1 Nov 2017 on the Denominators of Harmonic Numbers

Home , Harmonic number

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Eswarathasan and Levine considered 0 ∈ Ip and 0 ∈ Jp. It is clear that Jp ⊆ Ip.

In 1991, Eswarathasan and Levine [2] conjectured that Jp is ﬁnite for any

prime number p. In 1994, Boyd [1] conﬁrmed that Jp is ﬁnite for p ≤ 547 except 83, 127, 397. For any set S of positive integers, let S(x) = |S ∩ [1, x]|. In 2016, Sanna [3] proved that 2 0.765 Jp(x) ≤ 129p 3 x . Recently, Wu and Chen [4] proved that

2 + 1 p Jp(x) ≤ 3x 3 25 log . (1.1)

For any positive integer m, let Im be the set of positive integers n such that

m ∤ vn. In this paper, the following results are proved.

Theorem 1.1. The set of positive integers n such that vn is divisible by the least common multiple of 1, 2, ··· , ⌊n1/4⌋ has density one.

Theorem 1.2. For any positive integer m and any positive real number x, we have 1 2 + 1 q Im(x) ≤ 4m 3 x 3 25 log m ,

where qm is the least prime factor of m.

From Theorem 1.1 or Theorem 1.2, we immediately have the following corollary.

Corollary 1.3. For any positive integer m, the set of positive integers n such that m | vn has density one.

2 Proofs

We always use p to denote a prime. Firstly, we give the following two lemmas.

Lemma 2.1. For any prime p and any positive integer k, we have

k k Ipk = {p n1 + r : n1 ∈ Jp ∪{0}, 0 ≤ r ≤ p − 1}\{0}.

2 Proof. For any integer a, let νp(a) be the p-adic valuation of a. For any rational a number α = b , let νp(α) = νp(a) − νp(b). It is clear that n ∈ Ipk if and only if νp(Hn) > −k. k If n

−k. So n ∈ Ipk . In the following, we assume that n ≥ pk. Let

k k n = p n1 + r, 0 ≤ r ≤ p − 1, n1,r ∈ Z.

Then n1 ≥ 1. Write

n 1 1 b u pbv + au H = + H = + n1 = n1 n1 , (2.1) n m pk n1 pk−1a pkv pkav =1 k n1 n1 m X,p ∤m where p ∤ a and (un1 , vn1 ) = 1. k If n1 ∈ Jp, then p | un1 and p ∤ vn1 . Thus p | aun1 + pbvn1 and νp(p avn1 )= k.

By (2.1), νp(Hn) > −k. So n ∈ Ipk .

If n1 ∈/ Jp, then p ∤ un1 . Thus p ∤ aun1 + pbvn1 . It follows from (2.1) that

νp(Hn) ≤ −k. So n∈ / Ipk .

Now we have proved that n ∈ Ipk if and only if n1 ∈ Jp ∪{0}. This completes the proof of Lemma 2.1.

Lemma 2.2. For any prime power pk and any positive number x, we have

1 − 1 2 + 1 k 3 25 log p 3 25 log p Ipk (x) ≤ 4(p ) x .

Proof. If x ≤ pk, then

1 − 1 2 + 1 1 − 1 2 + 1 3 25 log p 3 25 log p k 3 25 log p 3 25 log p Ipk (x) ≤ x< 4x x ≤ 4(p ) x .

Now we assume that x>pk. By Lemma 2.1 and (1.1), we have

k k Ipk (x) = |{p n1 + r ≤ x : n1 ∈ Jp ∪{0}, 0 ≤ r ≤ p − 1}| − 1

x 1 − 1 2 + 1 ≤ pk J ( )+1 ≤ 4(pk) 3 25 log p x 3 25 log p .. p pk This completes the proof of Lemma 2.2.

θ Proof of Theorem 1.1. Let mn be the least common multiple of 1, 2, ··· , ⌊n ⌋ and let ⌊nθ⌋ denote the greatest integer not exceeding the real number nθ, where

0 <θ< 1 which will be given later. Let T = {n : mn ∤ vn}. For any prime p and

3 θ αp θ any positive number x with p ≤ x , let αp be the integer such that p ≤ x < pαp+1.

By the deﬁnition of mn and T ,

T (x) ≤ Ipαp (x). θ pX≤x Thus, by Lemma 2.2, we have

1 2 + 1 αp p Ipαp (x) ≤ 4 (p ) 3 x 3 25 log := I1 + I2, θ θ pX≤x pX≤x where 1 2 1 1 2 1 αp + αp + I1 =4 (p ) 3 x 3 25 log p , I2 =4 (p ) 3 x 3 25 log p δ θ δ x xδ, then

1 log x log x 1 x 25 log p = e 25 log p ≤ e 25δ log x = e 25δ .

It follows from pαp ≤ xθ that

1 2 1 1 θ 2 4θ 2 αp + + 1 + I1 =4 (p ) 3 x 3 25 log p ≤ 4e 25δ x 3 3 ≪ x 3 3 . δ θ δ θ log x x

1 2 1 αp + I2 = 4 (p ) 3 x 3 25 log p δ pX≤x θ + 2 + 1 ≤ 4 x 3 3 25 log 2 δ pX≤x 1 δ+ θ + 2 + 1 ≪ x 3 3 25 log 2 . log x 1 We choose θ = 4 and δ =0.1. Then

x 0.91 I1 ≪ , I2 ≪ x . log x Therefore, x T (x) ≤ Ipαp (x)= I1 + I2 ≪ . θ log x pX≤x It follows that the set of positive integers n such that vn is divisible by the least common multiple of 1, 2, ··· , ⌊n1/4⌋ has density one. This completes the proof of Theorem 1.1.

4 Proof of Theorem 1.2. We use induction on m to prove Theorem 1.2. By Lemma 2.2, Theorem 1.2 is true for m = 2. Suppose that Theorem 1.2 is true for all integers less than m (m> 2). If x ≤ m, then

1 2 + 1 1 2 + 1 q q Im(x) ≤ x< 4x 3 x 3 25 log m ≤ 4m 3 x 3 25 log m .

In the following, we always assume that x > m. If m is a prime power, then, by Lemma 2.2, Theorem 1.2 is true. Now we assume that m is not a prime power. Let pα be the least prime power divisor α α of m such that m = p m1 with p ∤ m1. Then m1 > p . It is clear that Im = α α α Im1 (Ip \Im1 ). By Lemma 2.1 and the deﬁnition of p , {1, 2, ··· ,p −1} ⊆ Im1 . HenceS α α α Im(x)= Im1 (x)+(Ip \ Im1 )(x) ≤ Im1 (x)+ Ip (x) − (p − 1).

By the inductive hypothesis, we have

1 2 + 1 1 2 + 1 3 3 25 log qm1 3 3 25 log qm Im1 (x) ≤ 4m1 x ≤ 4m1 x .

It follows that

1 2 + 1 3 3 25 log q α Im(x) ≤ 4m1 x m + Ipα (x) − (p − 1). (2.2)

We divide into the following three cases: α α Case 1: p ≥ 8. Then m1 >p ≥ 8. By Lemma 2.2, we have

1 2 + 1 α q Ipα (x) ≤ 4(p ) 3 x 3 25 log m .

It follows from (2.2) that

1 2 + 1 1 2 + 1 3 q α q Im(x) ≤ 4m1 x 3 25 log m + 4(p ) 3 x 3 25 log m

1 1 1 2 + 1 3 3 25 log qm = 4 1 + 1 m x α 3 3 (p ) m1 ! 1 2 + 1 ≤ 4m 3 x 3 25 log qm .

Case 2: pα < 8, p = 2. Then pα = 2 or 4 and x > m ≥ 2 × 3 = 6. By

Lemma 2.1 and J2 = ∅, we have I4 = {1, 2, 3} and I2 = {1}. It is clear that α Ipα (x) − (p − 1) = 0. It follows from (2.2) that

1 2 + 1 1 2 + 1 3 q q Im(x) ≤ 4m1 x 3 25 log m < 4m 3 x 3 25 log m .

5 Case 3: pα < 8, p =6 2. Then α = 1 and p = 3, 5 or 7. In addition, 1 1 1 α 1 1 1 3 3 3 3 3 3 x > m ≥ 3 × 4 = 12. Noting that m − m1 = m1 (p − 1) ≥ 4 (3 − 1) > 2 , 2 by (2.2), it is enough to prove that Ip(x) − (p − 1) ≤ 2x 3 . By Lemma 2.1, we have

Ip = {pn1 + r : n1 ∈ Jp ∪{0}, 0 ≤ r ≤ p − 1}\{0}.

By [2], J3 = {2, 7, 22}, J5 = {4, 20, 24} and

J7 = {6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735, 102728}.

3 2 3 If x ≥ 7 , then Ip(x) − (p − 1) ≤ 91 ≤ 2x 3 . If35

References

[1] D.W. Boyd, A p-adic study of the partial sums of the harmonic series, Exp. Math. 3 (4) (1994) 287–302.

[2] A. Eswarathasan, E. Levine, p-integral harmonic sums, Discrete Math. 91(3) (1991) 249–257.

[3] C. Sanna, On the p-adic valuation of harmonic numbers, J. Number Theory 166 (2016) 41–46.

[4] B.-L. Wu, Y.-G. Chen, On certain properties of harmonic numbers, J. Num- ber Theory. 175 (2017) 66–86.