Additive Functions on Shifted Primes

Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes. A real-valued function / defined on the positive integers is additive if it satisfies f(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determined by their values on the prime-powers. For additive arithmetic function /, let Q denote the frequency amongst the integers « not exceeding x of those for which « 2 [9]. This result is best possible in the sense that for each of a wide class of additive functions there is a value of « so that the inequality goes the other way. From a number theoretical point of view it is desirable to possess analogs of Ruzsa's result in which the additive function / is confined to a particular sequence of integers of arithmetic interest. In this announcement I consider shifted primes. Let a be a nonzero integer. Let Q„ denote the frequency amongst the primes p not exceeding x of those for which « < f(p + a) < h + 1 . Theorem 1. The estimate Qh « W(x)~ll2 holds uniformly in h, f, and x > 2. If for an integer N > 3 we define Sn to be the frequency amongst the primes Received by the editors November 26, 1991. Presented at the 1992 Illinois Number Theory Conference, University of Illinois at Urbana-Champaign, April 3-4, 1992. 1991 Mathematics Subject Classification. Primary 11K65, 11L20, 11N37, 11N60, 11N64. 273 © 1992 American Mathematical Society 0273-0979/92 $1.00+ $.25 per page 274 P. D. T. A. ELLIOTT p less than N of those for which h < f(N p) < h + 1, and set / X2+ 52 \ min(l, |/(/>)-Alogpl)2 p 3. The estimates given in these two theorems are of the same quality as Ruzsa's and again best possible. In particular, Theorem 1 improves the bound Qn < W(x)-ll2(\o%W(x))2 ofTimofeev[10]. If E(x) = 4+ 52 ~, p• oo if and only if the three series vI ^ m vArt ^ p' ¿S p ' ^ p I/(P)I>1 \f(p)\)l where the X¿ are real, zero if d > z, X\ = 1. Expanding and interchanging the order of summation gives (2) ôa^/V-i^Eaa E *<»+«)*+^. v ; •/_I dj\R n 0 define /?,(«)= E h^i(m)g(p)^-, «<(logx)2/< p<(loèx)6A+[s ß2(n)= 52 h(u)gi(r)g(p)^urp=n i^J&'y u<(\ogx)2A r<{logxfA+l5 and set ß(n) = g(n) -ßx(n) -ß2(n). Note that ßj(n) «; 1 uniformly in n, j Lemma 1. Let 0 < ô < 1/2. Then y^ max max n TTls(r,D,D2)=\ y oo. Of importance here is the quality of the error term. For w > (logx)3A+* it is as good as that of Bombieri and Vinogradov. To this end the functions ßj were introduced, manifesting the assertion of [5, p. 408], already in view in [3, p. 178], that for general multiplicative functions a change of form would be required. In particular, ß2(n) is largely supported on the primes and cannot be removed without further information concerning g . Most integers « will have few prime divisors, so that effectively the ßj(n) are < log log x/log x over the range 2 < « < x. The functions ßj run through the treatment of the integral at (2) along with the central function g. A notable feature of the method is the casting of the Selberg square functions on the multiplicative integers in a rôle, which on the additive group of reals, is traditionally played by a Féjer kernel. The outcome is the estimate (4) Qh <£ x~l logw [ (I \t\)e~ith E c?(«+ö)^ + (logx)-1(loglogx)2. (n,P)=l The complications introduced by the exceptional modulus Do mentioned earlier must now be dealt with. To this end [4] or [6] may be applied. For simplicity of exposition I appeal to Theorem 1 of [6]. Lemma 2. Let 0 < y < 1, 0 < ô < 1/8, 2 < log/V < Q < N. Then any multiplicative function g with values in the complex unit disc satisfies E m-^ E s<»>-fe(§§r)