Additive functions on arithmetic progressions with large moduli

In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott.     

A note on Mertens' formula for arithmetic progressions

We study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results.     

Primes in quadratic progressions on average

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

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli and other small moduli.

     

Group automorphisms with prescribed growth of periodic points,and small primes in arithmetic progressions in intervals

We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik?s Theorem, concerning small prime numbers in arithmetic progressions which lie in intervals.     

B-free numbers in short arithmetic progressions

Given a sequence B of relatively prime positive integers with the sum of inverses finite, we investigate the problem of finding B-free numbers in short arithmetic progressions.     

Squares of Primes and Powers of 2

 As an extension of the Linnik-Gallagher results on the “almost Goldbach” problem, we prove, among other things, that there exists a positive integer k ₀ such that every large even integer is a sum of four squares of primes and k ₀ powers of 2. (Received 7 September 1998; in revised form 3 May 1999)     

Primes in arithmetic progressions with friable indices

We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.     

Primes in progressions to moduli with a large power factor

On average, primes are uniformly distributed in short arithmetic progressions whose moduli may be divisible by high-powers of a given integer. In celebration of the seventieth birthday of Richard Askey. 2000 Mathematics Subject Classification Primary—11N13     

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     

General divisor functions in arithmetic progressions to large moduli

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.     

Primes in short arithmetic progressions with rapidly increasing differences

Primes are, on average, well distributed in short segments of arithmetic progressions, even if the associated moduli grow rapidly.

     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

On the number of representations in the ternary Goldbach problem with one prime number in a given residue class

For

     

The average value of Fourier coefficients of cusp forms in arithmetic progressions

Recently Blomer showed that if α(n) denote the normalized Fourier coefficients of any holomorphic cusp form f with integral weight, then

     

Biases in the prime number race of function fields

We derive a formula for the density of positive integers satisfying a certain system of inequality, often referred as prime number races, in the case of the polynomial rings over finite fields. This is a function field analog of the work of Feuerverger and Martin, who established such formula in the number field case, building up on the fundamental work of Rubinstein and Sarnak.