Two Topics in Number Theory: Sum of Divisors of the Factorial and a Formula for Primes

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1 Iteratioal Mathematical Forum, Vol. 2, 207, o. 9, HIKARI Ltd, Two Topics i Number Theory: Sum of Divisors of the Factorial ad a Formula for Primes Rafael Jaimczu Divisió Matemática, Uiversidad Nacioal de Lujá Bueos Aires, Argetia Copyright c 207 Rafael Jaimczu. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. Abstract A asymptotic formula for the sum of the divisors of the factorial is obtaied. A formula for primes is also obtaied. Mathematics Subject Classificatio: A99, B99 Keywords: Sum of divisors, factorial, primes Sum of Divisors of the Factorial I this article, as usual, d deotes the umber of positive divisors of, σ deotes the sum of the positive divisors of ad p deotes a positive prime. Let Ep be the multiplicity expoet of the prime p i the prime factorizatio of!. Therefore! = p p Ep The logarithm of the umber of divisors of!, amely logd! = logep + p

2 930 Rafael Jaimczu was studied by P. Erdős, S. W. Graham, A. Ivić ad C. Pomerace i [2]. These authors obtaied amog aother results the asymptotic formula log d! c 0 2 log where the costa c 0 is c 0 = =2 log R. Jaimczu [4] studied the sum of the logarithms of the expoets of!, amely log Ep p ad obtaied the formula log Ep c log where the costa c is p c = =2 log + A simple chage i the reasoig give i [4] proves equatio 2 with the costat c 0 3. I this sectio we study the sum of the divisors of!, amely σ! ad the logarithm of σ!. We shall eed the followig lemmas. Lemma. The followig two power series are well-ow x = + x + x2 + x 3 + x < 4 log + x = x x2 2 + x3 3 x4 + 4 x < 5 We also have the followig asymptotic formulae. x 3 = + fxx 6 where fx whe x 0 use the L Hospital s rule. log + x = fxx 7 where fx whe x 0 use the L Hospital s rule. e x = + fxx 8 where fx whe x 0 use the L Hospital s rule.

3 Divisors of the factorial ad primes 93 Lemma.2 The followig asymptotic formulae hold j= log j = log + 2 log + log 2π + O, 9! = 2π e + O 0 Proof. The proof of equatio 9 is as follows log j = log x + log j= 2 log c i = log + i= 2 log + c i + c i = log + i= i= 2 log + C + O, where C = i= c i ad i= c i = O. Note that the area j+ j log xdx is the sum of three areas, the area of the rectagle of basis ad height log j, the area of the rectagle triagle of basis ad height logj + log j ad the area c j betwee the choord ad the curve log x. Note also that the derivative of log x, amely /x is strictly decreasig ad cosequetly the area i= c i is cotaied i the rectagle triagle of basis ad height /. The value of the costat C is obtaied from the Stirlig s formula! 2π. Fially, equatio 0 is a immediate cosequece of equatios 9 e ad 8. The lemma is proved. Lemma.3 The followig two Mertes s formulae are well-ow see, for example, [5] p x p = log log x + M + O log x where M is called Mertes s costat. = e γ + O p log x p x log x 2 where γ = is Euler s costat. The followig is a well-ow Chevyshev iequality see, for example, [], [3] or [5] πx c x log x where πx is the prime coutig fuctio ad c is a positive costat. 3

4 932 Rafael Jaimczu The followig Legedre s theore is well-ow see for example [], [3] or [5] Ep = = where, as usual, x deotes the iteger part of x. p 4 Theorem.4 The followig asymptotic formulae hold. σ! = e γ! log σ! = e γ 2π e + O log log + O log 5 6 log σ! = log + 2 log + log log + γ + log 2π + O 7 log Proof. We have σ! = p p Ep+ p = p p p Ep p p p Ep+ 8 If 0 < x < the we have see equatios 5 ad 4 = 0 < log x = x + x2 2 + x3 3 + x + x2 + x 3 + x x = 9 x Substitutig x = p Ep+ ito 9 we obtai see 4 0 < log p Ep+ p Ep+ = 2 p Ep+ p Ep+ p Ep+ 2 p 2 Ep pep 2 p 20 p Note that p Ep+ 2 implies p Ep+ 2 ad if ad 2 the by mathematical iductio we ca prove easily the iequality.

5 Divisors of the factorial ad primes 933 O the other had, we have 0 p p = p p p + 2 p = p + p p 2 2 <p p 2 <p p 2 p + p p 2 <p 2 Note that if < p the 2 p =, besides x x. Equatio ad the mea value theorem give us p = log log log log + O 2 2 <p log = O log 22 O the other had, we have see equatio 3 0 p 2 p 2 = 2 π 2 p 2 c log where c is a positive costat. Note that if p 2 the p p 2 p = O log 2. Therefore 23 Equatios 23, 22, 2 ad 20 give us log = O p Ep+ p ad cosequetly see 8 = + O p Ep+ p log log Fially, substitutig equatios, 2 ad 25 ito 8 ad usig equatio 6 we obtai 5. Equatio 6 is a immediate cosequece of 5 ad 0. Equatio 7 is a immediate cosequece of equatios 6 ad 7. The theorem is proved. Now, ote the followig observatio. Amog the divisors of! are the divisors! =,...,. The cotributio of these divisors to σ! is see 5 =! σ! e γ =

6 934 Rafael Jaimczu where we have used the well-ow formula see, for example, [5] = log O the other had, the cotributio of the rest of the divisors to σ! is σ! =! σ! = =! σ! That is, a less cotributio tha the former. 2 A Formula for Primes = e γ The followig theorem is sometimes called either the priciple of cross-classificatio or the iclusio-exclusio priciple. We ow euciate the priciple. Theorem 2. Iclusio-exclusio priciplelet S be a set of N distict elemets, ad let S,..., S r be arbitrary subsets of S cotaiig N,..., N r elemets, respectively. For i < j <... < l r, let S ij...l be the itersectio of S i, S j,..., S l ad let N ij...l be the umber of elemets of S ij...l. The the umber K of elemets of S ot i ay of S,..., S r is K = N i r N i + i<j r N ij Proof. See, for example, [], [3] or [5]. i<j< r N ij r N 2...r I the followig theorem we establish a formula for the -th prime. Theorem 2.2 Let p be the -th prime ad let be a arbitrary but fixed positive iteger. The followig formula holds where the fuctio A x is lim x A x = p 26 x A x = x µ x ad where deotes a squarefree ot multiple of p icludig, µ is the Möbius fuctio ad x deotes the iteger part of x. Therefore, if we ow the primes differet of p the we ca determiate the prime p usig limit 26. That is, if we have as iformatio all primes except oe, we ca determiate this oe.

7 Divisors of the factorial ad primes 935 Proof. The umber of prime powers of p ot exceedig x is clearly log x +. log p By the iclusio-exclusio priciple this umber is x x + x p pi x i p i p j p i p Therefore we have A x = p i <p j x p i,p j p log x log p = x µ = A x x + = log x { } log x + log p log p where {y} = y y is the fractioal part of y ad cosequetly The theorem is proved. A x lim x log x = log p Acowledgemets. The author is very grateful to Uiversidad Nacioal de Lujá. Refereces [] T. M. Apostol, Itroductio to Aalytic Number Theory, Spriger, [2] P. Erdős, S. W. Graham, A. Ivić ad C. Pomerace, O the umber of divisors of!, Chapter i Aalytic Number Theory, Spriger, 996, [3] G. H. Hardy ad E. M. Wright, A Itroductio to the Theory of Numbers, Oxford, 960. [4] R. Jaimczu, Logarithm of the expoets i the prime factorizatio of the factorial, Iteratioal Mathematical Forum, 2 207, o. 3, [5] W. J. LeVeque, Topics i Number Theory, Volume I, Addiso-Wesley, 956. Received: November 5, 207; Published: November 2, 207