zzfi») = O - if p = 0 or (Rp < 0,

Transcription

1 A THEOREM ON MULTIPLICATIVE ARITHMETIC FUNCTIONS HUBERT DELANGE 1. Introduction. The following result is contained in a theorem that we proved in a previous paper:1 Let f be a real or complex valued multiplicative arithmetic function. Suppose that \f(n) : 1 for every n and that, as x tends to infinity, (1) zz f(p) log p = Px+o[x],2 PS* where p is a constant other than 1. Then, as x tends to infinity, zz*sxfin) =o[x]. The proof of the theorem used a classical tauberian theorem of Hardy and Littlewood. It is indeed possible to prove the above result without using any tauberian theorem. Besides, this new method enables us to obtain a more precise result. We shall actually prove here the following theorem. Theorem. Let f be a bounded multiplicative arithmetic function. Suppose that, as x tends to infinity, (2) IZfip) log p = Px + 0[xLix)], psx where p is a constant y* 1 and L a positive function defined for x ^ 1 and nonincreasing, and L(x) =o[l]. Set L*(x) = f\(l(t)/t)dt (x^l). Then, as x tends to infinity, i r l*(x)~\ zzfi») = O - if p = 0 or (Rp < 0, x nsx L logx J and fftpsio. = o[(logx)^/2i-(-^ -r^ i/p 0 Received by the editors April 26, Un thioreme sur les fonctions arithmetiques multiplicatives et ses applications, Ann. Sci. Ecole Norm. Sup. (3) 78 (1961), Throughout this paper p ranges over the primes while m, n, q range over the positive integers. A sum which contains no term is zero and a product which has no factor is one. 743

2 744 HUBERT DELANGE [August It is to be noticed that \p\ must be ^ 1 and, since p^l, fftp must be <1. In fact the hypothesis that/be bounded obviously implies that for every e>0 there can be at most a finite number of p's for which \/(p)\ =l+«, and this in turn implies lim sup ^ f(p) log p ^ 1-!->+» X pix It is also obvious that the theorem implies the above quoted result for, if (1) holds, we have (2) with L(x) = l.u.b. /(/>) log/. - p usz U psu The idea of trying to improve this result originates from a remark of J. P. Tull, and some points are due to him as indicated in footnotes. We end the paper by an example of application of the theorem. 2. We need some lemmas Lemma 1. Let L(t) be a real positive/unction defined /or t^ 1 and nonincr easing. Then, /or x^l, nixn \n J J i t Lemma 2. S^s*:*2/*. log p = 0 [x]. The proofs are easy and we leave them to the reader Lemma 3.3 Let g(m) = Y\.v\ -.ph P (so ihai * =g(m) 1km)- Then X)-"s* log(m/g(m)) =0[x]. Proof. This follows from 23 log -= X) log/>+ T, log p. m$x gyjrt) msi;j!/m m X;pq/m;q>l 2.3. Lemma 4.4 Let / be a bounded multiplicative arithmetic /unction. Set 6,(x) = Y,vs*f(P) log P- Then 3 This lemma is due to J. P. Tull. 4 This lemma is due to J. P. Tull and is an improvement of the preliminary theorem in our paper quoted in footnote 1. There we supposed /(«) g 1 for every n and in the conclusion we had o [l ] instead of 0 [l/log x] although the method of proof could yield this just as well.

3 1967] A THEOREM ON MULTIPLICATIVE ARITHMETIC FUNCTIONS E «w /«<?/ (-) + 0 [-i-l x Si xlogx SI \m/ LlogxJ Proof. Let F(x) = nsi/(w). Then we have x- C x F^> /. fin) log n = Fix) log x I -dt = F(x) log x + 0[x],»SI " I t which yields Therefore 1 1 ^ r 1 1 Fix) = -2Zf(n) log n x x log Xnix Llog x_ we have only to prove that Now Zfin) logn = zzf(n)of( ) + 0[x\. n$x n&x \ ft / zzf(n)6;(-) = zzf(n)fip)logp n^z \ ft / np$x = zz finp) logp+ 1Z fmfip) log p np x;p](n np^x\pyn = D /(«)iog#+ E / Ml \ /(-)/(#) log # x'lp\m;p \m m%x;p Am \ p / = 2^ /(»») log w 2^ /(w) log 7 + zz fc^)f(p) log p m%x;p \m \ p / and the desired result follows by Lemmas 2 and 3 since, if /(w) j ^ M for every «, and 2^ /(>») log - ^ m 2^ log msx gim) mix gim) E /(^-W)iog# ^m2 1Z \ogp. m$x;p \m \ P / m x;p m 3. Proof of the theorem If /(«) ^M for every n, we have

4 746 HUBERT DELANGE [August Ztm (-) - * S ^ - S /w [» (-) -, -I I n^x \ X / n%x n n^x L \n / H _1 ^ll But there exists a positive 7C such that Thus we get «/( )-p df(x) - px\ ^ ita;z(a;) for x ^ 1. Z/to«/(-) - A* E ^ M7.x - l(-) nsx \»/ nsx n nix n \n / and it follows by Lemma 1 that Z/(*)»/(-) = P* E + 0[xL*(x)}. ngx \ n / n$x n By Lemma 4 this yields 1 _ P _ /(») r 7*^)1 (3) -E/w = r^z + 0 ^ ck Sx log a; nsx n L log a; J 3.2. Now set as in the proof of Lemma 4 F(x) = nsx/(«). We have 2^ = -f(.)+ -^-*- -^-<ft + 0[l]. Sx» a; Jit2 J i t2 Thus (3) gives i P rx F(t) rl*(x)i (4) -F(*) =-- -^-dt + O ^ a: log xj i t- L log a: _ If we define G(x) for *> 1 by rx p(f) G(x) = (log*)-p -f- <B, we see that G is continuous for x > 1 and differentiable for all nonintegral values of x with 1 rf(ao p f*f(0 "I G'(x) =-^.!^ -AI* x(logx)"l a; log a; J i /2 J T L*(x) 1 = 0 ' +1 by (4). La;(loga:)(K'>+1J

5 1967] A THEOREM ON MULTIPLICATIVE ARITHMETIC FUNCTIONS 747 This implies that is r r* L*it) i Gix) = 0\ -~ dl, Ui tilogt) -»+x J f'^-dt which with (4) yields = o\q.ogx)<b> f"- 0- dt\, Ji t2 L J2 tiiogt) p+x J 1 v^ T C* L*it) "I rl*ix)l -zzf(n)=p0 (log x) 01^ \' dt +0 ^ X BSI L J 2 /(l0g/)(r"+1 J LlogX To complete the proof it suffices to show that 1. If Otp<0, 2. If ffip^o, C' L*jf) = V L*jx) 1 J 2 tilogl) -'+x " L (logx)^ J ' a*(x) = o <\ogx)<*> I ;; (ft J 2 ^logo^""1" The first assertion is obvious for Z* is increasing If fftp^o we have for x>4 where Cx L*it) r* dt L*ixx'2) I --^ dt ^ L*ixx'2) I- = C-, J2 tilogt)<r"+x Jxvi tilogt)^+x (logx)«" C = (2^0 - l)/(rp if (Rp > 0, = log 2 if (Rp = 0. Moreover it is easy to see that L*(x) rg2a*(x1/2). Thus if Sip ^ 0 we have for x > 4 f» L*it) C L*ix) - dt>--^ J2 tilogt)&p+x ' 2 (log x)01" 4. Remark. If we assume ftmilit)/t)dt< + <x>, the conclusion the theorem obviously becomes of s This is due to J. P. Tull.

6 748 HUBERT DELANGE [August 1 v- r 1 1 E f(n) = 0 - if p = 0 or (Rp < 0, x Sx Llogx. flog log x~ = 0 - if p 5= 0 and (Rp = 0, L log a; _ = 0[(log a;)*-1] if (Rp > Application. We shall now give an example of application of our theorem. Let u(n) be the number of prime divisors of n. If/(n) =z"c") where z is any complex number whose modulus is ^ 1, then / is a multiplicative function and \/(n) \ :js 1 for every n. Moreover ~52Pix/(p) log p = z9(x), where d(x) = ZPSx log p. If we know that (5) 6(x) = x + 0[x/(log a;)"] with a > 1, we can apply our theorem when z 9^ 1, with p = z and L(t) = i/(i+log;)«. By the above remark we conclude that, for \z\ ^1 and Z9^\, 1 v- r 1 1 X, 3"(") = 0 - if (Rz < 0 or z = 0, X nsx Ll0gX_ "log log X~ = 0 - if (Rz = 0 and z 9^ 0, _ log x _ = 0 [(log*) to*-1] if (Rz > 0. Now let q be any integer > 1 and r any integer. Denote by vg,r(x) the number of w's 1=x for which co(w) =r (mod g). It is plain that vq,r(x) «{r* St'"w}» q j=0 V nsx - where 7 = exp [2iri/q]. Since, for l^j^q 1, 617'^cos (2ir/q), we obtain x v9,r(x)-= 0[x/log x] if q ^ 3, 9 = 0[x log log x/log x] if q = 4, = 0[x(log x)-2 sin2<t'5'] if g > 4. 6 See e.g. H. Delange, Sur des formules dues d Atle Selberg, Bull. Sci. Math (2) 83 (1959), See e.g. E. Wirsing, Elementare Beweise des Primzahlsatzes mil Restglied. II, J. Reine Angew. Math. 214/215 (1964), 1-18.

7 1967] SOME PROPERTIES OF INTEGRAL CLOSURE 749 A better result is known,6 but it is interesting to notice that this can be proved by elementary means since (5) can.7 Facult des Sciences d'orsay, France SOME PROPERTIES OF INTEGRAL CLOSURE WILLIAM J. HEINZER Let D be an integrally closed domain with identity having quotient field K, let L be an algebraic extension field of K, and let D be the integral closure of D in L. We prove here that the following five ideal theoretic structure properties of D are inherited by D, namely: (a) D is a Priifer domain, (b) D is an almost Dedekind domain,1 (c) D is a Dedekind domain, (d) D has the <2i?-property,2 (e) D has property (#).3 The converse of (a) is true (that is, D Priifer implies that D is Priifer) and was established by Priifer in [ll, p. 31]. In case L is finite-dimensional over K, Noether [9, p. 37] proved the converse of (c) and Butts and Phillips [2, p. 270] proved the converse of (b). In the general case it is well-known or easy to see that the converses of (b), (c) and (e) are false. The converse of (d) is false (see [6, p. 102]) even when L is finite-dimensional over K. Our statements concerning (a), (b) and (c) will be obtained as corollaries to the following. Theorem 1. Let M be a prime ideal o/ D and let P = MC\D. Then DMr\K = DP. Proof. It is clear that DP is contained in DMC\K. To obtain the reverse inclusion we first observe that L may be assumed to be a normal extension of K. For let be a normal closure of L over K, D* the integral closure of D in E, and N a prime ideal of D* lying over 717. Then NC\D=P and DMQD^r\L_so that DMr\K^D*Nr\K. Let {Ma} be the set of prime ideals of D lying over P. By a well-known Received by the editors July 26, D is almost Dedekind if for each maximal ideal P of D, Dp is a discrete rank one valuation ring [4, p. 813]. 2 D has the QR-property if each integral domain between D and its quotient field is a quotient ring of D [6, p. 97]. 8 D is said to have property (#) if for Ai and A2 distinct subsets of the set of maximal ideals of D we have flmeafim^flmeafim [5, p. 33l].