On products of primes and almost primes in arithmetic progressions

We show that for any integers a and m with m ≥ 1 and gcd(a,m) = 1, there is a solution to the congruence pr ≡ a (modm) where p is prime, r is a product of at most k = 17 prime factors and p, r ≤ m. This is a relaxed version of the still open question, studied by P. Erd?s, A. M. Odlyzko and A. Sárközy, that corresponds to k = 1 (that is, to products of two primes).     

Strings of special primes in arithmetic progressions

The Green–Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu’s Theorem to certain subsets of the primes such as primes of the form ${\lfloor{\pi n}\rfloor}$ ? π n ? and some of arithmetic density zero such as primes of the form ${\lfloor{n\log\log n}\rfloor}$ ? n log log n ? .     

算术数列中的素变数指数和问题

旨在应用初等方法研究指数和问题,给出了算术数列中素变数非线性指数和的一个上界估计.     

Explicit relations between pair correlation of zeros and primes in short intervals

In this paper we obtain a quantitative version of the well known theorem by Goldston and Montgomery about the equivalence between the asymptotic behaviors of the mean-square of primes in short intervals, and of the pair-correlation function of the zeros of the Riemann zeta function.     

The segmented sieve of eratosthenes and primes in arithmetic progressions to 1012

The sieve of Eratosthenes, a well known tool for finding primes, is presented in several algorithmic forms. The algorithms are analyzed, with theoretical and actual computation times given. The authors use the sieve in a refined form (the dual sieve) to find the distribution of primes in twenty arithmetic progressions to 10¹². Tables of values are included.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

Short effective intervals containing primes in arithmetic progressions and the seven cubes problem

For any and any non-exceptional modulus , we prove that, for large enough ( ), the interval contains a prime in any of the arithmetic progressions modulo . We apply this result to establish that every integer larger than is a sum of seven cubes.

     

Primes representable by polynomials and the lower bound of the least primes in arithmetic progressions

Letf (m) be an irreducible quadratic polynomial with integral coefficients and positive leading coefficient. Under the assumption of Extended Riemann Hypothesis, we obtain new remainder terms in the upper bounds on primes represented byf(m) orf(p) which greatly improve Bantle's recent results. As an application, we obtain, in the second part of the paper, a new result on the lower bound of the least primes in arithmetic progressions with some difference.     

Discrepancy in arithmetic progressions

It is proven that there is a two-coloring of the first integers for which all arithmetic progressions have discrepancy less than . This shows that a 1964 result of K. F. Roth is, up to constants, best possible.

     

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.