Arithmetic Functions and Primes

调和分析将用组合思想加以加强，并应用于发展乘积和商在素数处的表示。最近的目标包括移位素数的正有理的完全乘法生成，以及解决30年前关于等差数列的加性函数和的Katai问题。一个终极目标是著名的哥德巴赫问题和孪生素数的无穷问题。原始根的值分布也有待研究。该项目旨在在数论中发展一种灵活的算术函数的一般理论，这种理论足够强大，可以解决某些经典难题。要考虑的所有三个主题都直接或间接地涉及由给定的有限形式的理性乘法生成的理性。积极的结果与构造分解整数的算法和基于分解大数的困难的加密理论研究直接相关。

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.