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This generalizes a result of Cho. In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Horizons of Fractal Geometry and Complex Dimensions. In celebration of the 60th birthday of Michel Lapidus. Niemeyer, E. Pearse, J. Rock, T. Samuel, eds. Abstract Consider the Laplacian operator on a bounded open domain in Euclidean space with Dirichlet boundary conditions.

Read Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday book edition (23 August ); Language: English; ISBN Buy Analytic Number Theory: In Honor of Helmut Maier's 60th Birthday on edition (August 23, ); Language: English; ISBN

The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser--Sarnak and Katz--Schaps--Vishne.

Algebra 45 , Abstract We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements.

The proofs involve an interplay of commutative algebra, algebraic number theory and additive number theory. We also study -- in part, out of necessity -- the upper order of the size of the torsion subgroup of various restricted classes of CM elliptic curves over number fields.

Finite Fields and their Applications 47 , Today, several proofs are known for both of these theorems. In , Jameson gave a short, simple proof of Lagrange's theorem based on an extension of Thue's lemma to the Gaussian integers. We establish a simple-to-check sufficient condition for a linear combination of multiplicative functions to be nonclustering , meaning not clustering anywhere.

This provides a means of generating new families of arithmetic functions possessing continuous distribution functions. As a specific application, we resolve a problem posed recently by Luca and Pomerance. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner. Mathematika 64 , In , Dirichlet gave a beautifully simple proof of this result using only basic facts about ternary quadratic forms.

A random algorithm for this problem of similar complexity was proposed by Rabin and Shallit in ; however, their analysis depended on both the ERH and on certain conjectures of Hardy--Littlewood type. Quaestiones Mathematicae 42 , Here we study properties of Golomb topologies.

Abstract We formulate the notion of typical boundedness of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the family all abelian varieties defined over number fields of degree lying in a set of arbitrarily small density. Integers 18A , article A15, 16 pages electronic. The other two algorithms admit slightly worse runtime estimates but are unconditional, in the sense that no unproved hypotheses are used in the proof of correctness or the running-time analysis.

In this paper we explain how to modify their algorithms to do slightly better. Abstract The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths resp. By work of Reid, this spectrum determines the commensurability class of the 2â€”manifold resp. We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets resp. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3â€”manifolds.

These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest. Abstract We prove a nontrivial estimate for the size of the least rational prime that splits completely in the ring of integers of certain families of nonabelian Galois number fields.

Our result complements results of Linnik and Vinogradov and of Pollack who studied this problem in the quadratic and abelian number field settings, respectively. This lower bound is implicit in work of Banks--Friedlander--Pomerance--Shparlinski. The proof builds on methods recently introduced by the author to study "popular subsets" for Euler's function. Glasgow Math. Abstract An oft-cited result of Peter Shiu bounds the mean value of a nonnegative multiplicative function over a coprime arithmetic progression. We prove a variant where the arithmetic progression is replaced by a sifted set.

Integers 19 , article A54, 7 pages. Abstract A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. In the present paper, that theorem is improved to what is likely to be the best possible result. We also establish that there exist infinitely many symmetric pairs of primes. Integers to appear. In a recent Math Horizons note, Matthew McMullen suggested studying triangular sums of consecutive triangular numbers.

American Institute of Mathematics, November Integers Conference.

Oberlin Pizza Lunch talk, March Stanford, February Alladi's 60th birthday, March UGA number theory seminar. Slides from my thesis defense. See also: Rational! Paul Pollack. What do you read, my lord?

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Words, words, words. Accepted Papers. An explicit approach to Hypothesis H for polynomials over finite fields Anatomy of integers.

Mixing for progressions in nonabelian groups. Quirk, Meghan H. The ranges of some familiar functions , December 8, Analytic number theory: essays in honour of Klaus Roth, W. Adleman, C. Chernykh, T.

Gallardo and O. Rahavandrainy Finite Fields and their Applications 14 , no. Simultaneous prime specializations of polynomials over finite fields Proc. A polynomial analogue of the twin prime conjecture Proc. Arithmetic properties of polynomial specializations over finite fields Acta Arith. Kobayashi and C. Pomerance J.

A remark on sociable numbers of odd order J. Revisiting Gauss's analogue of the prime number theorem for polynomials over a finite field Finite Fields and their Applications 16 , no.

Andrew Granville - 1/3 The pretentious approach to analytic number theory

On polynomial rings with a Goldbach property American Math. On Dickson's theorem concerning odd perfect numbers American Math. Luca J. Long gaps between deficient numbers Acta Arith. Hypothesis H and an impossibility theorem of Ram Murty Rend. On Hilbert's solution of Waring's problem Cent.

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Powerful amicable numbers Colloq. Values of the Euler and Carmichael functions which are sums of three squares Integers 11 , article A13, 16 pages electronic. On some friends of the sociable numbers Monatsh. The greatest common divisor of a number and its sum of divisors Michigan Math.

Quasi-amicable numbers are rare J. The exceptional set in the polynomial Goldbach problem Int. Ford Acta Math. Two remarks on iterates of Euler's totient function Arch. A more correct description of his results would be that they make effective earlier work of Dahlquist; see G. On the parity of the number of multiplicative partitions and related problems Proc.

Shevelev J. Finiteness theorems for perfect numbers and their kin American Math. PDF Errata Corrected.