants. For an odd prime number , we describe an algorithm which either proves the conjecture for all degree dihedral extensions of the rational numbers or finds a counterexample. We apply this to show the conjecture for all degree dihedral extensions of . The correctness of the algorithm follows from a finiteness property of the conjecture which we prove in full generality.

     

A sensitive algorithm for detecting the inequivalence of Hadamard matrices

A Hadamard matrix of side is an matrix with every entry either or , which satisfies . Two Hadamard matrices are called equivalent if one can be obtained from the other by some sequence of row and column permutations and negations. To identify the equivalence of two Hadamard matrices by a complete search is known to be an NP hard problem when increases. In this paper, a new algorithm for detecting inequivalence of two Hadamard matrices is proposed, which is more sensitive than those known in the literature and which has a close relation with several measures of uniformity. As an application, we apply the new algorithm to verify the inequivalence of the known inequivalent Hadamard matrices of order ; furthermore, we show that there are at least pairwise inequivalent Hadamard matrices of order . The latter is a new discovery.

     

On the total number of prime factors of an odd perfect number

We say is perfect if , where denotes the sum of the positive divisors of . No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form , where , , ..., are distinct primes and . We prove that if or for all , , then . We also prove as our main result that , where . This improves a result of Sayers given in 1986.

     

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.

     

Biquadratic reciprocity and a Lucasian primality test

Let be the sequence defined from a given initial value, the seed, , by the recurrence . Then, for a suitable seed , the number (where is odd) is prime iff . In general depends both on and on . We describe a slight modification of this test which determines primality of numbers with a seed which depends only on , provided . In particular, when , odd, we have a test with a single seed depending only on , in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for , Math. Comp. 61 (1993), 97-109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity.

     

A complete description of Golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length with complementary autocorrelation is at worst a problem. Andres, in his 1977 master's thesis, developed an algorithm which reduced this to a search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length generated by a primitive pair of length . Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.