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Highly Composite Numbers

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Highly Composite Numbers Highly composite numbers Proceedings of the London Mathematical Society, 2, XIV, 1915, 347 – 409 CONTENTS I. 1. Introduction and summary of results. §§ II. 2–5. Elementary results concerning the order of d(N). 2. The order of d(N) when the prime divisors of N are known. 3. The order of d(N) when the number of prime divisors is known. 4–5. The order of d(N) when nothing is known about N. III. 6–31. The structure of highly composite numbers. 6. Definition of a highly composite number. 7. A table of the first 102 highly composite numbers. 8. First standard form of a highly composite number. The index of the largest prime divisor. 9. Alternative standard form of a highly composite number. 10–24. The indices of the prime divisors in the first standard form. 25–27. The final primes in the second standard form. 28. The ratio of two consecutive highly composite numbers. 29. The form of d(N) for highly composite values of N. 30. The order of dd(N) for highly composite values of N. 31. Some apparently paradoxical forms of d(N) in the table. IV. 32–38. Superior highly composite numbers. 32. Definition of a superior highly composite number. 33. The exact form of a superior highly composite number. 34. The number of divisors of a superior highly composite number. 35. The maximum value of d(N)N 1/x for a given value of x. 36. Consecutive superior highly composite numbers. 37. The number of superior highly composite numbers less than a given number. A table of the first 50 superior highly composite numbers. 38. Superior highly composite numbers and the maximum order of d(N). V. 39–45. Application to the determination of the order of d(N). 39. The maximum order of d(N), assuming the prime number theorem. 40–43. The maximum order of d(N), assuming the Riemann hypothesis. 44. The order of the number of superior highly composite numbers less than N. 45. Highly composite numbers between consecutive superior highly composite numbers. General remarks. Highly composite numbers 99 VI. 46–51. Special forms of N. §§ 46. The maximum order of d(N) for a special form of N. 47. The maximum order of d(N) when d(N) is a power of 2. 48. The maximum order of d(N) when the indices of the prime divisors of N are never less than a given integer. 49. The maximum order of d(N) when N is a perfect nth power. 50. The number of divisors of the least common multiple of the first N natural numbers. 51. The number of divisors of N!. 52. On dd(n) and ddd(n). This paper, as it appeared orginally, is not complete. Since the London Mathematical Society was in some financial difficulty at that time, Ramanujan had to suppress part of what he had written in order to save expense. The unpublished part of the manuscript with annotation has been reproduced: Highly Composite Numbers by Srinivasa Ramanujan, Jean-Louis Nicolas, Guy Robin, The Ramanujan Journal, Volume 1, Issue 2, 1997, pp. 119 – 153. 100 Paper 15 I. Introduction and Summary of Results 1. The number d(N) of divisors of N varies with extreme irregularity as N tends to infinity, tending itself to infinity or remaining small according to the form of N. In this paper I prove a large number of results which add a good deal to our knowledge of the behaviour of d(N). It was proved by Dirichlet ∗ that d(1) + d(2) + d(3) + + d(N) 1 · · · = log N + 2γ 1+ O N − √ † N where γ is the Eulerian constant. Voron¨oi ‡ and Landau § have shewn that the error term 2 +ǫ 2 may be replaced by O(N − 3 ), or indeed O(N − 3 log N). It seems not unlikely that the real 3 +ǫ value of the error is of the form O(N − 4 ), but this is as yet unproved. Mr. Hardy has, however, shewn recently ¶ that the equation d(1) + d(2) + d(3) + d(N) 3 · · · = log N + 2γ 1+ o(N − 4 ) N − is certainly false. He has also proved that d(1) + d(2)) + + d(N 1) + 1 d(N)) N log N (2γ 1)N 1 · · · − 2 − − − − 4 ∞ d(n) = √N [H 4π (nN) Y 4π (nN) ], √n 1{ }− 1{ } 1 X p p where Yn is the ordinary second solution of Bessel’s equation, and ∞ xw 2 we− dw H1(x)= ; π (w2 1) Z1 − p and that the series on the right-hand side is the sum of the series 1 N 4 ∞ d(n) cos 4π (nN) 1 π , √ 3 { − 4 } π 2 1 n 4 X p and an absolutely and uniformly convergent series. ∗Werke, Vol.2, p.49. †f = O(φ) means that a constant exists such that |f| < Kφ : f = o(φ) means that f/φ → 0. ‡Crelle’s Journal, Vol. 126. p. 241. §G¨ottinger Nachrichten, 1912. ¶Comptes Rendus May 10, 1915. Highly composite numbers 101 The “average” order of d(N)) is thus known with considerable accuracy. In this paper I consider, not the average order of d(N)), but its maximum order. This problem has been much less studied. It is obvious that d(N) < 2√N. It was shewn by Wigert∗ that log N (1+ǫ) d(N) < 2 log log N (i) for all positive values of ǫ and all sufficiently large values of N, and that log N (1 ǫ) d(N) > 2 log log N − (ii) for an infinity of values of N. From (i) it follows in particular that d(N) < N δ for all positive values of δ and all sufficiently large values of N. Wigert proves (i) by purely elementary reasoning, but uses the “Prime Number Theo- rem”†to prove (ii). This is, however, unnecessary, the inequality (ii) being also capable of elementary proof. In 5 I shew, by elementary reasoning, that § log N +O log N d(N) < 2 log log N (log log N)2 for all values of N, and log N +O log N d(N) > 2 log log N (log log N)2 for an infinity of values of N. I also shew later on that, if we assume all known results concerning the distribution of primes, then a√(log log N) d(N) < 2Li(log N)+O[log Ne− ] for all values of N, and a√(log log N) d(N) > 2Li(log N)+O[log Ne− ] for an infinity of values of N, where a is a positive constant. I then adopt a different point of view, I define a highly composite number as a number whose number of divisors exceeds that of all its predecessors. Writing such a number in the form N = 2a2 3a3 5a5 pap , · · · · · ∗Arkiv f¨or Matematik, Vol. 3, No.18 x †The theorem that π(x) ∼ log x , π(x) being the number of primes not exceeding x. 102 Paper 15 I prove that a a a a , 2 ≥ 3 ≥ 5 ≥···≥ p and that ap = 1, for all highly composite values of N except 4 and 36. I then go on to prove that the indices near the beginning form a decreasing sequence in the stricter sense, i.e., that a > a > a > > a , 2 3 5 · · · λ where λ is a certain function of p. Near the end groups of equal indices may occur, and I prove that there are actually groups of indices equal to 1, 2, 3, 4,...,µ, where µ again is a certain function of p. I also prove that if λ is fairly small in comparison with p, then log p a log λ ; λ ∼ log 2 and that the later indices can be assigned with an error of at most unity. I prove also that two successive highly composite numbers are asymptotically equivalent, i.e., that the ratio of two consecutive such numbers tends to unity. These are the most striking results. More precise ones will be found in the body of the paper. These results give us a fairly accurate idea of the structure of a highly composite number. I then select from the general aggregate of highly composite numbers a special set which I call “superior highly composite numbers”. I determine completely the general form of all such numbers, and I shew how a combination of the idea of a superior highly composite number with the assumption of the truth of the Riemann hypothesis concerning the roots of the ζ-function leads to even more precise results concerning the maximum order of d(N). These results naturally differ from all which precede in that they depend on the truth of a hitherto unproved hypothesis. II. Elementary Results concerning the Order of d(N). 2. Let d(N) denote the number of divisors of N, and let N = pa1 pa2 pa3 pan , (1) 1 2 3 · · · n where p1,p2,p3,...,pn are a given set of n primes. Then d(N)=(1+ a )(1 + a )(1 + a ) (1 + a ). (2) 1 2 3 · · · n Highly composite numbers 103 From (1) we see that (1/n) log(p p p p N) 1 2 3 · · · n = (1/n) (1 + a ) log p +(1+ a ) log p + +(1+ a ) log p { 1 1 2 2 · · · n n} > (1 + a )(1 + a )(1 + a ) (1 + a ) log p log p log p 1/n. { 1 2 3 · · · n 1 2 · · · n} Hence we have (1/n) log(p p p p N) n d(N) < { 1 2 3 · · · n } , (3) log p log p log p log p 1 2 3 · · · n for all values of N.
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