International Congress of Mathematicians

210 T. D. Wooleycircumstances one has card(„4(F, Rj) ~ c(n)P, where the positive number c(n) isgiven by the Dickman function, and it follows that St(P,R) >• F* + p 2t — k . It isconjectured that in fact St(P, R) -C P e (P t + p 2 *-*). We refer to the exponent Xt asbeing permissible when, for each e > 0, there exists a positive number n = n(t, k,e)with the property that whenever F < P v , one has St(P,R) -C P Xt+e . One expectsthat the exponent Xt = max{£, 2t — k} should be permissible, and with this in mindwe say that 5t is an associated exponent when Xt = t + 5t is permissible, and thatA* is an admissible exponent when Xt = 2t — k + A t is permissible.The computations required to determine sharp permissible exponents for aspecific value of k are substantial (see [20]), but for larger k one may summarise somegeneral features of these exponents. First, for 0 < t < 2 and k > 2, it is essentiallyclassicalthat the exponent 5t = 0 is associated, and recent work of Heath-Brown[6] provides the same conclusion also when t = 3 and k > 238,607,918. Whent = o(Vk), one finds that associated exponents exhibit quasi-diagonal behaviour,and satisfy the property that 5t —t 0 as k —¥ oo. To be precise, Theorem 1.3 of [28]shows that whenever k > 3 and 2 < t < 2e~ 1 k 1 / 2 , then the exponentA4fcV2(A k \ (OUis associated. For larger t, methods based on repeated efficient differencing yield thesharpest estimates. Thus, the corollary to Theorem 2.1 of [26] establishes that fork > 4, an admissible exponent A t is given by the positive solution of the equationA t e At^k = fee 1-2 */*. The exponent Xt = 2t — k + fee 1-2 */* is therefore always permissible.Previous to repeated efficient differencing, analogues of these permissibleexponents had a term of size fee - */* in place of fee 1-2 */* (see [15]), so that in asense, the modern theory is twice as powerful as that available hitherto.The above discussion provides a useable analogue of the mean-value estimatein (1.3). We turn next to localised minor arc estimates. Take Q = P, and define mas in the introduction. Suppose that s, t and w are parameters with 2« > k + 1 forwhich A s , A* and A w are admissible exponents, and definemk^A t^A s A wa(k) 2(s(k + A W^ A t ) + tw(l + A s j) 'Then Corollary 1 to Theorem 4.2 of [27] shows that sup QGm \h(a)\ -C p 1^cr (* ! )+' ! ) andfor large k this estimate holds with a(k)^1= k(logk + O(loglogfc)). Applying ananalogue of (1.2) with h in place of/, and taking 3 t = [|fc(logfc + loglogfc+ 1)] ands = 2t+k+ [Ak log log k/ log k], for a suitable A > 0, we deduce from our discussionof permissible exponents that J m h(a) s e(—na)da = o(n s / k^r). By considering therepresentations of a given integer n with all of the fcth powers F-smooth, it is now3 We write [z] to denote max{n e Z : n < z}.