Arithmetic Functions and Primes

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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.