advertisement

WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS BY EUGENE E. KOHLBECKERO) 1. Introduction. Let A be a denumerable set of distinct positive numbers without finite limit point. Then the elements of the set can be arranged in a sequence Xi<X2<X3< • • ■ , where X* tends to infinity with k. Let 0 = j'oOi<*'2< ■ • • be the elements of the additive semi-group generated by A. Let p(vi) denote the number of ways of expressing pt in the form WiXi + m2X2+OT3X3+ • • • , where the mj are non-negative integers. The generating function f(s) of p(vm) is given by oo oo /(*) = II (1 - e-*")-1= E pMe-'~. We are concerned only with sets A which are such that both the product and the sum converge for all s>0. More generally we can consider the weighted partition function p(vm) defined by the generating function oo oo /(*) = II (1 ~ <r'x*)-**> = £ pMe-"-, k=l m=0 where ip(k) is a function on the positive integers into the non-negative reals such that the sum and product are both convergent for all s>0. (Actually convergence of either the sum or product implies convergence of the other.) Here p("i) E II ( ) where the summation is taken over non-negative integers mk, and is finite. The &th factor in the product is equal to 1 for all k such that \k>V{. When ip(k) = 1 for all positive integers k, this reduces to the case discussed in the preceding paragraph. Letting n(u) = £xjtsu \//(k), we have Presented to the Society, under the title Asymptotic properties of partitions, November 1957; received by the editors December 20, 1956. (') Author is now at the University of Utah. 346 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 16, WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 347 log/W = JC *(*) log (1 -e-'^)-1 k=l 00 = E {»(**) - »(Xt-i)} log (1 - *-**)-> + re(X0log (1 - e-^)-1 k=2 00 = X) »(X0{log (1 - er'^)-1 - log (1 - e-'^+i)-1} k-l M /» \h-i = E »(Xfc)*I k-i /. g-"8 - Jh l-e-«° ^M -n(u)du, x,00 1 -0—U8e-«" where the above computation is justified if re(X*_i) log (1— e~,x*)-1 tends to zero as k tends to infinity. This is in fact the case since i-l 0 g w(X*_i) log (1 - e~s^)-1 = }~2 4s(m) log (1 - e"8^)-1 m=l ka oo ^ X) ^(w) log (1 - e-s**)-i + m=l X) ^(m) log (1 - cr8*")-1, m=kQ+l where k0 can be chosen so that X^m-*0+i *P(m) l°g (1 —e~8>"»)_I<e (in view of our assumption about \f/) and where 2=,f(ffl) log (1 —e~eX*)-1—>0as &—>°°. Now letting P(u) = E'«s« />(»**)»vve have 00 00 2Zp{v™)e-"°>= £ m=0 {^0v> - P(vm-i)}e-^ m=0 CO /» 00 = X) P("m) {e-8"" - e~"m+1} = s j m—0 P(u)e~audu, J 0 where the computation is justified as in the preceding Thus we have the following basic relation between paragraph. the functions n(u) and P(u): (1) exp <s I -n(u)du\ (Jo 1 - e-8" = s I j P(u)e^sudu. J o The object of this paper is to show that an asymptotic relation of the form n(u)~uaL(u) as w—>oo, where a is a positive constant and L is a slowly oscillating function in the sense of Karamata [6](2), is equivalent to an asymptotic relation of the form log P(u)~ua"-a+1)L*(u), where L* is a slowly (2) Numbers in brackets refer to the bibliography. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 348 E. E. KOHLBECKER [July oscillating function, related to L by a certain implicit formula which can be solved for L* in the cases usually encountered. The special case of this where L(u) is a constant was given by Knopp [7] and Erdos [3]. When L(u) is a power of log u the deduction in one direction (from n(u) to P(u)) was made by Brigham [l] using results of Hardy and Ramanujan [5]. When we start from the assumption about n(u) we get an asymptotic formula only for log P(u) and not for P(u) itself, so that the results are weak in this sense. However it should be noted that the assumption made on n(u) is a very simple and natural one, and the resulting asymptotic law which we obtain for log P(u) is equivalent to the original assumption made for n(u). The proof of the above equivalence uses only the obvious monotonicity of n(u) and P(u) and the basic relation (1). Thus, more generally, starting with equation (1), where n(u) and P(u) are functions on the non-negative real numbers such that r* n(u) I -du, J o u rR I Jo n(u) log * -du, u and fa„, x. I P(u)du Jo exist in the Lebesgue sense for every positive R, and a and L*(u) are as before, we obtain the following: (i) If «(tt)~w"L(w) as m—>oc and P(u) is nondecreasing then log P(u) ~tt«/(«+i>I*(tt) as m—>oo. (ii) If log P(u)^jual{a+l)L*(u) as m—»<» and n(u) is nondecreasing, then n(u)~u"L(u) as u—*<*>. Thus most of our results will be stated for functions n(u) and P(u) satisfying the foregoing conditions (for all positive s), and we shall specialize to the case of partitions only at the end of the paper. To prove (i) we go from the assumption on n(u) to a resulting property of the generating function f(s) by an Abelian argument, and then from this property of f(s) to the assertion about P(u) by a Tauberian argument. The latter argument follows the method developed by Hardy and Ramanujan [5], for the special case in which the slowly oscillating function is a power of the logarithm. To prove (ii) we go from the assumption on P(u) to a resulting property of f(s) by an Abelian argument and then from this property of f(s) to the assertion about n(u) by a Tauberian argument. The latter step is accomplished in two stages, in order to make use of a known Tauberian Theorem [4, Theorem 108]. The material in this paper is essentially the same as that contained in the author's doctoral dissertation at the University of Illinois under the direction of P. T. Bateman, to whom the author is deeply grateful. 2. Definitions and properties of slowly oscillating functions. Let the function L(x) be defined and positive valued for all sufficiently License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use large x, and let it 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 349 be continuous and such that L(cx)/L(x)—A as x—»°o for every fixed positive c. Then L(x) is called a slowly oscillating function. We shall make use of the following properties of slowly oscillating functions: (i) L(x)=K(l-\-r(x)) exp fla(b(t)/t)dt where K is a positive constant, while r(x) and 8(x) are continuous functions, which tend to zero as x tends to infinity. See [6; 8]. In the sequel in order to assure the differentiability of L(x), we will occasionally assume that it is exactly (and not merely asymptotically) of the form K exp flJJ>(t)/t)dt. Whenever this assumption is made it will involve no loss of generality. A slowly oscillating function of this special form will be called a normalized slowly oscillating function and will be de- noted by /. (ii) The convergence finite interval L(cx)/L(x)—>1 [a, b] with 0<a<b. It is easy to see that as x—>oo is uniform in c, for c in any See [2; 6; 8]. the following properties are consequences of the above. (iii) x'L(x)—>oo and x~«Z,(x)—>0 as x—»°° for every fixed positive e. (iv) log Z(x)/log x—>0 as x—>». A proof of this is also to be found in Polya-Szego [10, p. 68]. There L(x) is assumed to be nondecreasing, but this is not necessary for the proof. We will require the following lemma. Lemma 1. If a is a positive constant and L(x) is a slowly oscillating function defined for xSixo, then I ta~1L(t)dt ~ x"L(x)/a as x —>oo. Proof. By (iii) both sides of the proposed asymptotic relation tend to infinity with x. Hence it suffices to prove this relation with L replaced by /, where I is the normalized slowly oscillating function associated with L as in (i). But on integration by parts we have a j t"~lJ(t)dt = x"j(x) - x°J(xo) - I o(t)Vl J(t)dt and so a I f-Vtydt - x"J(x) (2) « g xll(xo) + | | 5(t)Ir-V^dt + \ max | 8(t)\ \ f t?-lJ(.t)dt where x0<Xi. Now given an arbitrary positive number e, we can choose Xi, so that | d(t)\ ^€ for t^Xi. Thus the right hand side of (2) is less than License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 350 E. E. KOHLBECKER 2* I for sufficiently [July ta~lJ(t)dt large x. Since e is arbitrary, the lemma is established. 3. The connection between n(u) and log/(s): Theorem 1. If fon(u)u~ldu exists in the Lebesgue sense for every R>0 f(s)=exp {sfo(e~su/(l—e~su))n(u)du}, for n(u)~uaL(u) as u—*<x>implies the relation all positive log f(s) ~ T(a + l)f(c* + l)(lA)<"L(l/j) where a is a positive constant s, then and the relation as s -+ 0, and L is a slowly oscillating function for « Srwo- (Here f(a + l)= E»=i l/ma+l and Y(a + l) =f0*e-'t<'dt.) Proof. We first remark that if G(u)=f£(v"/(ev — l))dv constant Ki, such that G(u)<K\Uae~u for W—'1. For /» 2u 2u •» oo e-"^ + 2e~u J u Now let e be an arbitrary positive eu — 1 we may choose a positive vae~vl2dv g KiWer". J 2u " Ir = X, -du is a "J 2u /» o there vae~vdv+ 2 I/» M e-"/2(e-«/V)dji u /ua then number, less than min (1, l/3A"i). Since We-mudu = i-(a + l)r(ct + 1) m=i J o 17so small that Sr)a/a<e and a positive 7 so large that (3) f(a+l)r(a+l)- C —du <e. J , e" - 1 Yet us write "eM- U:+^+0^id" We now choose t0 so large that /"/8 0 »(/) est - 1 -rf/ /><o I n(t) \ - at Thus for sufficiently oscillating function, | n(t)\ ^2taL(t) I I = /«i/» I n<i) I ' ' Jo « r*i' dt + 2 I - ''+h+'for /^/0. Then dt r,/s ta~lL(t)dt = 7v2+ 2 I J t„ small 5, we have by Lemma and the choice of n f>-lL(t)dl. J <0 1, the definition License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use of a slowly 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 351 l*l«+T(fy^)«4'(7H7)<-(7)X7> Let 5 be so small that for u ^ 17we have Then ,1 u/^r1 (1 — e)LI — ) I - \s/J, («/5)-l(«a) j (««- 1)L(1A) AM WJ, Now, since L(u/s)/L(l/s)-^l ass—*0, uniformly («•- 1)L(1A) for n^u^y, we have for 5 sufficiently small / 1V / 1\ r y ua <1-2<,(7)i(7)J, 7=7* /1 v /1 \ r7 «a *r.*(i + «(7)i(7)/f 7^7*. Using (3) this becomes (1 - 2e)(^-JL(^j {f(« + l)r(a + 1) - e} ^/iS(i By property small s, + 2e)(~) (i) of a slowly oscillating /' °° (u/s)"L(u/s) y ^^^^^— eu — 1 L(~) function / 1 \a d« g 3 ( — ) \ s / where J is the normalized slowly oscillating property (i) of a slowly oscillating function. /"° -j( w y e" — 1 /u\ — )du=-G(u)j[ \ s/ i^a + ^^ r °° J y + !) + el • we have for sufficiently -j[ «" eu — 1 function Now / M\ — )du, \j/ associated with L as in /«\r r°° G(u)8 (/u\ (u\ — )\ +1 — Jm-1/! — Jdw \5/l7 J 7 \ s/ \ 5/ g G(t)/( —J + j max 5(—j 1 f G(«)7f —j ir^ « / 1\ <w(T) + */t /•" W ^'(7)* (u\ License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 352 E. E. KOHLBECKER [July for 5 sufficiently small, since G(u)<Ki uae~u<KiW/(eu — 1) for m=1 and since |G(/y)| ^e by (3). Hence, by the restriction e<l/37vi, we have for s sufficiently small f —-—j(— )dug- J 7 e" - 1 \s J —j(—)<3ij(—)<4tL(—\ l - Kie \s / \sj \s ) so that h^l2e(l/s)"L(l/s). Since e is arbitrary, our result follows from the estimates The Tauberian counterpart of Theorem 1 is: Theorem 2. If fgn(u)u~1du and f^n^u^1 for 7i, 72 and I>. log udu exist in the Lebesgue sense, for every positive R, f(s) = exp {sf£ (e~su/(e~'u — l))n(u)du} s, and n(u) is nondecreasing, then the relation for all positive log/to ~J"(« + l)r(« + 1)(1/«)«L(1/*) as s—+0 implies the relation n(u)<^uaL(u) as u—>°o, where a is a positive constant and L is a slowly oscillating function for u^u0. Proof. Tauberian (4) We give the proof in two stages in order to make use of a known Theorem. Define N(u) = /,"=i (l/m)n(u/m). This exists, since " 1 / u\ £_„(_)<( m_2 m ("° 1 /u\ —n[ — )dx=\ \m/ J1 x \ x / r n(t) —dt, Jo the non-negativity of n(u) being implied by the hypothesis. a non-negative, nondecreasing function of u. Now /l 00 Clearly N(u) is p—su -n(u)du 0 e~°u - 00 f = 55 s I m=l t 1 M e~m""n(u)du J 0 = X, 5 I e~s" — n[ — )du m_i Jo /, m \m/ ao e~>uN(u)du. 0 To this last expression we apply [4, Theorem 108] and obtain N(u) ^(aArl)uaL(u) as u—■»«>. It remains to show that the relation N(u)^(aArl)u"L(u) as w—>oo implies the relation n(u)~uaL(u) as «—>o°, under the assumptions of the theorem. From the definition of N(u) and the properties of the Mobius function we obtain License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS n(u) = X) -N[ „_i This requires proving - * p(m) m-i*-i convergent. — ). m \m) that l^lu is absolutely 353 ( u\ —r n[—r) mk \mk/ This is the case, since by (4) we have it±.(JL)s±SLJJL)+lf^A \ mk) k-i\k \k / k J o k-i m-i mk = N(u)+ C( £ / ) ~)—dt g N(u) + J M + log—j -^- cf/, and these integrals exist by hypothesis. Let e be an arbitrary positive number less than l/f(a so large that | 2^°=roo+i p(k)/ka+1\ <e and (l/ra0)°<€/3. + l). Choose Wo=^2 Choose w0 so large that (1" '](i)"L(i)n"+"s *(i) a (I-'-'(tHt)"*+» and (1 - 2e) -Z,(m) 1 —e /M\ (l + 2«) g Ll — I ±s -L(m) \k/ 1+ e for u ^ Uo and 1 ^ k g w0. Then *_i * \ * / *_i £a+1 A Im(£)I ^ 2e X) ^-T1 *_l Since X*°=i (u(k)/ka+1) ">o u(k) £ k=l - = l/f(a £a+1 «"i(«)f(« + 1) < 2ef2(a + 1)««L(«). + l), our choice of m0 guarantees / U\ N[ — ) - U"L(u) k Choose to so large that that < 3et2(a + l)u«L(u). \ k/ | N(t)\ ^2taL(t)'C(a + l) ior t>t0. Then for sufficiently large u License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 354 E. E. KOHLBECKER " u(k) Z k=m0+l /U\\ -±-Ln( —)\<\ k /' \ k/ I {•" N(U/X) Jmo X ' !o o /*'» dx= [ Jo /* tt'm° N(t)t~idt+ 2f(a + 1) | J („ [July +1 fulmo N(t)t-Ht J«0 3 / M \° / u \ a \mo/ \mo/ /-iZ(/)d/ = — ( — ) L[ — Jf(« + 1) < eu"L(u){2(cx + 1), by Lemma 1 and the choice of m0. (The above computation is valid if f0N(u)u~ldu exists in the Lebesgue sense for every positive R. This is indeed the case under the assumptions of the theorem since rR | Jo N(u) rR -du = I u J o /'R /'R o n(u) — uaL(u) /•" n(t) u \ -^-^duAr u n(u) -du rR/ ( J o \J \ du FR du\n(t) t — \—dt u / t R + IrB n(t) - log — dt, u Jot Combining = -dt) Jot) n(u) 0 which exist by assumption.) 1 ( — I n(u) + j the preceding y, -N I results we obtain ( — ) - uaL(u) 1% k \k/ < 4ef2(a + l)waT,(M). Since e is arbitrary, our theorem 4. Additional properties follows. of slowly oscillating functions. For the purposes of completing the following theorems it will be necessary to introduce a relationship between the variable u which tends to infinity and the variable 5 which tends to zero. That is we must define the variable s in terms of u. To this end we prove the following lemmas. Lemma 2. Suppose A and a are positive constants and J(u) is a normalized slowly oscillating function defined for m = Wo, r" -dt, 8(0 J(u) = K exp I J«o ' where K is a positive constant and h(u) is a continuous function which tends to zero as u—>oo. Then for every sufficiently large u there is a unique positive number au such that In addition we have the following: License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 355 (a) au tends monotonically to zero as u—>oo, (b) ffu is a differentiable function of u and dcru -~-> du —o-u d / a — (uau) '—' <r«( - u(a +1) du \ \ a + 1/ (c) M-1'(°+I){^/(l/0}1/<a+1)=o-K J as u —* oo, = w-1/fa+1>Z,1(M) where Li(u) is a slowly oscillating function, (d) If au is defined for u^u0, then /cr i<// '—' I 1 -\-1 k0 \ U(TUOS M —> oo . a / Proof. Since the expression A(l/a)a+1J(l/o-) is a continuous function of a which tends to oo as cr—>0(by property (iii) of a slowly oscillating function) and since for small a, the existence and uniqueness of au for large u, as well as the properties (a) and (b) follow immediately from the inverse function theorem. To prove (c) we note that by (b) we have for arbitrary positive e and large u — (1 + e)o-u do-u — (1 — e)cr„ du u(a + 1) u(a +1) Hence for fixed H^ -(l 1 and large u, we have + e) rHu du a + 1 J u u rHu J u l do-u au du -(l-«) rHu du a + 1 J u U or (1 - e) log ffW<«+n ^ log— g (1 + e) log ffWOfi). 0~Hu Ii II<1 we reverse the limits on the integrals, obtaining the inequality signs reversed. Since e is arbitrary, = log Hll(-a+1) or o-Hu/o-u~H-ll(a+» ior large u. We now let Lx(u) = u1Ka+1)ffu. It is then clear that Lx(u) is a slowly oscillating In order to establish (d) we write f " Cu (a + 1) I o-tdt= I l("ic+1»-1Li(t)di~-- the same result with lim,,^ log (cr„/cr//„) function. / 1\ W^a+^Li(u) = ( 1 + — W as u —■>oo, License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 356 E. E. KOHLBECKER [July where we have made use of property (c) to establish the first and third relations and of Lemma 1 to establish the second. In the case of a more general slowly oscillating function than the one considered above we are able to assert only that su is asymptotic and not necessarily equal to u~1!(a+1)Li(u), where Lx(u) is a slowly oscillating function. This is because ull'-a+1)su may not be continuous A(l/s)a+lL(l/s) may not be monotonic in the small. in general, since Lemma 3. Suppose A and a are positive constants and L(u) is any slowly oscillating function. Then for every sufficiently large u, there exists a positive number su such that \ su / \ su / In addition we have the following: (a) 5a tends to zero as u—>oo, (b) su is determined up to a factor which tends to one as «—>oo , (c) There is a slowly oscillating function L\(u) such that ( / IV) su = u-vte+vlALl ,/(a+1) — )> ~w-1'<<"+1)L1(w) fli«->». In fact, if L(u) = (lArr(u))J(u) where J(u) is a normalized slowly oscillating function and if cru is as defined in Lemma 2 relative to J, then w_1/(a+1)Li(w) = <7„. Proof. Since the expression which tends to infinity .<4(l/.s)a+1L(l/s) as 5 tends to zero, is a continuous the existence function of su for large of 5 u and the property (a) are immediate. By property (i) of a slowly oscillating function, L(u) must have the form (l+r(u))J(u) where J(u) is of the normalized form discussed in the preceding lemma. Hence V Su ) \ \ Su )) \ Su) Thus for large u (u \-l/(a+l) . , ... J / u \ lA — ——)~u-^Lx(u) 1 + r(l/su)/ \ 1 + r(l/Su)/ as u —> oo where Li(u) is as obtained in the previous lemma. This shows the validity of (b) and (c). 5. The connection between log/(s) and log P(u): Theorem 3. Suppose foP(u)du exists in the Lebesgue sense for every posi- License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 357 live R, f(s) = sf" P(u)e~'udu, for all positive s, B and a are positive constants, L is a slowly oscillating function, and su is a positive number defined for large u such that u = Ba(l/su)a+1L(l/su). Then if P(u) is nondecreasing, the relation logf(s)~B(l/s)aL(l/s) as m—>oo. as s—>0 implies Proof. Since adding a positive to f(s), we may assume P(w)S;0 oscillating function and Lemma 3 in assuming L(l/s) of the special the relation + l/a)usu constant to P(u) merely adds a constant for all positive u. Property (i) of a slowly (c) shows that there is no loss of generality form '(t) -KtXff We shall make this assumption throughout positive number less than 1. For any u>0, T* this proof. Let « be an arbitrary s>0 we have P(u)e~'zdx g 5 If% OOP(x)e~"dx /.CO u since u^x log P(u)~(l " u and P(u) is nondecreasing. /» Thus, since P(u) is non-negative oo P(x)e-"xdx = /(*). o Hence, by the hypothesis on/(5), we have for all sufficiently P(u) < exp |(1 + €)b(—\ l(—\ small s + su\ . The estimate will be about the best possible if we choose 5 so that (l/s)aL(l/s) -\-su is about as small as possible. Since 5(1/$)—»0 as s—*0 and this minimum occurs, for large u, when s is near that value such that u = Ba(l/s)a+lL(l/s). This is the value of 5 denoted by su in the statement of the theorem. Using this value of s, and replacing B(l/su)"L(l/su) by usu/a we obtain P(u) < exp j(l for large u. To get a lower estimate split f(su) =sufoP(x)e~x,udx for P(u) 4- «) (1 + — j usu\ ior large u we also consider f(su). We into five parts License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 358 E. E. KOHLBECKER /» ulH r* (1—f)u + Su I 0 /» (l-ff)u + 5U I «7ti/H where 77 is a fixed positive 77 =■ max and f is a positive number [July /» Hu J (l-f)u number /* oo + 5„ I + Su I J (l+f)n J Hu such that {l, (8a + 8)1+1'a, (8c*-1 + 8)<"+1} less than 1, an exact choice of which (in terms of e) will be made later (f = e/3). The five shall use the tributions to Now P(x) x (by Lemma parts of f(su) we denote by Ji, J2, Jz, Ji, J&, respectively. We upper estimate for P(u) obtained above to show that the conJi, J2, Ji, and Jf, to f(su) are comparatively small. <exp \2(lArl/a)xsx} for large x and xsx is increasing for large 2(b)) and tends to infinity with x. Hence we have for large u rulH Ji = su I ( ( P(x)e-xa"dx = exp <2l g exp h(l since stJ///~771/(a+1).?t, by Lemma 2(c). l\u 1 -\-j— ) rx su/h> su I e~x^dx H-jH-a'^+l)uSu\ Hence Ji^exp {usu/2a}, since 77 has been chosen so that 77=- (8(a + l))1+1/Q. For x^Hu, where wis sufficiently large, we have P(x) <exp \2(lArl/a)xsu}. We claim P(x) ^exp {xs„/2} for x = 77m. It suffices to show that 2(1 + l/a)xsx ^xSu/2 for x^Hu. Since sx is a decreasing function of x for large x, and hence Shu^sx, for x = 77w, it suffices to show 4(lArl/a)sHu^su. Since by Lemma 2(c), shu^H-v^+Vsu and thus 5ffu/5u^277-1/(a+1) for large u, we need only show 277-1/("+1) g 1/4(1+l/cv). This clearly = (8(l+l/a))a+l. Thus we have /» 00 Hu f* 00 P(x)e-X'"dx =: su I holds since 77 has been chosen (* oo e"»'2<r"«<fo g s„ I J Hu e-x"'l2dx = 2. Jo We now define $(x) so that$'(x) =5,,, that is we take &(x) =f^stdt, where to is such that st is defined for all t^t0. Now by Lemma 2, #(x)~(l +l/a)x.sI and t?"(x)'~ —^/x(a + l). We assume w so large that #(x) ^ (l+2?/)/(l+17) • (1 + l/a)xsx and P(x) ^exp {(l+-n)§(x)} for x^u/II, where 77 is a positive number less than 1, to be chosen later in terms of f, 77 and a. The integrand in J2 and J4 does not exceed exp {(1 Arr))§(x) —xsu} on the interval [m/77, Hu]. Yet <p(x) = (1 +7j)t?(x) —xsu. Applying Taylor's Theorem to c6(x) around x = u (the maximum value of c6(x) occurs near x = u), we have {X - <f>(x)= <j>(u)+ (x - u)4>'(u) -\-4>"(u ti)^ + 6(x - «)) where 0 < 0 < 1. Now License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 4>(u) = (1 + ri)d(u) — usu ^ (1 + 2n) I 1 -\-Jusu usu =-V ( 1\ \ a / 2ij I 1 -|-1 a — usu usu and (p'(u) = (l-r--n)§'(u)—su = nsu. Assume u so large that t?"(x) ^ (-1/(1+t?))(si/x(o!+1)) Then if u/H^x^Hu 359 for x^u/II. we have <p"(x) = (1 + r,)d"(x) gNow SHu~H~lla+1su — Si —Shu —Shu ^ <x(a + 1) x(a + 1) Hu(a + 1) as u—>oo and so for sufficiently SHu ^ Then if u/H^x^Hu large m, we have ff_15u. we have — su <p"(x) g- H2u(a + 1) Thus if x is in [(w/ff), ffw] we have <p(X) ^-h usu / a 2-qI H-) \ 1\ If in addition |x —m| ^u / 1\ \ a / 2i7( 1 H-) a usu S-h (x — u)2 2 then usu c6(x) g-h su USU+ 77?;M5K- ————-a / H2u(a +1) f 2usu usu + ffi/ws,, - ——-—2H'(a (/ 2 \ \\ a J <[2 + — + H)r)-}usu a USu f2 + 1) ) 2H2(a + 1)1 f2 =S-USu a 3H2(a + 1) provided we take =_r_ 77 6ff2(a + 1)(2 + (2/a) + ff) Hence if u is sufficiently large, we have n (1—f)u J2 + Ii ^ Su | n Hu e^Ux •^ M/ff + su I e*<*^x *> (l+f)M ^ ffM5„ exp < (-J y \\a 3H2(a+l)J License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use usu t ■ ) • 360 E. E. KOHLBECKER Combining our estimates, we obtain J i + J 2 + J, + /. S 3 exp {£} [July for u sufficiently + Husu exP {(1 large - JIJ~~> •*.} ^(ff + 3)^exp{(l-3^+i))M,„} < exp t\7 ~ m2(a + i))) < exp < (-25 ) USur , F IVa Now by hypothesis ) where 5 = -. V we have for su sufficiently 8H2(a+ 1) small, i.e., for u sufficiently large /(*.) ^ exp |(1 - <x5)b(-\ =^exp<(l— ^(y)} ab) —-> = exp U-5Jm.su> Thus J* = f(su) - (Ji + J2 + J, + Ii + Jt) ^ exp i(-5 — exp <(-28Jusu for u sufficiently > ^ exp <(-25 large. Now by the monotonicity /i J us\ Jusu> of P(x) (l+f)u we have /• (l+f)u P(x)e-f«dx ^ P((l + £)u)su I (l-f)u e-"»c/x ^ (l-f)U < P((l +f)«) exp {-(1 Hence for every sufficiently -f)w*B}. large u, we have P((l + f)«) = /a exp {(1 - f)M5u} ^ exp |f-25 + 1 - M us\ ^ exp|m + —J(1- 25- fiusX ^ expif 1+ —J(1- 2f)«5B|• Thus P((l+f)w)^exp ^5(i+f)„. On replacing S(l+l/a)(l-3f)(l+f)«Ju} (l+f)« for u large, since s„ by y we have, for y sufficiently ^exp {(l-3r)(l+l/«)y*,}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use large, P(y) 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS Now we take f = e/3. Then for large y, P(y) >exp{(l and this completes Theorem tive R, f(s) = L is a slowly such that u = 361 -e)(l + l/a)ysv}, the proof. 4. Suppose fRP(u)du exists in the Lebesgue sense for every posisfo P(u)e~'udu, for all positive s, B and a are positive constants, oscillating function, and su is a positive number defined for large u Ba(l/su)a+1L(l/su). Then the relation log P(u)~(lArl/a)usu as w—>ooimplies the relation log/to~i?(l/.s)aL(l/.s) as s—*0. Proof. As in the proof of Theorem 3, we may assume that L is a normalized slowly oscillating function, so that Lemma 2 can be applied. We may also assume that P(x)^0 for all positive x. Let 0<e<l/(2cv+3) be given. Then P(x)>exp {(1 —e)(lArl/a)xsx} for sufficiently large .r. Hence if u is sufficiently large /.(l+Ot. / / 1\ f(su) ^ su I exp < (1 — «) I 1 H-jxsx = exp < (1 — e) f 1 H-I usu> su 1 = exp < (1 — e) I 1 H-jusuf exp { — xsu}c7x exp { —usu}(l = exp < (1 — 2e) ( 1 H-) since exp { —eusu}^>0 1 — xsufdx — exp{ — eusu}) usu — usuf , as u—»oo. Thus for u sufficiently log/(s„) = {1 - (2a + 3)c| — = {1- large (2a + 3)e}B(—) a l(— Y \ Su/ \ Su/ We proceed now to obtain the estimate from above. We split f(su) into the same five parts as in the proof of Theorem 3, and use here the estimates from above for Ji, J2, J4, and Ji, where 77 and f are chosen as before. This is permissible since nowhere in the computation of these estimates did we make use of the monotonicity of P(u). Indeed we used only the fact that P(u) <exp {(l+e)(l+l/a)M5u| for sufficiently large n, which is now immediate by hypothesis. Here /' (l+f)u /• (l+f)u P(x)e~x,udx (l-f)M for u sufficiently ^ Su I J (l-f)" exp { (1 + e)(l large. Hence License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use + l/a)xsx — xsu}dx 362 E. E. KOHLBECKER J3 =Sexp <(1 + e)f H-](1 + f)^(i+f)»|i« ^ exp |(1 + e)fl + — j (1 + f)i«.| since Sa+r)u = s„ for w sufficiently [July I exp { —xsu}dx exp { -(1 - £)usu} large. With f = e/3 this gives /, ^ exp {(l/a)(l + (2a + 2)4)«5„|. Now, using this inequality and the estimates for Ji, 72, JA, J$, obtained in the proof of Theorem 3, we have for sufficiently large u, f(su) ^ exp i(-25) g2exp|(l usu\ + exp |(1 + (2« + 2)«) —i + (2c*+2)e)-j- f£ exp |(1 + (2« + 3)e) —| , log/(j«)=g{l + (2a + 3)€}—= {l + (2«+ 3)€|73(—Yl(-Y a \su / \ su / This inequality, together with the estimation of log/(s„) from below gives the desired result, since when u runs through all sufficiently large positive values, su takes all sufficiently small positive values by Lemma 2. 6. Statement of results. The results obtained may be summarized in the following manner. Main Theorem. Suppose that n(u) and P(u) are functions on the nonnegative reals and that f*n(u)u~^du and fon(u)u~1 log udu and f§P(u)du exist in the Lebesgue sense for every positive R. Suppose further that I n oo exp <s | Uo e~su -n(u)du> 1-e"8" \ j /.oo = s I Jo P(u)e~"'du for all positive s. Suppose that a is a positive constant and that L(u) is a slowly oscillating function. For large u let su be a positive number such that u = aT(a + l)f (a + 1) (—) L (-) \Su/ \Su/ and let L*(u) be a slowly oscillating function defined for large u such that L*(u)~(l+l/a){aY(a + l)^a + l)L(l/su)}1Ka+l), the existence of L*(u) being guaranteed by Lemma 3(c). (i) If n(u)<^,uaL(u) as u—>oo and P(u) is nondecreasing, then log P(u) ~(1 +l/a)usu~uai(a+1)L*(u) as w—»». License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS (ii) If log P(u)~(l-r-l/a)uSu~ualla+1)L*(u) decreasing, then n(u)~uaL(u) Proof, (i) follows Theorems 363 as u—><» and if n(u) is non- as u—><x>. 1 and 3, while (ii) is the result of Theorems 4 and 2. In the case of partitions the functions n(u) and P(u) are automatically nondecreasing and trivially satisfy the required conditions of integrability. Thus we have the following corollary, as mentioned in the introduction. Corollary 1. Suppose n(u) and P(u) are defined from a set A of positive real numbers and a non-negative valued function \f/(k) on the positive integers as in the second paragraph of the introduction. Suppose a, su, L(u), and L*(u) are as in the Main Theorem. Then n(u)~uaL(u) as w—>eo if and only if log P(u)~(l+l/a)uSu~uala+1L*(u) as u^><x>. It should be pointed out that in any special case of the function L(u) it will usually be possible to "solve" the equation u=aY(a.Arl)%(ctJt-l) ■(l/su)a+1L(l/su), to obtain an asymptotic formula for su in terms of u. As an example, consider the function L(u) =7C(log w)ai(log log u)az ■ ■ ■ (\ogku)ak, logi u denoting the "&th iterated logarithm of u." This function is slowly oscillating. (See [10, p. 68].) Since log M~(a + 1) log (l/su) we have in this case L(l/su) = (l/(a + l))aiL(u). Thus if L(u) has this special form, the conclusion of the above corollary reduces to: Corollary 1*. Suppose n(u) and P(u) are defined real numbers and a non-negative valued function \p(k) as in the second paragraph of the introduction. Suppose Main Theorem while L(u) is of the form 7C(log w)ai(log Then, n(u)~uaL(u) as w—»°o if and only if logP(«)~f / i\ { 1 + —Kar(a+ / l)f(a+ l)f -J 1 V1) from a set A of positive on the positive integers a and su are as in the log u)^ • • • (log* u)ak. > 1/(a+1) ^/(-h-d^i/ch-i) as u—> oo. This includes the Knopp-Erdos theorem mentioned in the introduction as well as Brigham's theorem and its converse as very special cases. We remark that if Xi, X2, • • • , are positive integers and if the greatest common divisor of those Xi for which yf/(i) St 1 is unity, then an asymptotic relation for log P(u) of the sort given in Corollary 1 or Corollary 1* is equivalent to the same asymptotic relation for log p(n) as n goes to infinity through integral values. In fact, since the additive semigroup generated by a set of coprime positive integers contains all sufficiently large positive integers (see Knopp [7, pp. 60-63] and Ostmann [9, pp. 25-26]), it follows that there is a fixed positive integer c such that every integer not less than c is expressible in the form mi\,-1+m2\h+ • • • , where mi, m2, • • • are nonLicense or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 364 E. E. KOHLBECKER [July negative integers and \p(ji) ^ 1, ^(j'2) ^1, • ■ • . From this it is easy to see that p(n)^p(m) ior any integers re and m with n —c^m^O. Hence P(n - c) P(n) > p(n) ^ —-~ n —c + 1 for integral re^c, from which the asserted equivalence readily follows. When \f/(k) = 1 ior all positive integral values of k, the generating function f(s) = Tit- i (1 —e~'Xt)~'l'm discussed in this paper relates to partitions into parts taken from a set A = {Xi, X2, • • • }, repetitions being allowed. One is led to ask whether or not similar conclusions might be obtained for partitions into distinct parts taken from A. The generating function in the latter case is g(5) = IIi°=i (1 +e~'x*) = 5Zm-o q(vm)e~a"m, say, where va, vx, ■ ■ ■ are again the elements of the additive semi-group generated by A. More generally we have the following result. The special case in which L(u) is a constant was proved by Knopp [7]. Corollary 2. Suppose ip(k) is a function on the positive integers taking non-negative integral values and is such that '%2\kiuip(k)~uc'L(u) as u—><*>, where a is a positive constant and L(u) is a slowly oscillating function. Suppose IL"-i (1 +e-8X")^> = X;.o q(vm)e-"- and Q(u) = £,„*« ff("-)- For large u let Su be a positive number such that u= (1 — l/2a)aT(a and let L**(u) be a slowly oscillating function L**(w)~/l +— Wl - — Jal>+ + l){(a-r-l)(l/su)a+1L(l/su) such that l)f(a+ 1)l(— jji/<«+i) asM_> oo. Then log Q(u) ~ (1 + l/a)usu ~ ual<-a+»L**(u) as « -> oo . Proof. If g(s) is the generating function here and f(s) = IXt°_i (l—e~sXk)~*(-k) then g(s)=f(s)/f(2s). By Theorem 1 we can conclude that log g(s) ~(l-\/2a)r(a + l)£(a + l)(l/s)aL(l/s) as s->0. Under the assumptions on \p we note that Q(u) is nondecreasing and so, since g(s) =sf0"Q(u)e~''udu, the result follows from Theorem 3. Corollary 2*. Suppose \p, q, Q and su are as in Corollary 2 and L(u) is of the form K(\og u)ai(\og log m)"2 • • • (log* u)ak. Then if ^2,\k<,u4'(k)~uaL(u) as «—> oo, we have ioge(M)~(i ■Ml--\aT(a+ + -) l)t(«+ 1)[- ) \ m°/("+1)L(m)i»°+1>05M-> oo. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1958] WEAK ASYMPTOTIC PROPERTIES OF PARTITIONS 365 Bibliography 1. N. A. Brigham, A general asymptotic formula for partition functions, Proc. Amer. Math. Soc. vol. 1 (1950) pp. 182-191. 2. H. Delange, Sur un theoreme de Karamata, Bull. Sci. Math. ser. 2 vol. 79 (1955) pp. 9-12. 3. P. Erdos, On an elementary proof of some asymptotic formulas in the theory of partitions, Ann. of Math. vol. 43 (1942) pp. 437-450. 4. G. H. Hardy, Divergent Series, Oxford, Clarendom Press, 1949. 5. G. H. Hardy and S. Ramanujan, Asymptotic formulas concerning the distribution of in- tegers of various types, Proc. London Math. Soc. vol. 16 (1917) pp. 112-132. 6. J. Karamata, Sur un mode de croissance riguliire des functions, Mathematica. (Cluj) vol. 4 (1930) pp. 38-53. 7. K. Knopp, Asymptotische formeln der additiven Zahlentheorie, Schriften der Konigsberger Gelehrten Gesellschaft, Naturwissenschaftliche Klasse, 2 Jahr vol. 3 (1925) pp. 45-74. 8. J. Korevaar, T. Aardenne-Ahrenfest, and N. G. Debruijn, A note on slowly oscillating functions, Nieuw Archief voor Wiskunde vol. 23 (1949) pp. 77-86. 9. H. H. Ostmann, Additive Zahlentheorie, Erster Teil, Berlin, Springer, 1956. 10. G. P6lya and G. Szegb, Aufgaben und Lehrsatze aus der Analysis, Springer, vol. 1, 1925. University of Illinois, Urbana, III. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Berlin, Julius