特殊数在算术级数上的分布

The Distribution of Special Numbers in Arithmetic Progressions

设正整数n的素数分解为n=p_(1)^(a1)p_(2)^(a2)…p_(k)^(ak)。若所有的ai均不相同,那么称n为特殊数。特殊数是近几年提出的新概念,Aktas Kevser和Murty Ram计算了特殊数个数的渐进公式。利用Siegel-Walfisz定理、Abel求和等一系列工具可以解决特殊数在算数级数上的分布,并且在广义黎曼假设的基础上,可以缩小对模q的限制。在此基础上,未来可解决特殊数的Titchmarsh除数函数问题。

Let n=p_(1)^(a1)p_(2)^(a2)…p_(k)^(ak) be the canonical prime factorization of n,ai>0.Then n is a special number if all the ai are distinct.The concept of the special number is newly put forward in these years.Aktas Kevser and Murty Ram compute the number of special numbers.By using a series of tools such as the Siegel-Walfisz Theoremand Abel summation,one can solve the distribution of the special numbers in arithmetic progressions,and by using GRH,one can reduce restrictions of mod q,and,based on the above,one can solve the Titchmarsh divisor problemfor the special numbers in the future.