Primes Represented by Binary Cubic Forms

Let f(x, y) be a binary cubic form with integral rational coefficients,and suppose that the polynomial f(x, y) is irreducible in Q[x,y] and no prime divides all the coefficients of f. We provethat the set f Z⁽²⁾ contains infinitely many primes unless f(a,b) is even for each (a,b) in Z², in which case the set contains infinitely many primes. 2000Mathematical Subject Classification: primary 11N32; secondary11N36, 11R44.     

A Three Squares Theorem with almost Primes

As an application of the vector sieve and uniform estimateson the Fourier coefficients of cusp forms of half-integral weight,it is shown that any sufficiently large number n 3 (mod 24)with 5 n is expressible as a sum of three squares of integershaving at most 521 prime factors. 2000 Mathematics Subject Classification11P05, 11N36, 11N75, 11E25.     

On a Problem of Brocard

It is proved that, if P is a polynomial with integer coefficients,having degree 2, and 1 > > 0, then n(n – 1) ...(n – k + 1) = P(m) has only finitely many natural solutions(m,n,k), n k > n, provided that the abc conjecture is assumedto hold under Szpiro's formulation. 2000 Mathematics SubjectClassification 11D75, 11J25, 11N13.     

Traces of Singular Values and Borcherds Products

Let p be a prime for which the congruence group ₀(p)* is ofgenus zero, and let be the corresponding Hauptmodul. Let f be a nearly holomorphic modularform of weight 1/2 on ₀(4p) which satisfies some congruencecondition on its Fourier coefficients. We interpret f as a vectorvalued modular form. Applying Borcherds lifting of vector valuedmodular forms we construct infinite products associated to and relate them to Zagier's traceformula for the singular values of . 2000 Mathematics Subject Classification 11F03, 11F30 (primary);11F22, 11F37, 11F50 (secondary).     

Differences Between Consecutive Primes

Let p_n be the nth prime. Then this paper is concerned with provingthe following result on the distribution of consecutive primes. The exponent of x in this theorem improves on the work of Heath-Brownwho proved (1) with exponent . Under the Riemann hypothesisone can prove (1) with exponent .The proof of the theorem startswith the Heath-Brown–Linnik identity which leads to aformula giving the number of primes in an interval in termsof coefficients of certain Dirichlet series. I then estimatethe coefficients by using, among other things, the informationwhich can be gained from Montgomery's mean value theorem andHuxley's version of the Hal' asz lemma. Furthermore, by usingfamiliar sieve arguments I am able to discard some of the coefficientsallowing us to gain an improvement over the previous resultof Heath-Brown. 1991 Mathematics Subject Classification: 11N05.     

The Solubility of Diagonal Cubic Diophantine Equations

This paper proves conditional existence results for non-trivialsolutions of the equation where the coefficients a_i and the unknowns X_i are taken to berational integers. No such results were previously known for n6. The proofs useelementary facts about the 3-descent procedure for ellipticcurves of the form E_A: X³ + Y³ = AZ³. Thus, when n=4, and the a_i are each prime, and are all congruentto 2 modulo 3, it is shown that (*) will have non-trivial solutions,providing that the Selmer conjecture holds for the curves E_A.One may replace the Selmer conjecture by an appropriate formof the Generalized Riemann Hypothesis, when n=5 and the a_i areagain taken to be primes, all congruent to 8 modulo 9. Finally,when n=5, one may require only that the a_i be square-free andcoprime to 3, providing one assumes both the Selmer conjectureand a special case of Schinzel's conjecture (on the representationof primes by cubic polynomials). 1991 Mathematics Subject Classification:11D25, 11G05, 14G05.     

On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Duality involving the mock theta function f(q)

We show that the coefficients of Ramanujan's mock theta functionf(q) are the first non-trivial coefficients of a canonical sequenceof modular forms. This fact follows from a duality which equatescoefficients of the holomorphic projections of certain weight1/2 Maass forms with coefficients of certain weight 3/2 modularforms. This work depends on the theory of Poincaré series,and a modification of an argument of Goldfeld and Sarnak onKloosterman–Selberg zeta functions.     

On the Representation of Primes by Cubic Polynomials in Two Variables

In an earlier paper (see Proc. London Math. Soc. (3) 84 (2002)257–288) we showed that an irreducible integral binarycubic form f(x, y) attains infinitely many prime values, providingthat it has no fixed prime divisor. We now extend this resultby showing that f(m, n) still attains infinitely many primevalues if m and n are restricted by arbitrary congruence conditions,providing that there is still no fixed prime divisor. Two immediate consequences for the solvability of diagonal cubicDiophantine equations are given. 2000 Mathematics Subject Classification11N32 (primary), 11N36, 11R44 (secondary).     

Predictor-corrector Methods with Improved Absolute Stability Regions

Generalized predictor-corrector methods with an extended regionof absolute stability are studied. Choosing an extrapolation-predictorand a backward-differentiation-corrector, methods of ordersup to 6 are constructed. The real stability boundaries of thesemethods are of magnitude m², where m is the number of right-handside evaluations per integration step. The coefficients of themethods can be generated during the computation for arbitraryvalues of m. The storage requirements are limited and are independentof m.     

Asymptotic Expansions for the Coefficients of eP(z)

If P is a real polynomial with non-negative coefficients, thenone can obtain a complete asymptotic expansion for the nth coefficientof e^P(z). The proof requires the following result: the coefficientsin the Stirling expansion for the gamma function are all differentfrom zero. Department of Mathematics and Computer Science Drexel UniversityPhiladelphia PA 19104, USA     

Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

On Waring's Problem for Cubes and Smooth Weyl Sums

Non-trivial estimates for fractional moments of smooth cubicWeyl sums are developed. Complemented by bounds for such sumsof use on both the major and minor arcs in a Hardy-Littlewooddissection, these estimates are applied to derive an upper boundfor the sth moment of the smooth cubic Weyl sum of the expectedorder of magnitude as soon as s> 7.691. Related argumentsdemonstrate that all large integers n are represented as thesum of eight cubes of natural numbers, all of whose prime divisorsare at most exp (c(log nlog log n)^1/2}, for a suitable positivenumber c. This conclusion improves a previous result of G. Harcosin which nine cubes are required. 1991 Mathematics Subject Classification:11P05, 11L15, 11P55.     

The Mathieu Group M11 and the Modular Curve X(11)

In this paper, we prove that the modular curve X(11) over afield of characteristic 3 admits the Mathieu group M₁₁ as anautomorphism group. We also examine some aspects of the geometryof the curve X(11) in characteristic 3. In particular, we showthat every point of the curve is a point of inflection, thecurve has 110 hyperflexes and there are no inflectional trianglesand 11232 inflectional pentagons, of which 144 are self-conjugate.The hyperflexes correspond to the supersingular elliptic curves.We comment on the relationship of Ward's quadrilinear invariantfor M₁₂ to our work and announce for the first time the equationsfor Klein's A-curve of level 11. We also comment on the relationof our work to some unpublished work of Bott and Tate. 1991Mathematics Subject Classification: 11F32, 11G20, 14G10, 14H10,14N10, 20B25, 20C34.     

Spectral Analysis of Higher-Order Differential Operators II: Fourth-Order Equations

We investigate the location and nature of the spectrum of thefourth-order self-adjoint equation (p₀ y')'+(p₁ y')'+qy=zwy subject to certain asymptotic assumptions on the coefficients.The main tools are the theory of asymptotic integration andthe Titchmarsh–Weyl M-matrix. Asymptotic integration yieldsasymptotic formulae for the solutions of the differential equationwhich are then used to derive properties of the M-matrix. Thecharacterisation of spectral properties in terms of the boundarybehaviour of M leads to the desired results.     

On the Number of Carmichael Numbers up to x

It is shown that, for all large x, there are more than x^0.33Carmichael numbers up to x, improving on the ground-breakingwork of Alford, Granville and Pomerance, who were the firstto demonstrate that there are infinitely many such numbers.The same basic construction as that employed by these authorsis used, but a slight modification enables a stronger resulton primes in arithmetic progressions based on a sieve methodto be employed. 2000 Mathematics Subject Classification 11N13(primary), 11N36 (secondary).     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

Congruences De Sommes De Chiffres De Valeurs Polynomiales

Let m, g, q N with q 2 and (m, q – 1) = 1. For n N,denote by s_n(n) the sum of digits of n in the q-ary digitalexpansion. Given a polynomial f with integer coefficients, degreed 1, and such that f(N) N, it is shown that there exists C= C(f, m, q) > 0 such that for any g Z, and all large N, In the special case m = q = 2 and f(n)= n², the value C = 1/20 is admissible. 2000 Mathematics SubjectClassification 11B85 (primary), 11N37, 11N69 (secondary).     

On the Topology of Singularities of the Set of Supporting Hyperplanes of a Smooth Submanifold in an Affine Space

Many new universal relations are obtained between the Eulernumbers of manifolds of singular supporting hyperplanes of anarbitrary generic smooth closed k-dimensional submanifold inRⁿ where n 7 or k = 1. These relations are applied to Barner-convexcurves in an odd-dimensional space Rⁿ. A universal (nontrivial)linear relation is established between the numbers of singularsupporting hyperplanes of various types but of the same totalmultiplicity n of tangency with a given generic smooth closedconnected Barner-convex curve in Rⁿ. The coefficients of thisrelation are defined by Catalan numbers.     

Integrals and Areas Expressed as Bilinear Forms of Discrete Defining Data

An ordered set of discrete data-pairs (x_k, y_k) is supposed givenfor k = 1(1)n, derived perhaps as a consequence of experiment,and indicating some form of relationship between x and y. Itis required that these data be interpolated in some systematicway so as to establish this relationship in the form of a rectifiablecurve in the (x, y) plane, and that y is then to be integratedwith respect to x (or vice versa), thereby defining an areain this plane. We shall here consider interpolation schemeswhere by this integral can be expressed as a bilinear form _jKC_jkY_jx_k,the "weighting" coefficients C_jk being numbers independent ofthe data values (though not necessarily of their number, n). The expressions obtained sepcialize to Gregory formulae, andti Simpson's Rule and other forms of Cotes formulae in suitablecontexts, but the information is primarily of use in dealingwith unequally-spaced data values, or in estumating the areawithin a closed curve. The simplicity of the numerical algorithmis clearly unaffected by the geometric complexity of the interpolatedcurve. Values for the weighting coefficients are proposed, and theresults of a numerical experiment are described which teststhe applicability of the formulae described.