Farey arcs and series

Abstract

We find a finite subdivision of the interval [0,1] w The circle method of Hardy-Litlewood-Ramanujan [1] is based on the subdivision of the unit interval [0,1] by Farey's arcs. Large and small arcs are equally important for binary problems because the third (lowering) sum on the small arcs does not occur for them. For this reason the number-theorists axe aware of the problem of equalizing the arcs, which is equivalent to their precise construction for all rational Farey's points.Yurii Linnik was the first to introduce me to this problem in the beginning of the 50s, when he himself was busy with the binary Goldbach problem [2]. Linnik gave me this subject and suggested to continue his work [3] on addition of two primes with very scarce sequences. After that we had often discussed this problem. It was clear that the problem of constructing Farey's arcs could be solved in defferent ways, and the question was to find a simple and natural way of representing all Farey's arcs.This problem was solved for the first time by Kloosterman in paper [4], where his famous sums were introduced. However, Kloosterman's method of constructing Farey's arcs is rather complicated, and the reason for this is the uniform representation of the arcs for all points of the Faxey's series.If we give up the uniformity condition, then the construction of Farey's arcs can be considerably simplified. Here we suggest a nonuniform way of representing them, with special attention to the first half of Farey's series. This method can be considered as a modification of the principle of large and small arcs with explicit representation of both types of arcs. This method can be applied to binary as well as to ternary problems because in both cases the contribution of small arcs is included in the remainder term, but in binary problems (like Gotdbach's problem) we cannot as yet obtain adequate (in the order) estimates on these arcs. w The problem of constructing Farey's arcs is closely related to constructing a finite Farey series for a given parameter r > 1. The classical construction of this series starts with two rational points 0 1 1'17Adding the numerators and denominators of these points, we obtained Farey's series of the second level 0 1 I T ~2.