Primes of the form in short intervals

In this note, we prove that for every and , the short interval contains at least one prime number of the form with . This improves a similar result due to Huxley and Iwaniec, which requires .

     

On the finiteness properties of Matlis duals of local cohomology modules

Let R be a complete semi-local ring with respect to the topology defined by its Jacobson radical, a an ideal of R, and M a finitely generated R-module. Let D _R (−) := Hom _R (−, E), where E is the injective hull of the direct sum of all simple R-modules. If n is a positive integer such that Ext _R ^j (R/a, D _R (H _a ^t (M))) is finitely generated for all t > n and all j ⩾ 0, then we show that Hom _R (R/a, D _R (H _a ⁿ (M))) is also finitely generated. Specially, the set of prime ideals in Coass _R (H _a ⁿ (M)) which contains a is finite. Next, assume that (R, m) is a complete local ring. We study the finiteness properties of D _R (H _a ^r (R)) where r is the least integer i such that H _a ^r (R) is not Artinian.     

Properties of the Beurling generalized primes

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In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/.     

More Primes and Polynomials

Triplets of polynomials in four variables with rational coefficients and every term of arbitrarily high degree are exhibited. At least one polynomial in each triplet vanishes on infinitely many integer points whose coordinates are all prime.     

A note on two Erdősʼs proofs of the infinitude of primes

P. Erdős found numerous theorems, problems, results and conjectures in elementary number theory. Some of them are two Erdősʼs proofs of of the famous Euclidʼs theorem on the infinitude of primes. As noticed below, one of these proofs immediately implies the fact that the number of primes smaller than x is $⩾\log x/(2\log 2)$.     

关于奇数Goldbach问题

本文通过Ｌ函数非零区域的扩张及相应的素数分布均值问题的研究证明了：每一个奇数Ｎ≥ｅｘｐ（ｅｘｐ（９．７１５））都能够表示成为三个素数之和．     

On prime submodules and primary decomposition

We characterize prime submodules of R × R for a principal ideal domain R and investigate the primary decomposition of any submodule into primary submodules of R × R.     

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.

     

On the Convergence of Series of Reciprocals of Primes Related to the Fermat Numbers

We examine densities of several sets connected with the Fermat numbers F_m=2^2m+1. In particular, we prove that the series of reciprocals of all prime divisors of Fermat numbers is convergent. We also show that the series of reciprocals of elite primes is convergent.