A Bombieri-Vinogradov type exponential sum result with applications

We prove a Bombieri-Vinogradov type result for linear exponential sums over primes. Then we apply it to show that, for any irrational α and some θ>0, there are infinitely many primes p such that p+2 has at most two prime factors and ‖αp+β‖<p−θ.     

Primes in quadratic progressions on average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.     

Chebyshev?s bias in Galois extensions of global function fields

We study Chebyshev?s bias in a finite, possibly nonabelian, Galois extension of global function fields. We show that, when the extension is geometric and satisfies a certain property, called, Linear Independence (LI), the less square elements a conjugacy class of the Galois group has, the more primes there are whose Frobenius conjugacy classes are equal to the conjugacy class. Our results are in line with the previous work of Rubinstein and Sarnak in the number field case and that of the first-named author in the case of polynomial rings over finite fields. We also prove, under LI, the necessary and sufficient conditions for a certain limiting distribution to be symmetric, following the method of Rubinstein and Sarnak. Examples are provided where LI is proved to hold true and is violated. Also, we study the case when the Galois extension is a scalar field extension and describe the complete result of the prime number race in that case.     

Irregularities in the distributions of primes in function fields

We adapt the Maier matrix method to the polynomial ring Fq[t], and prove analogues of results of Maier [H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221-225] and Shiu [D.K.L. Shiu, Strings of congruent primes, J. London Math. Soc. 61 (2000) 359-373] concerning the distribution of primes in short intervals.     

On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψk(x;q,a) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression . This theorem not only gives the asymptotic formula for ψk(x;q,a) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.     

JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY,AND THE THEORY OF ARITHMETICAL FUNCTIONS

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

In this paper some important contributions of John Knopfmacher to ‘Abstract Analytic Number Theory’ are described. This theory investigates semigroups with countably many generators (generalized ‘primes’), with a norm map (or a ‘degree map’), and satisfying certain conditions on the number of elements with norm less than x (Axiom A resp. Axiom A#), and ‘arithmetical’ functions defined on these semigroups.

It is tried to show some of the impact of John Knopfmachers ideas to the future development of number theory, in particular for the topics ‘arithmetical functions’ and ‘asymptotics in additive arithmetical semigroups’.     

An application of algebraic sieve theory

Let K be a fixed totally real algebraic number field of finite degree over the rationals. The theme of this paper is the problem about the occurrence of algebraic almost-primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. The method is based on a weighted upper and lower linear Selberg-type sieve in K and makes use of a multidimensional algebraic version of Bombieris theorem on primes in arithmetic progressions.     

Primes in arithmetic progressions to spaced moduli

We prove two average results on the distribution of primes in arithmetic progressions to widely separated moduli, one of which improves upon Elliotts work [2].Received: 28 April 2004     

Some Applications of a Bombieri-Type Theoremin Totally Real Number Fields

 The primary concern of this paper is to present three further applications of a multi-dimensional version of Bombieri’s theorem on primes in arithmetic progressions in the setting of a totally real algebraic number field K. First, we deal with the order of magnitude of a greatest (relative to its norm) prime ideal factor of , where the product runs over prime arguments ω of a given irreducible polynomial F which lie in a certain lattice point region. Then, we turn our attention to the problem about the occurrence of algebraic primes in a polynomial sequence generated by an irreducible polynomial of K with prime arguments. Finally, we give further contributions to the binary Goldbach problem in K. (Received 11 January 2000; in revised form 4 December 2000)     

A generalization of the Barban-Davenport-Halberstam Theorem to number fields

For a fixed number field K, we consider the mean square error in estimating the number of primes with norm congruent to a modulo q by the Chebotarëv Density Theorem when averaging over all q?Q and all appropriate a. Using a large sieve inequality, we obtain an upper bound similar to the Barban-Davenport-Halberstam Theorem.     

Distribution of primes and dynamics of the w function

Let P be the set of all primes. The following result is proved: For any nonzero integer a, the set a+P contains arbitrarily long sequences which have the same largest prime factor. We give an application to the dynamics of the w function which extends the “seven” in Theorem 2.14 of [Wushi Goldring, Dynamics of the w function and primes, J. Number Theory 119 (2006) 86-98] to any positive integer. Beyond this we also establish a relation between a result of congruent covering systems and a question on the dynamics of the w function. This implies that the answer to Conjecture 2.16 of Goldring's paper is negative. Two conjectures are posed.     

On the Mean Value of the Product of Multiplicative Functions with Shifted Argument

In this paper, we consider the mean value of the product of multiplicative arithmetic functions with shifted argument. The investigated functions have to satisfy the following conditions: their moduli do not exceed 1; the values on the set of primes are close to 1 for one of the functions and close to a fixed complex number for the other function. Some consequences for the classical functions are given.     

An Additive Problem with Piatetski-Shapiro Primes and Almost-Primes

Suppose that . We prove a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes p = [n ^1/ and show that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.Received November 29, 2001; in revised form August 21, 2002 Published online October 15, 2003     

On the index of composition of integers from various sets

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Π _p|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Professor M.V. Subbarao passed away on February 15, 2006. Received: 3 March 2006 Revised: 28 October 2006     

Flat cyclotomic polynomials of order three

We say that a cyclotomic polynomial Φn has order three if n is the product of three distinct primes, p<q<r. Let A(n) be the largest absolute value of a coefficient of Φn. For each pair of primes p<q, we give an infinite family of r such that A(pqr)=1. We also prove that A(pqr)=A(pqs) whenever s>q is a prime congruent to .     

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

On Waring-Goldbach problem involving fourth powers

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that any sufficiently large integer N satisfying the congruence condition can be represented as the sum of twelve fourth powers of primes and the fourth power of one P₅. This result constitutes an improvement upon that of Ren and Tsang.     

On the coefficients of ternary cyclotomic polynomials

We study coefficients of ternary cyclotomic polynomials Φ_pqr(z)=∏_ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.     

Additive representations of integers

In this paper we shall mainly study additive representations of integers prime to the firstm primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers.