Arithmetic Functions and Primes

调和分析是加强与组合的想法和appliedto开发表示的产品和合格的评估在theprimes。 直接目标包括完整的乘法生成的正有理数的移位素数，并解决一个三十岁的问题Katai关于加法函数的算术级数的总和。 一个最终目标是著名的哥德巴赫问题和孪生素数的无限性。 原根的值分布也将被研究。该项目的目的是在数论中发展一种灵活的算术函数的一般理论，这种理论足以解决某些经典难题。 所有这三个主题都直接或间接地涉及到有理数由限制形式的给定有理数的乘法生成。 积极的结果有immediaterelevant建设的算法因式分解整数和理论研究的加密基础上的困难factorising大数字。

Harmonic Analysis is to be enhanced with combinatorial ideas and appliedto develop representations by products and quotients evaluated at theprimes. Immediate aims include the complete multiplicative generation ofthe positive rationals by shifted primes, and the solution of a thirtyyear old problem of Katai concerning sums of additive functions onarithmetic progressions. An ultimate aim is the celebrated problem ofGoldbach and the infinitude of twin primes. The value distribution ofprimitive roots is also to be studied.The project aims to develop in number theory a flexible general theory ofarithmetic functions strong enough to attack certain classicallydifficult problems. All three of the topics to be considered directly orindirectly involve the multiplicative generation of rationals by givenrationals of a restricted form. Positive results have immediaterelevance to the construction of algorithms to factorise integers and tothe theoretical study of encryption based upon the difficulties offactorising large numbers.