Two Topics in Number Theory: Sum of Divisors of the

International Mathematical Forum, Vol. 12, 2017, no. 19, 929 - 935 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.71088

Two Topics in Number Theory: Sum of Divisors of the Factorial and a Formula for Primes Rafael Jakimczuk Divisi´on Matem´atica, Universidad Nacional de Luj´an Buenos Aires, Argentina c 2017 Rafael Jakimczuk. This article is distributed under the Creative Copyright Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract A asymptotic formula for the sum of the divisors of the factorial is obtained. A formula for primes is also obtained.

Mathematics Subject Classification: 11A99, 11B99 Keywords: Sum of divisors, factorial, primes

1

Sum of Divisors of the Factorial

In this article, as usual, d(n) denotes the number of positive divisors of n, σ(n) denotes the sum of the positive divisors of n and p denotes a positive prime. Let E(p) be the multiplicity (exponent) of the prime p in the prime factorization of n!. Therefore n! =

pE(p)

Y p≤n

The logarithm of the number of divisors of n!, namely log(d(n!)) =

X p≤n

log(E(p) + 1)

(1)

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Rafael Jakimczuk

was studied by P. Erd˝os, S. W. Graham, A. Ivi´c and C. Pomerance in [2]. These authors obtained (among another results) the asymptotic formula n (2) log (d(n!)) ∼ c0 log n where the constan c0 is c0 =

∞ X

log k k=2 (k − 1)k

(3)

R. Jakimczuk [4] studied the sum of the logarithms of the exponents of n!, namely X

log E(p)

p≤n

and obtained the formula X

log E(p) ∼ c1

p≤n

n log n

where the constan c1 is c1 =

∞ X

log k k=2 (k + 1)k

A simple change in the reasoning given in [4] proves equation (2) with the constant c0 (3). In this section we study the sum of the divisors of n!, namely σ(n!) and the logarithm of σ(n!). We shall need the following lemmas. Lemma 1.1 The following two power series are well-known 1 = 1 + x + x2 + x3 + · · · |x| < 1 1−x x2 x3 x4 + − + ··· |x| < 1 2 3 4 We also have the following asymptotic formulae. 1 = 1 + f (x)x 1−x where f (x) → 1 when x → 0 (use the L’Hospital’s rule). log(1 + x) = x −

log(1 + x) = f (x)x

(4)

(5)

(6)

(7)

where f (x) → 1 when x → 0 (use the L’Hospital’s rule). ex = 1 + f (x)x where f (x) → 1 when x → 0 (use the L’Hospital’s rule).

(8)

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Divisors of the factorial and primes

Lemma 1.2 The following asymptotic formulae hold √ 1 1 , log j = n log n − n + log n + log 2π + O 2 n j=1 n X

 

√ n! =

(9)

√    2πnn n 1 1+O n e n

(10)

Proof. The proof of equation (9) is as follows n X

−

j=1 ∞ X i=1

log j =

Z n

log x + log n −

1

ci +

∞ X

ci = n log n − n +

i=n

n−1 X 1 1 log n − ci = n log n − n + log n + 1 2 2 i=1

1 1 log n + C + O , 2 n  

 

j+1 ∞ 1 where C = 1 − ∞ log xdx is i=n ci = O n . Note that the area j i=1 ci and the sum of three areas, the area of the rectangle of basis 1 and height log j, the area of the rectangle triangle of basis 1 and height log(j + 1) − log j and the area cj between the choord and the curve log x. Note also that the derivative P of log x, namely 1/x is strictly decreasing and consequently the area ∞ i=n ci is contained in the rectangle triangle of basis 1 and height 1/n. √ value of the constant C is obtained from the Stirling’s formula n! ∼ √ The nn n 2π en . Finally, equation (10) is an immediate consequence of equations (9) and (8). The lemma is proved.

P

P

R

Lemma 1.3 The following two Mertens’s formulae are well-known (see, for example, [5]) 1 1 = log log x + M + O log x p≤x p

!

X

(11)

where M is called Mertens’s constant. Y p≤x

1 1− p

!

e−γ 1 = 1+O log x log x

!!

(12)

where γ = 0.5772156649 . . . is Euler’s constant. The following is a well-known Chevyshev inequality (see, for example, [1], [3] or [5]) π(x) ≤ c

x log x

where π(x) is the prime counting function and c is a positive constant.

(13)

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Rafael Jakimczuk

The following Legendre’s theoren is well-known (see for example [1], [3] or [5]) E(p) =

∞ X k=1

$

n pk

%

(14)

where, as usual, bxc denotes the integer part of x. Theorem 1.4 The following asymptotic formulae hold. 1 σ(n!) = e n! log n 1 + O log n

!!

γ

σ(n!) = e

γ

√

(15)

!! √ nn n 1 2π n log n 1 + O e log n

(16)

√ 1 1 (17) log (σ(n!)) = n log n − n + log n + log log n + γ + log 2π + O 2 log n !

Proof. We have Y Y pE(p)+1 − 1 1 1 E(p)   σ(n!) = =Q p 1 − 1 p−1 pE(p)+1 p≤n p≤n p≤n p≤n 1 −

!

Y

(18)

p

If 0 < x < 1 then we have (see equations (5) and (4)) 0 < − log(1 − x) = x + =

x = 1−x

Substituting x = 0 < − log(1 − ≤

2 pE(p)

≤

1 x

x 2 x3 + + · · · ≤ x + x2 + x3 + · · · 2 3

1 −1

1 pE(p)+1

1 pE(p)+1

(19) into (19) we obtain (see (14)) )≤

1 pE(p)+1

−1

=

1 pE(p)+1



1−

1



≤2

pE(p)+1

2 2 ≤ jnk pE(p) p p

Note that pE(p)+1 ≥ 2 implies

1 pE(p)+1 (20)

1 1−

1 pE(p)+1

≤ 2 and if k ≥ 1 and n ≥ 2 then by

mathematical induction we can prove easily the inequality nk ≥ kn.

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Divisors of the factorial and primes

On the other hand, we have 0 ≤

p≤n

=

1

X

p

=

j k n p

1

X p≤ n 2

1 X 1 n p≤ n 1 −

p n

2

+

p

j k n p

X X 1 1 1  + ≤ p p≤ n p n − 1 p n n