Two Topics in Number Theory: Sum of Divisors of the Primorial and

International Mathematical Forum, Vol. 12, 2017, no. 7, 331 - 338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7113

Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts 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 Let pn be the n-th prime and σ(n) the sum of the positive divisors of n. Let us consider the primorial Pn = p1 .p2 . . . pn , the sum of its Q positive divisors is σ(Pn ) = ni=1 (1 + pi ). In the first section we prove the following asymptotic formula σ(Pn ) =

n Y

(1 + pi ) =

i=1

6 γ e Pn log pn + O (Pn ) . π2

Let a(k) be the squarefree part of k, in the second section we prove the formula X 1≤k≤x

a(k) =

π2 2 x + o(x2 ). 30

We also study integers with restricted squarefree parts and generalize these results to s-th free parts.

Mathematics Subject Classification: 11A99, 11B99 Keywords: Primorial, divisors, squarefree parts, s-th free parts, average of arithmetical functions

332

1

Rafael Jakimczuk

Sum of Divisors of the Primorial

In this section p denotes a positive prime and pn denotes the n-th prime. The following Mertens’s formulae are well-known (see [5, Chapter VI]) !

1 1 = log log x + M + O , log x p≤x p X

(1)

where M is called Mertens’s constant. !

1 log 1 − p p≤x X

!

1 = − log log x − γ + O , log x

(2)

where γ is the Euler’s constant. 1 1− p

Y p≤x

 Y  p≤x

!

e−γ 1 = +O , log x log2 x !

(3)



1  = eγ log x + O(1). 1 1− p

(4)

Note that equation (4) is an immediate consequence of (3) if we use the formula 1 = 1 − x(1 + o(1)) 1+x

(x → 0),

since then !

1 1+O



1 log x

1  =1+O . log x 





In this section we examine the sum p≤x log 1 + p1 and the product p≤x 1 + Then, we apply the results to the sum of the divisors of the primorial. Our main theorem is the following. P

Q

Theorem 1.1 The following asymptotic formulae hold. 1 log 1 + p p≤x

!

X

Y p≤x

!

6 1 = log log x + γ + log 2 + O , π log x 

1 1+ p

!

=



6 γ e log x + O(1). π2

(5)

(6)

1 p



.

333

Two topics in number theory: The primorial and squarefree parts

Proof. If we consider the formula ex = 1 + x(1 + o(1))

(x → 0)

then equation (6) is an immediate consequence of equation (5). Since we have 1 exp O log x

!!

!

1 =1+O . log x

Therefore we have that to prove equation (5). We have the equation 1 log(1 + x) = x − x2 (1 + o(1)) 2

(x → 0)

Consequently 1 log 1 + p p≤x

!

X

=

1 1 X 1 + o(1) − . 2 p≤x p2 p≤x p X

(7)

Now, we have 1 1 + o(1) X 1 + o(1) X 1 + o(1) X 1 + o(1) = − = +O 2 2 2 2 p p p p x p p p>x p≤x

 

X

(8)

Note that we have used the well-known formula (see [1, Chapter 3]) 1 1 =O , 2 x n>x n  

X

where n denotes a positive integer. Substituting (1) and (8) into (7) we obtain almost (5) 1 log 1 + p p≤x

!

X

!

1 = log log x + C + O , log x

where C is a constant. Therefore we have almost (6) Y p≤x

1 1+ p

!

= eC log x + O(1).

It is well-known the limit (Euler’s product formula)(see [1, Chapter 11]) lim

n→∞ 2

n Y i=1

1 1− 2 pi

!

=

6 . π2

(9)

Note that ζ(2) = π6 , where ζ(s) denotes the Riemann zeta function. Equations (3) and (9) give eC = π62 eγ . The theorem is proved.

334

Rafael Jakimczuk

Theorem 1.2 Let us consider the primorial Pn = p1 .p2 . . . pn , the sum of Q its positive divisors is σ(Pn ) = ni=1 (1 + pi ). We have the following asymptotic formula σ(Pn ) =

n Y

(1 + pi ) =

i=1

6 γ e (p1 .p2 . . . pn ) log pn + O ((p1 .p2 . . . pn )) π2

6 γ = e Pn log pn + O(Pn ) π2

(10)

Note that the primorial Pn = p1 .p2 . . . pn is its greatest divisor. Proof. Put x = pn in equation (6). The theorem is proved. It is well-known (see [2, Chapter XVIII]) that lim sup

σ(n) = eγ . n log log n

We have Theorem 1.3 The following limit holds lim

n→∞

σ(Pn ) 6 = 2 eγ . Pn log log Pn π

(11)

Proof. It is well-known that Pn = e(1+o(1))pn , therefore log log Pn ∼ log pn . Now, limit (11) is an immediate consequence of (10). The theorem is proved.

2

Sum of Squarefree Parts and Generalization

Every positive integer k can be written in the form k = q.n2 where q is a squarefree or quadratfrei number and n2 is the greatest square that divides k. The positive integer q is called the squarefree part of k and it is denoted a(k) (see [6]). Therefore if k is a square we have a(k) = 1, in contrary case a(k) is the product of primes which have odd exponent in the prime factorization of k. For exampe, if k = 24 .53 .112 .235 then a(k) = 5.23. If q is relatively prime with n in the decomposition k = q.n2 , that is (q, n2 ) = 1, we shall call q the relatively prime squarefree part of k and will be denoted a0 (k), besides we shall say that k has a relatively prime squarefree part. The average of the arithmetic function a(k) is studied in the following theorem.

335

Two topics in number theory: The primorial and squarefree parts

Theorem 2.1 The following formula holds X

a(k) =

1≤k≤x

π2 2 x + o(x2 ). 30

(12)

Proof. The number of squarefree integers q not exceeding x (denoted Q(x)) is well-known (see either [2, Chapter XVIII] or [4]). We have Q(x) =

X

1=

q≤x

1 x + o(x), ζ(2)

(13)

where ζ(s) denotes the Riemann zeta function. From here, we can obtain easily (see [2, Chapter XXII]) that Q1 (x) =

X

q=

q≤x

1 x2 + o(x2 ). 2ζ(2)

(14)

Let us consider the set Sn such that 

Sn = q : q ≤

x n2



(15)

Let us consider  > 0. We choose the positive integer N such that 1 1