Highly Composite Numbers by Srinivasa Ramanujan

Transcription

1 THE RAMANUJAN JOURNAL, c 997 Kluwer Academic Publishers. Manufactured in The Netherlands. Highly Composite Numbers by Srinivasa Ramanujan Annotated by JEAN-LOUIS NICOLAS [email protected] Institut Girard Desargues, UPRES-A-508, Mathématiques, Bâtiment 0, Université Claude Bernard LYON, F-696 Villeurbanne cédex, France GUY ROBIN UPRES-A-6090, Théorie des nombres, calcul formel et optimisation, Université de Limoges, F Limoges cédex, France [email protected] Received December 5, 995; Accepted April, 996 Abstract. In 95, the London Mathematical Society published in its Proceedings a paper of Ramanujan entitled Highly Composite Numbers. But it was not the whole work on the subject, and in The lost notebook and other unpublished papers, one can find a manuscript, handwritten by Ramanujan, which is the continuation of the paper published by the London Mathematical Society. This paper is the typed version of the above mentioned manuscript with some notes, mainly explaining the link between the work of Ramanujan and works published after 95 on the subject. A number N is said highly composite if M < N implies dm <dn, where dn is the number of divisors of N. In this paper, Ramanujan extends the notion of highly composite number to other arithmetic functions, mainly to Q k N for k 4 where Q k N is the number of representations of N as a sum of k squares and σ s N where σ s N is the sum of the sth powers of the divisors of N. Moreover, the maximal orders of these functions are given. Key words: highly composite number, arithmetical function, maximal order, divisors 99 Mathematics Subject Classification: Primary N56; Secondary M6. Foreword In 95, the London Mathematical Society published in its Proceedings a paper of Srinivasa Ramanujan entitled Highly Composite Numbers. cf. [6]. In the Collected Papers of Ramanujan, this article has number 5, and in the notes cf. [7], p. 9, it is stated: The paper, long as it is, is not complete. The London Math. Soc. was in some financial difficulty at the time and Ramanujan suppressed part of what he had written in order to save expenses. This suppressed part had been known to Hardy, who mentioned it in a letter to Watson, in 90 cf. [8], p. 9. Most of this suppressed part can be now found in the lost notebook and other unpublished papers cf. [8], p. 80 to. An analysis of this book has been done by Rankin, who has written several lines about the pages concerning highly composite numbers cf. [9], p. 6. Also, some information about this subject has already been published in [], pp. 8 9 and []. Robin cf. [5] has given detailed

2 0 NICOLAS AND ROBIN proofs of some of the results dealing with complex variables, and Riemann zeta function, since as usual, Ramanujan sometimes gives formulas which probably were obvious to him, but not to most mathematicians. The article below is essentially the end of the paper written by Ramanujan which was not published in [6], but can be read in [8]. For convenience, we have kept on the numbering both of paragraphs which start from 5 to 75 and formulas from 68 to 408, so that references to preceding paragraphs or formulas can easily be found in [6]. There is just a small overlap: the last paragraph of [6] is numbered 5, and contains formulas 68 and 69. This last paragraph was probably added by Ramanujan to the first part after he had decided to suppress the second part. However this overlap does not imply any misunderstanding. There are two gaps in the manuscript of Ramanujan, as presented in the lost notebook. The first one is just at the beginning, where the definition of Q n is missing. Probably this definition was sent to the London Math. Soc. in 95 with the manuscript of Highly Composite Numbers. It has been reformulated in the same terms as the definition of Q n given in Section 55. The second gap is more difficult to explain: Section 57 is complete and appears on pp. 89 and 90 of [8]. But the lower half of p. 90 is empty, and p. 9 starts with the end of Section 58. We have completed Section 58 by giving the definition of σ s N, and the proof of formula 0. All these completions are written in italics in the text below. It should be noted that in [8] pp are not handwritten by Ramanujan, and, as observed by Rankin cf. [9], p. 6 were probably copied by Watson, but that does not create any gap in the text. Pages 8 and 8 of [8] do not belong to number theory, and clearly the text of p. 84 follows p. 8. On the other hand, pp. 09 deal with highly composite numbers. With the notation of [6], Section 9, Ramanujan proves in pp that log p r log + /r = log p log + Or holds, while on pp., he attempts to extend the above formula by replacing p by p s. More precise results can now be found in [7]. Pages 09 do not belong to the paper Highly Composite Numbers and are not included in the paper below. In the following paper, Ramanujan studies the maximal order of some classical functions, which resemble the number, or the sum, of the divisors of an integer. In Section 5 54, Q N, the number of representations of N as a sum of two squares is studied, and its maximal order is given under the Riemann hypothesis, or without assuming the Riemann hypothesis. In Section 55 56, a similar work is done for Q N the number of representation of N by the form m + mn + n. In Section 57, the number of ways of writing N as a product of + r factors is briefly investigated. Between Section 58 and Section 7, there is a deep study of the maximal order of σ s N = d s d N under the Riemann hypothesis, by introducing generalised superior highly composite numbers. In Section 7 74, Q 4 N, Q 6 N and Q 8 N the numbers of representations of N as a sum of 4, 6 or 8 squares are studied, and also their maximal orders. In the last paragraph

3 HIGHLY COMPOSITE NUMBERS 75, the number of representations of N by some other quadratic forms is considered, but no longer its maximal order. One feels that Ramanujan is ready to leave the subject of highly composite numbers, and to come back to another favourite topic, identities. The table on p. 50 occurs on p. 80 in [8]. It should be compared with the table of largely composite numbers p. 5, namely the numbers n such that m n dm dn. Several results obtained by Ramanujan in 95, but kept unpublished, have been rediscovered and published by other mathematicians. The references for these works are given in the notes at the end of this paper. However, there remain in the paper of Ramanujan, some never published results, for instance, the maximal order of Q N cf. Section 54 or of σ s N cf. Section 7 whenever s. The case s = has been studied by Robin, cf. []. A few misprints or mistakes were found in the manuscript of Ramanujan. Finally, it puts one somewhat at ease that even Ramanujan could make mistakes. These mistakes have been corrected in the text, but are also pointed out in the notes. Hardy did not much like highly composite numbers. In the preface to the Collected Works cf. [7], p. XXXIV he writes that The long memoir [6] represents work, perhaps, in a backwater of mathematics, but a few lines later, he does recognize that it shews very clearly Ramanujan s extraordinary mastery over the algebra of inequalities. One of us can remember Freeman Dyson in Urbana in 987 saying that when he was a research student of Hardy, he wanted to do research on highly composite numbers but Hardy dissuaded him as he thought the subject was not sufficiently interesting or important. However, after Ramanujan, several authors have written about them, as can be seen in the survey paper []. We think that the manuscript of Ramanujan should be published, since he wrote it with this aim, and we hope that our notes will help readers to a better understanding. We are indebted to Berndt, and Rankin for much valuable information, to Massias for calculating largely composite numbers and finding the meaning of the table occurring in [8], p. 80 and to Lydia Szyszko for typing this manuscript. We thank also Narosa Publishing House, New Delhi, for granting permission to print in typed form the handwritten manuscript on Highly Composite Numbers which can be found in pages 80 of [8].. The text of Ramanujan 5. Let Q N denote the number of ways in which N can be expressed as m + n. Let us agree to consider m + n as two ways if m and n are unequal and as one way if they are equal or one of them is zero. Then it can be shown that From this it easily follows that + q + q 4 + q 9 + q 6 + q = +4 q q q + q5 q 5 q7 q + 7 = +4Q q + Q q + Q q + } 68 ζsζ s = Q s + Q s + Q s +, 69

4 NICOLAS AND ROBIN where ζ s = s s + 5 s 7 s +. Since q q + q q + q q + = dq +dq +dq +, it follows from 68 that for all values of N. Let Q N dn 70 N = a. a.5 a5 p a p, where a λ 0. Then we see that, if any one of a, a 7, a,..., be odd, where, 7,,..., are the primes of the form 4n, then But, if a, a 7, a,...be even or zero, then Q N = 0. 7 Q N = + a 5 + a + a 7 7 where 5,, 7,...are the primes of the form 4n +. It is clear that 70 is a consequence of 7 and From 7 it is easy to see that, in order that Q N should be of maximum order, N must be of the form where p is a prime of the form 4n +, and 5 a 5. a.7 a 7 p a p, a 5 a a 7 a p. Let π x denote the number of primes of the form 4n + which do not exceed x, and let ϑ x = log 5 + log + log 7 + +log p, where p is the largest prime of the form 4n +, not greater than x. Then by arguments similar to those of Section we can show that π x π Q N N x x 4 π 4 x 7 e x θ x e x θ x e x θ 4 x for all values of N and x. From this we can show by arguments similar to those of Section 8 that, in order that Q N should be of maximum order, N must be of the form e ϑ x +ϑ x +ϑ 4 x +

5 HIGHLY COMPOSITE NUMBERS and Q N of the form π x π x 4 π 4 x.... Then, without assuming the prime number theorem, we can show that the maximum order of Q N is log N log log N + O log log N }. 74 Assuming the prime number theorem we can show that the maximum order of Q N is Li log N+Olog log N Ne a } 75 where a is a positive constant. 54. We shall now assume the Riemann Hypothesis and its analogue for the function ζ s. Let ρ be a complex root of ζ s. Then it can be shown that = γ log π + log + 4 log Ɣ ρ 4, so that ρ + ρ = + γ log π + 4 log Ɣ It can also be shown that ϑ x = x x x ρ /ρ x ρ /ρ + O x π x = Lix Li x Lix ρ Lix ρ + O 77 x so that ϑ x = x + O xlog x π x = Lix + O x log x. 78 Now π x = Lix x + x ρ log x ρ + x ρ ρ 4 x + x ρ log x ρ + x ρ + O x log x. ρ But by Taylor s Theorem we have Liϑ x}=lix x + x ρ log x ρ + x ρ + Olog x }. ρ

6 4 NICOLAS AND ROBIN Hence where } x π x = Liϑ x} R x+o log x R x = It can easily be shown that x + ρ + ρ x + x ρ log x ρ + x ρ ρ. R xlog x x ρ ρ 79 and so from 76 we see that } x + γ log π + 4 log Ɣ R xlog x 4 } x. γ + log π 4 log Ɣ 80 4 It can easily be verified that + γ log π + 4 log Ɣ 4 =.0, γ + log π 4 log Ɣ 8 4 =.899, approximately. Proceeding as in Section 4 we can show that the maximun order of Q N is where N = log log Li Li log N+ N 8 } } log Nlog/ log log N log Nlog/ 4 log log N R log N + O. log log N 55. Let Q N denote the number of ways in which N can be expressed as m + mn +n. Let us agree to consider m + mn +n as two ways if m and n are unequal, and as one way if they are equal or one of them is zero. Then it can be shown that q 4 + q 4 + q q 4 + q 4 + q q 4 +q 4 q 4 + q 4 + q 4 q q = +6 q q q + q4 q 4 q5 q + 5 = +6 Q q + Q q + Q q + } 8

7 HIGHLY COMPOSITE NUMBERS 5 where,, 4, 5,...are the natural numbers without the multiples of. From this it follows that where It also follows that for all values of N. Let ζsζ s = s Q + s Q + s Q + 84 ζ s = s s + 4 s 5 s + Q N dn 85 N = a. a.5 a5 p a p, where a λ 0. Then, if any one of a, a 5, a,...be odd, where, 5,,...are the primes of the form n, then But, if a, a 5, a be even or zero, then Q N = Q N = + a 7 + a + a 9 + a 87 where 7,, 9,...are the primes of the form 6n +. Let π x be the number of primes of the form 6n + which do not exceed x, and let ϑ x = log 7 + log + log 9 + +log p, where p is the largest prime of the form 6n + not greater that x. Then we can show that, in order that Q N should be of maximum order, N must be of the form and Q N of the form π x e ϑ x +ϑ x +ϑ 4 x + π x 4 π 4 x.... Without assuming the prime number theorem we can show that the maximum order of Q N is log N log log N + O log log N }. 88 Assuming the prime number theorem we can show that the maximum order of Q N is Li log N+Olog Ne a log N}. 89

8 6 NICOLAS AND ROBIN 56. We shall now assume the Riemann hypothesis and its analogue for the function ζ s. Then we can show that π x = Liϑ x} R x+o x/log x } 90 where R x = x + x ρ log x ρ + x ρ } ρ where ρ is a complex root of ζ s. It can also be shown that ρ + = + γ + Ɣ log + log ρ Ɣ 9 and so + γ + log + log Ɣ x } Ɣ R xlog x γ log log Ɣ x. } Ɣ 9 It can easily be verified that + γ + log + log Ɣ Ɣ =.080, γ log log Ɣ 9 Ɣ =.90, approximately. Then we can show that the maximum order of Q N is Li log N+ N 94 where N = log/ } log Li log Nlog/ log log N log Nlog/ } 4 log log N R log N+O. log log N 57. Let d r N denote the coefficient of N s in the expansion of ζs} +r as a Dirichlet series. Then since it is easy to see that, if ζs} = s s 5 s p s..., N = p a pa pa...pa n n,

9 HIGHLY COMPOSITE NUMBERS 7 where p, p, p...are any primes, then ν=n d r N = λ=a ν ν= λ= + r λ 95 provided that r >. It is evident that and that, if r 0, then d N = 0, d 0 N =, d N = dn; for all values of N. It is also evident that, if N is a prime then d r N + r 96 d r N = + r for all values of r. It is easy to see from 95 that, if r > 0, then d r N is not bounded when N becomes infinite. Now, if r is positive, it can easily be shown that, in order that d r N should be of maximum order, N must be of the form e ϑx +ϑx +ϑx +, and consequently d r N of the form + r πx + r πx + r πx... and proceeding as in Section 46 we can show that N must be of the form e ϑ+rx +ϑ+ r x +ϑ+ r x + 97 and d r N of the form + r π+rx + r π+ r x + r π+ r x From 97 and 98 we can easily find the maximum order of d r N as in Section 4. It may be interesting to note that numbers of the form 97 which may also be written in the form approach the form e ϑx r log+r }+ϑx r log+ r }+ϑx r log+ r }+ e ϑx+ϑ x+ϑx / + as r 0. That is to say, they approach the form of the least common multiple of the natural numbers as r 0.

10 8 NICOLAS AND ROBIN 58. Let s be a non negative real number, and let σ s N denote the sum of the inverses of the sth powers of the divisors of N. If N denotes N = p a pa pa...pa n n where p, p, p,...are any primes, then σ s N = + p s + p s + p s + + p a s + p s + p s + p s + + p a s + p s n + pn s + pn s + + p a ns n. For s = 0,σ 0 N=dNis the number of divisors of N. For s > 0, we have: σ s N = p a +s p s p a +s p s Now, from the concavity of the function log e t, we see that p a n+s n pn s. 99 n log e t + log e t + +log e t n } log exp t } + t + +t n. 00 n Choosing t = a +s log p, t = a +s log p,...,t n =a n +s log p n in 00, formula 99 gives σ s N < p p p p n N s/n} n p s p s p s. 0 n By arguments similar to those of Section we can show that it is possible to choose the indices a, a, a,...,a n so that σ s N = p p p p n N s/n } n p s p s p s ON s/n log N /n }}. 0 n There are of course results corresponding to 4 and 5 also. 59. A number N may be said to be a generalised highly composite number if σ s N > σ s N for all values of N less than N. We can easily show that, in order that N should be a generalised highly composite number, N must be the form a a 5 a5 p a p 0

11 HIGHLY COMPOSITE NUMBERS 9 where a a a 5 a p =, the exceptional numbers being 6, for the values of s which satisfy the inequality s + 4 s + 8 s > s + 9 s, and 4 in all cases. A number N may be said to be a generalised superior highly composite number if there is a positive number ε such that for all values of N less than N, and σ s N N ε σ sn N ε 04 σ s N N ε > σ sn N ε 05 for all values of N greater than N. It is easily seen that all generalised superior highly composite numbers are generalised highly composite numbers. We shall use the expression and the expression a a 5 a5 p a p p p 5... p.... as the standard forms of a generalised superior highly composite number. 60. Let N = N λ where λ p. Then from 04 it follows that or λ s+a λ λ sa λ λ ε, λ ε λ s+a λ. 06 λ sa λ Again let N = Nλ. Then from 05 we see that λ s+a λ > λ s+a λ } λ ε

12 0 NICOLAS AND ROBIN or λ ε > λ s+a λ λ. 07 s+a λ Now let us suppose that λ = p, in 06 and λ = P in 07. Then we see that log + p s ε> log + P s. 08 log p log P From this it follows that, if 0 <ε log + s, log then there is a unique value of p corresponding to each value of ε. It follows from 06 that λε λ s log λ a λ ε, 09 s log λ and from 07 that + a λ > λε λ s log λ ε. 0 s log λ From 09 and 0 it is clear that [ log λ ] ε λ s λ a λ = s log λ. Hence N is of the form [ log ε s ε s log ] log ε s [ ε s log ]...p where p is the prime defined by the inequalities Let us consider the nature of p r. Putting λ = p r in 06, and remembering that a pr r, we obtain pr ε p s+a pr r p sa pr r p sr+ r pr sr. Again, putting λ = P r in 07, and remembering that a Pr r, we obtain r > P s+apr r P s+a Pr P ε r P sr+ r Pr sr. 4

13 HIGHLY COMPOSITE NUMBERS It follows from and 4 that, if x r be the value of x satisfying the equation x ε = x sr+ x sr 5 then p r is the largest prime not greater than x r. Hence N is of the form e ϑx +ϑx +ϑx + 6 where x r is defined in 5; and σ s N is of the form where x x x a x a 7 r x = sr+ sr sr+ sr p sr+ p sr. and p is the largest prime not greater than x. It follows from 04 and 05 that σ s N N ε x e εϑx x e εϑx x e εϑx 8 for all values of N, where x, x, x,...are functions of ε defined by the equation xr ε = x r sr+ xr sr, 9 and σ s N is equal to the right hand side of 8 when 6. In 6 let us suppose that Then we see that N = e ϑx +ϑx +ϑx + x = log x sr+ x sr. log r x r = πx r log x r sr+ πx r d log x r sr+ xr sr xr sr = πx r log xr ε πx r d log xr ε επxr = επx r log x r πx r log x r dε dx r x r

14 NICOLAS AND ROBIN in virtue of 9. Hence επxr log r x r εϑx r = επx r log x r ϑx r } πx r log x r dε dx r x r πxr επxr = ε dx r πx r log x r dε dx r x r x r πxr = dε dx r πx r log x r dε x r } πxr = dx r πx r log x r dε x r = ϑx r dε. 0 It follows from 8 and 0 that σ s N N ε e ϑx +ϑx +ϑx + }dε for all values of N. By arguments similar to those of Section 8 we can show that the right hand side of is a minimum when ε is a function of N defined by the equation N = e ϑx +ϑx +ϑx +. Now let s N be a function of N defined by the equation N = s x x x where ε is a function of N defined by the Eq.. Then it follows from 8 that the order of σ s N s N for all values of N and σ s N = s N for all generalised superior highly composite values of N. In other words σ s N is of maximum order when N is of the form of a generalised superior highly composite number. 6. We shall now consider some important series which are not only useful in finding the maximum order of σ s N but also interesting in themselves. Proceeding as in 6 we can easily show that, if x be continuous, then log + log + 5 log p log p = xθx x tθt dt 4 where p is the largest prime not exceeding x. Since x dx = x x x x dx, we have xϑx xϑx dx = x dx x ϑx} x + xx ϑx} dx. 5

15 HIGHLY COMPOSITE NUMBERS Remembering that x ϑx = O xlog x }, we have by Taylor s Theorem θx t dt = x dx x ϑx} x + x ϑx} x + O xlog x }. 6 It follows from 4 6 that log + log + 5 log p log p θx = C + t dt + xx ϑx} dx x ϑx} x + O xlog x } 7 where C is a constant and p is the largest prime not exceeding x. 64. Now let us assume that x = where s > 0. Then from 7 we see that, if p x s is the largest prime not greater than x, then log s + log s + log 5 5 s = C + But it is known that θx dx x s s log p + + p s x ϑx x s x s dx + Ox s log x 4 }. 8 x θx = x ρ x + x + ρ x ρ ρ + O x 5 9 where ρ is a complex root of ζs. By arguments similar to those of Section 4 we can show that x ρ s ρ ρ s = x s x ρ ρ dx. Hence x ρ ρ x s x s dx = O x s x ρ ρ } } x ρ s dx = O ρ ρ s = O x s 4. Similarly x ρ ρ x s x s dx = x ρ s ρρ s +O x ρ s = x ρ s ρρ s ρρ s +O x s.

16 4 NICOLAS AND ROBIN Hence 8 may be replaced by log s + log s + log 5 log p s p s θx dt = C + t s s x + x x s x s dx s x ρ s ρρ s + O x s + x s 4. 0 It can easily be shown that C = ζ s ζs when the error term is o. 65. Let S s x = s x ρ s ρρ s. Then S s x s x ρ s ρρ s = s x s. ρ ρρ s ρ s} If m and n are any two positive numbers, then it is evident that / mn lies between m and n. Hence ρ ρρ s ρ s} lies between χ and χs where χs = ρ s ρ s = ρ ρ + s s = s ρ s + = s ρ ρ s s /. We can show as in Section 4 that Hence so that s ρ = s s s log π + χs = s s + } Ɣ s s Ɣ + ζ s s ζs log π Ɣ s Ɣ + ζ s s ζs. 4 5 χ0 = χ = +γ log 4π. 6

17 HIGHLY COMPOSITE NUMBERS 5 By elementary algebra, it can easily be shown that if m r and n r be not negative and G r be the geometric mean between m r and n r then G + G + G + < m +m +m + }n + n + } 7 unless m n = m n = m n = From this it follows that ρ ρρ s ρ s} < χχs}. 8 The following method leads to still closer approximation. It is easy to see that if m and n are positive, then / mn is the geometric mean between m + 8 m + n and n + 8 m + n 9 and so mn lies between both. Hence ρ ρρ s ρ s} lies between ρ ρ + ρ s ρ s + ρ ρ + 4 s s and ρ ρ + 4 s s 40 and is also less than the geometric mean between these two in virtue of 7 } + s + ρ ρ + 4 s s = χ s and } + s + ρ ρ + 4 s s = χ s. = ρ ρρ s ρ s} ρ ρ s s ρ ρ + 8 ρ ρ + ρ ρ + 4 s s = ρ ρ s s ρ ρ + 8 ρ ρ + s s + ρ ρ + 4 s s = ρ ρ s s ρ ρ + 8 s s ρ ρ s s ρ ρ s s ρ ρ } 0 } 9 } s s ρ ρ s s ρ ρ s s ρ ρ } + } + } + Since the first value of ρ ρ is about 00 we see that the geometric mean is a much closer approximation than either.

18 6 NICOLAS AND ROBIN Hence lies between ρ ρρ s ρ s} χ+ } + s χ +s and χs+ } + s χ +s 4 and is also less than the geometric mean between these two. 66. In this and the following few sections it is always understood that p is the largest prime not greater than x. It can easily be shown that θx dt t s s x + x θx} s dx = + θx} s x s x s s s + x s s + x 4s x ns + + 4s ns sx s s sx s s 4sx s 4s + O x s 4 where n = [ + s ]. It follows from 0 and 4 that if s > 0, then log s + log s + log 5 log p s p s = ζ s ζs + ϑx} s + ϑx} s + x s s s s + x 4s x ns + + 4s ns sx s s sx s s 4sx s 4s + S sx + O x s + x s 4 4 where n = [ + s ]. When s =,, or we must take the limit of the right hand side when s approaches 4,, or. We shall consider the following cases: 4 Case I. 0 < s < 4 where n = [ + s ]. log s + log s + log 5 log p s p s = ϑx} s + ϑx} s s s + x ns ns sx s + x s s + x 4s 4s + s s sx s + S sx + O x s, 44

19 HIGHLY COMPOSITE NUMBERS 7 Case II. s = 4 Case III. s > 4 log log log 5 log p p 4 = 4 ϑx} 4 + ϑx}+x 4 x + log x + S x + O log s + log s + log 5 log p s p s = ζ s ζs + ϑx} s + x s sx s + x s sx s + S s x + O x s 4. s s s Making s in 46, and remembering that v s lim s s ζ } s = log v γ ζs where γ is the Eulerian constant, we have log + log + log 5 log p p = log ϑx γ + x + x + S x + O x From we know that x S x +γ log4π = approximately, for all positive values of x. When s >, 46 reduces to log s + log s + log 5 log p s p s = ζ s ζs + ϑx} s + sx s s s + sx s s + S sx + O x s 4 49 Writing Ox s for S s x in 4, we see that, if s > 0, then log s + log s + log 5 log p s p s = ζ s ζs + ϑx} s + x s s s + x s s + when n = [ + s ]. ns sx s s + O x s 50 + x ns

20 8 NICOLAS AND ROBIN Now the following three cases arise: Case I. 0 < s < log s + log s + log 5 log p s p s = ϑx} s s where n = [ + s ]. Case II. s = + x s s + x s x ns + + s ns + O x s 5 log + log + log log p = ϑx}+ 5 p log x + O. 5 Case III. s > log s + log s + log 5 log p s s p s = ζ ζs + ϑx} s + O x s. s We shall now consider the product It can easily be shown that where Lix is the principal value of x 0 s s 5 s p s. x a+bs a + bs ds = b Lixa+bs 54 dt ; and that log t S s x ds = S sx x log x + O s }. 55 log x Now remembering 54 and 55 and integrating 4 with respect to s, we see that if s > 0, then log s s 5 s p s } = log ζs Liϑx} s Lix s Lix s n Lix ns + Li x s x s + S s x x s } + O log x log x 56 where n = [ + s ].

21 HIGHLY COMPOSITE NUMBERS 9 Now the following three cases arise. Case I. 0 < s < log s s 5 s p s } = Liϑx} s Lix s Lix s sx s n Lix ns + slog x S sx log x + O x s /log x } 57 where n = [ + s ]. Making s in 56 and remembering that limli + h log h }=γ 58 h 0 where γ is the Eulerian constant, we have Case II. s = 5 = ζ It may be observed that ζ exp p Li + S x θx + + O log x log x }. 59 = Case III. s > s s 5 s p s [ = ζs exp Liθx s }+ sx s s log x + S sx log x + O x s log x }]. 6 Remembering 58 and making s in 6 we obtain =e 5 γ log ϑx + + S x + O }. p x x log x 6 It follows from this and 47 that e γ 5 p = γ + log + log log p + + p + O p log p. 6

22 40 NICOLAS AND ROBIN 69. We shall consider the order of x r. Putting r = in 9 we have ε = log + x s ; log x and so Let Then we have x r log+x s log x + x s From this we can easily deduce that Hence and so t r = + = x r sr+ xr sr. 64 x r = x t r /r. tr /r x st r+ r = x st. r log r + O s log x log x } x r = x /r r /rs + O ; 65 log x }. x r r /s x /r. 66 Putting λ = in we see that the greatest possible value of r is a = log ε s log + O = log x log + log log x + O. 67 s log Again log N = ϑx + ϑx + ϑx + =ϑx + x + O x / 68 in virtue of 66. It follows from Section 68 and the definition of r x, that, if sr and sr + are not equal to, then r x = ζsr eox sr ζsr + } ; and consequently r x r = ζsr ζsr + } eo r x s 69

23 HIGHLY COMPOSITE NUMBERS 4 in virtue of 66. But if sr or sr + is unity, it can easily be shown that r x r r x r r+ x r+ = ζ sr } ζ sr + } eox r s We shall now consider the order of s N i.e., the maximum order of σ sn. It follows from 7 that if s, then N = s x x ζs e Ox s 7 in virtue of 67, 69 and 70. But if s =, we can easily show, by using 6, that N = s x x e Olog log x. 7 It follows from Section 68 that log x = log ζs ζs + Liθx } s Liϑx } s + Liϑx } s n Liϑx } ns n Li x s x s + + S s x x s } + O log x log x 7 where n = [ + ]; and also that, if s, then, s log x = log ζs ζs + Li x s x s } + O ; 74 log x and when s = log x = Li x s x s } + O. 75 log x It follows from 7 75 that log s N = log ζs +Liϑx } s Liϑx } s + Liϑx } s n Liϑx } ns n Li x s + Li x s x s + + S s x x s } + O log x log x 76

24 4 NICOLAS AND ROBIN where n = [ + ]. But from 68 it is clear that, if m > 0 then s Liϑx } ms = Li log N x + O x / } ms = Li log N ms msx log N ms + O x ms = Lilog N ms x log N ms log log N + O x ms. By arguments similar to those of Section 4 we can show that Hence S s x = S s log N + O x log x } = S s log N + O x s log x 4}. } log N = log ζs +Lilog s N s Lilog N s + Lilog N s n n + log N s + S s log N log log N x log N s log log N Lilog N ns Lilog N s + O + Li x s } log N s log log N 77 where n = [ + s ] and x = /s x x + O = /s } log N log N + O log x log log N 78 in virtue of Let us consider the order of s N in the following three cases. Case I. 0 < s <. Here we have Lilog N s = log N s log N s log log N + O s log log N } Li x s x s = x s + O slog x log x log N s = /s log N s s log log N + O log log N x log N s log log N = /s log N s log N s + O log log N log log N }. }. }.

25 HIGHLY COMPOSITE NUMBERS 4 It follows from these and 77 that log N = Lilog s N s Lilog N s + Lilog N s n n Lilog N ns + s/s log N s s log log N + S slog N log N log log N + O s } log log N 79 where n = [ + s ]. Remembering 58 and 78 and making s in 77 we have Case II. s =. N = ζ Case III. s >. exp Li log N + log + S log N + log log N N = ζs exp Lilog N s s/s s s + S slog N log N log log N + O s log log N Now making s in this we have N = eγ log log N log N + S log N + } O log log N log N s } log log N 80 }. 8 O log N log log N }. 8 Hence } log Lim N eγ log log N N e γ +γ log 4π =.558 approximately and } log Lim N eγ log log N N e γ 4 γ+log 4π =.9 approximately. The maximum order of σ s N is easily obtained by multiplying the values of s N by N s. It may be interesting to see that x r x /r as s ; and ultimately N assumes the form e ϑx +ϑ x +ϑx / +

26 44 NICOLAS AND ROBIN that is to say the form of a generalised superior highly composite number approaches that of the least common multiple of the natural numbers when s becomes infinitely large. The maximum order of σ s N without assuming the prime number theorem is obtained by changing log N to log Ne O in all the preceding results. In particular N = eγ log log N + O} Let + q + q 4 + q =+8Q 4 q + Q 4 q + Q 4 q + }. Then, by means of elliptic functions, we can show that Q 4 q + Q 4 q + Q 4 q + 84 = q q + q + q + q q + 4q4 + q + 4 = q q + q q + q q + 4q4 +q + 4 4q 4 q + 8q8 4 q + q +. 8 q But q q + q q + q q + =σ q +σ q +σ q +. It follows that for all values of N. It also follows from 84 that Let Q 4 N σ N 85 4 s ζsζs = s Q 4 + s Q 4 + s Q where a λ 0. Then, the coefficient of q N in N = a a 5 a5 p a p is and that in q q + q q + q q + N a a 5 a5 p ap ; 5 p 4q 4 q + 8q8 4 q + q + 8 q

27 HIGHLY COMPOSITE NUMBERS 45 is 0 when N is not a multiple of 4 and N a a 5 a5 p ap 5 p when N is a multiple of 4. From this and 84 it follows that, if N is not a multiple of 4, then Q 4 N = N a a 5 a5 p ap ; 87 5 p and if N is a multiple of 4, then Q 4 N = N a a 5 a5 p ap p It is easy to see from 87 and 88 that, in order that Q 4 N should be of maximum order, a must be. From 8 we see that the maximum order of Q 4 N is 4 eγ log log N log N + S log N + = 4 eγ log log N + O }. log N It may be observed that, if N is not a multiple of 4, then Q 4 N = σ N; } O log N log log N 89 and if N is a multiple of 4, then 7. Let Q 4 N = σ N a +. + q + q 4 + q =+Q 6 q + Q 6 q + Q 6 q + }. Then, by means of elliptic functions, we can show that Q 6 q + Q 6 q + Q 6 q + = 4 q +q + q +q + q 4 +q + 6 But q q q q + 5 q 5 q σ q + σ q + σ q + } = 4 q q + q q + q } q + + q q + q q + q } q +.

28 46 NICOLAS AND ROBIN It follows that Q 6 N 5σ N 9 for all values of N. It also follows from 90 that Let 4 ζs ζ s ζsζ s = s Q 6 + s Q 6 + s Q N = a a 5 a5 p a p, where a λ 0. Then from 90 we can show, as in the previous section, that if a N be of the form 4n +, then Q 6 N = N a a and if a N be of the form 4n, then Q 6 N = N + a a 5 a a5 + 5 p p } a p ; p p 9 p p } a p. p p 94 It follows from 9 and 94 that, in order that Q 6 N should be of maximum order, a N must be of the form 4n and a, a, a 7, a,... must be 0;, 7,,... being primes of the form 4n. But all these cannot be satisfied at the same time since a N cannot be of the form 4n, when a, a 7, a,...are all zeros. So let us retain a single prime of the form 4n in the end, that is to say, the largest prime of the form 4n not exceeding p. Thus we see that, in order that Q 6 N should be of maximum order, N must be of the form 5 a 5. a.7 a 7 p a p.p where p is a prime of the form 4n + and p is the prime of the form 4n next above or below p; and consequently Q 6 N = 5 N 5 a 5+ a+ p ap+ p }. 5 p

29 HIGHLY COMPOSITE NUMBERS 47 From this we can show that the maximum order of Q 6 N is 5N e Li log N +O log log N } log N log N = where 5,, 7,...are the primes of the form 4n Let + q + q 4 + q N + Li } log N + Olog log N log N log N = +6Q 8 q + Q 8 q + Q 8 q + }. Then, by means of elliptic functions, we can show that But It follows that Q 8 q + Q 8 q + Q 8 q + 96 = q + q + q q + q + q + 4 q 4 q +. 4 σ q + σ q + σ q + = q q + q q + q q +. for all values of N. It can also be shown from 96 that Let Q 8 N σ N 97 s + 4 s ζsζs = Q 8 s + Q 8 s + Q 8 s N = a. a.5 a5 p a p, where a λ 0. Then from 96 we can easily show that, if N is odd, then and if N is even then Q 8 N = N a + a+ p ap+ ; 99 p Q 8 N = N 5. a + a+ p ap p

30 48 NICOLAS AND ROBIN Hence the maximum order of Q 8 N is or more precisely log N 5/ ζn e Lilog N +O loglog N log N = ζn + Lilog N 5/ } + O log log N ζn + Lilog N 6/6 log N 5/ + S log N log N 5/ } 5 log log N log log N + O. log log N There are of course results corresponding to those of Sections 7 74 for the various powers of Q where q Q = + 6 q q q + q4 q q5 4 q +. 5 Thus for example q Q = + q + q q + 4q4 q + 5q5 4 q +, 40 5 Q q = q q q q + 4 q 4 q 5 q 5 4 q + 5 q +7 + q + q + q + q + q + q 4 + q + q +, 40 6 Q 4 q = + 4 q + q q + q q + q +8 q + 6 q 6 q + 9 q 9 6 q The number of ways in which a number can be expressed in the forms m +n, k +l + m + n, m + n, and k + l + m + n can be found from the following formulae. + q + q 4 + q q + q 8 + q 8 + q = + q + q q q5 q q7 5 q +, q + q 4 + q 9 + +q +q 8 +q 8 + q = +4 q + q q + q 4 q + 4q4 6 q +, q + q 4 + q q + q + q 7 + q = + q q +q + q4 +q q5 4 q + q7 5 q, 407 7

31 HIGHLY COMPOSITE NUMBERS 49 + q + q 4 + q 9 + +q +q + q 7 + q = +4 +q + q q + 4q4 q + 5q5 4 +q + 7q7 5 +q where,, 4, 5...are the natural numbers without the multiples of. Notes 5. The definition of Q N given in italics is missing in [8]. It has been formulated in the same terms as the definition of Q N given in Section 55. For N 0, 4Q N is the number of pairs x, yɛ Z such that x + y = N. Formula 69 links together Dirichlet s series and Lambert s series see [5], p Effective upper bounds for Q N can be found in [], p. 50 for instance: log Q N log log N log log N + log log log N log log N The maximal order of Q N is studied in [8], but not so deeply as here. See also [], pp For a proof of 76, see [5], p.. In 76, we remind the reader that ρ is a zero of the Riemann zeta-function. Formula 79 has been rediscovered and extended to all arithmetical progressions []. 56. For a proof of 9, see [5], p.. In the definition of R x, between formulas 90 and 9, and in the definition of N, after formula 94, three misprints in [8] have been corrected, namely x ρ ρ and x ρ ρ have been written instead of x ρ ρ and x ρ ρ, and R log N instead of R log N. 57. Effective upper bounds for d N can be found in [], p. 5, for instance: log d N + log log N log log N log log N log log N For a more general study of d k n, when k and n go to infinity, see [] and [4]. 58. The words in italics do not occur in [8] where the definition of σ s N and the proof of 0 were missing. It is not clear why Ramanujan considered σ s N only with s 0. Of course he knew that σ s N = N s σ s N, cf. for instance Section 7, after formula 8, but for s > 0 the generalised highly composite numbers for σ s N are quite different, and for instance property 0 does not hold for them. 59. It would be better to call these numbers s-generalised highly composite numbers, because their definition depends on s. Fors =, these numbers have been called superabundant by Alaoglu and Erdös cf. [, 4] and the generalised superior highly composite numbers have been called colossally abundant. The solution of s + 4 s + 8 s = s + 9 s is approximately For s =, the results of these sections are in [] and [4]. 6. The references given here, formula 6 and Section 8 are from [6]. For a geometrical interpretation of s N, see [], p. 0. Consider the piecewise linear function u f u such that for all generalised superior highly composite numbers N, f log N = log σ s N, then for all N, N = exp f log N. s Infinite integrals mean in fact definite integrals. For instance, in formula 0, επx r x r dx r should be read xr επt t dt. 64. Formula 9 is proved in [5] p. 9 from the classical explicit formula in prime number theory. 65. There is a misprint in the last term of formula 40 in [8], but, may be it is only a mistake of copying, since the next formula is correct. This section belongs to the part of the manuscript which is not handwritten by Ramanujan in [8]...

32 50 NICOLAS AND ROBIN

33 HIGHLY COMPOSITE NUMBERS 5 Table of largely composite numbers. n d n d n d * * * * * * * * * * * * *

34 5 NICOLAS AND ROBIN n d n d * This table has been built to explain the table handwritten by S. Ramanujan which is displayed on p. 50. An integer n is said largely composite if m n dm dn. The numbers marked with one asterisk are superior highly composite numbers. Notes Continued The approximations given for / mn comes from the Padé approximant of t in the neighborhood of t = : t+ t+ = / + 8 t There are two formulas 6 in [8], p. 99. Formula 6 can be found in []. As observed by Birch cf. [], p. 74, there is some similarity between the calculation of Section 6 to Section 68, and those appearing in [8], pp. 8. In formulas 56 and 57 Liθx} s should be read Liθx} s, the same for Li log N in 80 and for several other formulas. 7. There is a wrong sign in formula 79 of [8], and also in formulas 8 and 8. The two inequalities following formula 8 were also wrong. In formula 80, the right coefficient in the right hand side is ζ/ instead of ζ/in [8]. It follows from 8 that under the Riemann hypothesis, and for n 0 large enough, n > n 0 σn/n e γ log log n. It has been shown in [] that the above relation with n 0 = 5040 is equivalent to the Riemann hypothesis. 7. Formula 84 is due to Jacobi. For a proof see [5] p.. See also [6], pp. 60. In formula 89 of [8], the sign of the second term in the curly bracket was wrong. 7. Formula 90 is proved in [5], p It is true that if N = 5 a 5 a 7 a 7 p ap p with p p, then Q 6 N will have the maximal order 95. But, if we define a superior champion for Q 6, that is to say an N which maximises Q 6 NN ε for an ε>0, it will be of the above form, with p p log p. In 95, the error term was written O log N / in [8], cf. [5]. log log N 74. Formula 96 is proved in [5], p In formula 40 the sign of the third term in the curly bracket was wrong in [8]. In [8], the right hand side of 98 was written as the left hand side of 96.

35 HIGHLY COMPOSITE NUMBERS 5 Table, p. 50: This table calculated by Ramanujan occurs on p. 80 in [8]. It should be compared to the table of largely composite numbers, p The entry is not a largely composite number: = and d50840 = 48 while the four numbers 400, 500, 45800, are largely composite and do not appear in the table of Ramanujan. Largely composite numbers are studied in [9]. References. L. Alaoglu and P. Erdös, On highly composite and similar numbers, Trans. Amer. Math. Soc , B.J. Birch, A look back at Ramanujan s notebook, Math. Proc. Camb. Phil. Soc , J.L. Duras, J.L. Nicolas, and G. Robin, Majoration des fonctions d k n, to appear. 4. P. Erdös and J.L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France 0 975,. 5. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford, G.H. Hardy, Ramanujan, Cambridge Univ. Press, 940 and Chelsea, J.L. Nicolas, Répartition des nombres hautement composés de Ramanujan, Canadian J. Math. 97, J.L. Nicolas, Grandes Valeurs Des Fonctions Arithmétiques, Séminaire D.P.P. Paris, 6e année, 974/75, n o. G0, p J.L. Nicolas, Répartition des nombres largement composés, Acta Arithmetica 4 980, J.L. Nicolas and G. Robin, Majorations explicites pour le nombre de diviseurs de N, Canad. Math. Bull. 6 98, J.L. Nicolas, Petites valeurs de la fonction d Euler, Journal of Number Theory 7 98, J.L. Nicolas, On Highly Composite Numbers, Ramanujan revisited, Academic Press, 988, pp J.L. Nicolas, On Composite Numbers, Number Theory, Madras 987, Lecture Notes in Maths. 95, edited by K. Alladi, Springer Verlag, 989, pp K.K. Norton, Upper bounds for sums of powers of divisor functions, J. Number Theory 40 99, H. Rademacher, Topics in Analytic Number Theory, Springer Verlag, S. Ramanujan, Highly Composite Numbers, Proc. London Math. Soc. Serie 4 95, S. Ramanujan, Collected Papers, Cambridge University Press, 97, and Chelsea S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House and Springer Verlag, New Delhi, R.A. Rankin, Ramanujan s manuscripts and notebooks II, Bull. London Math. Soc. 989, G. Robin, Sur l ordre maximum de la fonction somme des diviseurs, Séminaire Delange-Pisot-Poitou. Paris, 98 98; Progress in Mathematics Birkhaüser, 8 98, 44.. G. Robin, Grandes valeurs de fonctions arithmétiques et problèmes d optimisation en nombres entiers, Thèse d Etat, Université de Limoges, France, 98.. G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures et Appl , 87.. G. Robin, Sur la différence Liθx πx, Ann. Fac. Sc. Toulouse 6 984, G. Robin, Grandes valeurs de la fonction somme des diviseurs dans les progressions arithmétiques, J. Math. Pures et Appl , G. Robin, Sur des Travaux non publiés de S. Ramanujan sur les Nombres Hautement Composés, Publications du département de Mathématiques de l Université de Limoges, France, 99, pp. 60.