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THE DISTRIBUTION of VALUES of the DIVISOR FUNCTION D(N) (L-1)

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THE DISTRIBUTION of VALUES of the DIVISOR FUNCTION D(N) (L-1) THE DISTRIBUTION OF VALUES OF THE DIVISOR FUNCTION d(n) By P. ERDiiS and L. MIRSKY [Received 14 March 1X51.-Read 15 March 19511 1. THROUGHOUT this note the letters p, q will be reserved for primes, and pV will denote the vth prime; cr, q,,... are to stand for absolute positive constants. Let d(n) denote, as usual, the number of positive divisors of n, and D(x) the number of distinct values assumed by d(n) in the range 1 < n < x. Our principal object is to estimate the order of magnitude of D(x) for large values of x. The argument will be based on a result concerning A-numbers, which are de&red as integers having the form pyp;“.‘.pg”, where a1 > a2 > . 2 uk and k is arbitrary. If A(z) denotes the number of A-numbers not exceeding 2, then, as was shown by Hardy and Rama- nujan,? logA N 2 ~ 1%X t (x+c(j) (l-1) 113( loglogx 1 We shall be led to consider a certain subclass of the A-numbers, namely the B-numbers, defined as integers having Dhe form 91-l 92-l Pl Pz .*.pp-1, where q1 > q2 Z . >, qk and k is arbitrary. Making use of the one-one correspondence between A-numbers and B-numbers specified by the scheme j$?..p$ tf p~~-~...$+-l (al > . > a,) we shall obtain the following estimate for B(s), the number of B-numbers not exceeding x. THEOREM I. As x --f co, 2771~2 (log x)k log B(x) N - p. 2’3 loglog x This result will, in turn, lead to THEOREY II. As x --f m, 2d2 (log x)” logD(x) N __ p. 213 loglog x t G. H. Hardy and S. Ramanujan, ‘Asymptotic formulae for the distribution of integers of various types’, PTOC. Londo?z ,%fc&. Sot. (2), 16 (1917), 112-32. Re- printed in Collected Papers of S, Ramnnujan (Cambridge, 1927), 245-61. Proc. London Math. SW. (3) 2 (1952) 5388.3.2 S 258 P, ERD6S AND L. MIRSKP The main idea of the proof is as follows. Let m be called & D-number if d(n) # cl(m) for 0 < n < rn.? Then D(z) is evidently the number of D- numbers not exceeding X. We shall show that every D-number is either a B-number or else does not differ greatly from a B-number. This will imply a relation between B(x) and D(x) which will enable us to infer Theorem II from Theorem I. The problem of finding asymptotic formulae for B(z) and D(z) seems diffi- cult. We are, however, able to obtain some results concerning the relative behaviour of B(x) and D(z) for large values of x. Even this is not trivial, for the relation between B and D-the sets of B-numbers and D-numbers respectively-is complicated. Thus there exist infinitely many m such that m E B, m 3 D,$ and infinitely many n such that n g D, n 3 B.$ Some information on the relation between B(x) and D(x) is provided by THEOREM III. For all suficiently large values of x D(x) - B(z) > c1 logloglog x. A more interesting fact is that B(x) and D(z) have the same asymptotic behaviour. We shall, in fact, establish the following result. THEOREM IV. As x -+ co A number of further questions involving d(n) naturally suggest them- selves. Thus, let F(x) denote the greatest integer Tchaving the property that there exists a run of k consecutive integers, say n+ 1, n+2,..., n+k, such that nfk < x and d(n+l), d(n+2),..., d(n+k) are all distinct. We shall prove THEOREM V. For all suficiently lurge values of x F(x) > c$& As regards an upper bound for F(x) we can at present prove nothing better than F(x) -c exp(c3E), an estimate which follows trivially from Theorem II. We conjecture that the true order of magnitude of F(a) is (log x)Q. 7 It is almost obvious that a D-number is necessarily an A-number. $ The symbol a E X means that a is a member of the set X; a 3 X means the contrary. 8 For let k >, 3. If wzk = pl...pk, then WA, E B, WQ 3 D. Also, if nk E D, d(nk) = 2k, then nk 3 B, DISTRIBUTIO,N OF VALUES OF DIVISOR FUKCTION d(n) 259 A related problem consists in the estimation of the longest run of con- secutive integers < x all of which have the same number of divisors. This problem seems to be one of exceptional difficulty, and we have not been able to make any progress with it. We are not even able to prove that there exist infinitely many integers n for which d(n) = d(n+ 1). Let h(z) denote the least positive integer which does not occur among the numbers d(n), 1 < n < X. We shall conclude our note with the proof of the following result. TEEOREM VI. For x > 6, h(x) is equal to the least prime q satisfying the inequality .W-1 > 2. 2. The notation to be used below is as follows: The letter E denotes an arbitrarily small positive number; x denotes a sufficiently large number, i.e. a number exceeding a suitable absolute constant.7 The O-notation and the asymptotic formulae refer to the passage x -+ co. As usual r(x) stands for the number of primes not exceeding x, and 8(x) for the sum BzrclogpS The set of A-numbers will be denoted by A. Given any k, there obviously exists a unique m E D such that d(m) = k. Moreover, there exists a unique m* E B such that3 d(m*) = Ic. Hence a one-one correspondence can be set up between B and B, specified by the conditions d(m) = d(m*), mED, m*EB. If m E D, then m* will invariably denote the B-number corresponding to m. It is, of course, obvious that m* > m. If n = p’41...$+, (2.1) we shall write {n) = p$..p$, where a;,..., ai is a permutation of a, ,..., ak such that a; 3 . > a;. Evi- dently (n> < n. If the canonical representation of an integer n is written in the form (2.1), it will be referred to as its expansion. We shall also speak of the expansion of n with expbnents > 2, of the expansion of n with primes > z, and so on, when referring to the parts of the expansion of n having these properties, 3. We shall make frequent use of two simple lemmas. t Whenever z is required to exceed a bound depending on E, that fact will be stated explicitly. $ For, if k = qr,..qS, where q1 > . 2 qS, then m* = pp-l...p$-1. 260 P, ERD6S AND L. XIRSKY LEMMA 1. Let n, t be positive integers. If N(n, t) denotes the number of sets of integers ul,,.., u, such that t>uul> . > u, > 0, (3.1) then N(n, t) < (2n)2t. Consider the sets of integers k,, k,,.. ., kt such that k,+...+k, = n; &,...,Ic, > 0. (3.2) A one-one correspondence between the sets or,..., u, satisfying (3.1) and the sets k,,,..., kt satisfying (3.2) may be established by the requirement that k, should be the number of U’S having the value v. Hence fV(n, t) is equal to the number of sets of k’s satisfying (3.2), and therefore N(n, t) < (n+ l)t+l < (2n)U. LEMMA 2. If m = pp...pEk ED and, for some i, ai+ = tt’, where t > t’ > 2, then Pu4 G Pkil, (3.3) and, if m is suficiently large, t < 2 loglog m, a,+1 < 4(loglogm)2. Write ml = pi,. .pFL-i p f -lpF$+; . .pjjkpi ; :. Then d(m,) = d(m), and so rnr > m. Hence (3.3) follows at once. More- over, if m is sufficiently large, pk < #Mm, r)ktl < 210gm, log( 2 log m) < 2 loglog m, log 2 a,+1 < t2 < 4(loglogm)2. 4. In this and the next section we give a proof of Theorem I. We shall write y = ~~2---El/lO&m , and shall assume that x > ~~(6). Let m G A, m < y, and suppose that p > (logx)~/(loglogx)2 is a prime factor of m. Then the exponent of $J is < 3(log s)tloglog z; for otherwise y >, m 3 (,g y)3(‘ogz’*10g’og5, (gee) log > 3(log z)*loglog X. a(p) loglog X > 3(log s)~loglog x. $p 3 (logx)& 9 logz > 3(log z)~loglog x. - 4 (loglog X)2 = 4 loglog* DISTRIBUTION OF VALUES OF DIVISOR FUNCTION d(n) 261 Two A-numbers < y will be said to belong to the same class if they coincide in their expansions with prime factors > (logx)i/(loglogz)a. In view of the remark made above the corresponding exponents must all be < 3(log x)*loglog 5. An A -number m = p$..p? < y (al > . > ak) will be called restricted if a, < 3(logx)~loglogx. The number of such numbers will be denoted by A&). We note at once that each class contains at least one restricted number.t Hence A&) > C(x), where C(s) is the number of classes. If K,(x) denotes the number of numbers in the ith class, then 4y) = K,(x)+K,(x)+...+K,~,,(x).
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