PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 97 (111) (2015), 175–180 DOI: 10.2298/PIM140617001T

THE SUM OF THE UNITARY DIVISOR FUNCTION

Tim Trudgian

Abstract. We establish a new upper bound on the function σ^{∗}(n), the sum of all coprime divisors of n. The main result is that σ^{∗}(n) ⩽ 1.3007n log log n for all n ⩾ 570 571.

1. Introduction

1.1. The function σ(n). Let σ(n) denote the sum of the divisors of n; for example, σ(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. In 1913 Grönwall showed that

(1.1)  lim supσ(n)/(n loglogn) = e^{γ} = 1.78107 …,

where γ is Euler’s constant. A proof is given in [5, Thm. 322]. Robin showed that the manner in which (1.1) behaves is connected with the Riemann hypothesis. More precisely, he showed, in [8], that for n ⩾ 5041 the inequality

(1.2)  σ(n) < e^{γ}n loglogn

is equivalent to the Riemann hypothesis. Ivić [6] showed that

σ(n) < 2.59 n loglogn, (n ⩾ 7),

which was improved by Robin [op. cit.] to

(1.3)  σ(n) < \frac{σ(12)}{12 loglog12} n loglogn ⩽ 2.5634 n loglogn, (n ⩾ 7).

Akbary, Friggstad and Juricevic [1] improved this further, replacing the right-side of (1.3) with

(1.4)  \frac{σ(180)}{180 loglog180} n loglogn ⩽ 1.8414 n loglogn
⩽ 1.0339 e^{γ}n loglogn, (n ⩾ 121).

Given Robin’s criterion for the Riemann hypothesis in (1.2) it is reasonable to suggest that (1.4) is close to the best bound that one may hope to exhibit.

2010 Mathematics Subject Classification: 11A25; 11N56.

Key words and phrases: unitary divisors, exponential divisors, arithmetic functions.

Supported by Australian Research Council DECRA Grant DE120100173.

Communicated by Žarko Mijajlović.

1.2. The function σ^{∗}(n). We say that d is a unitary divisor of n if d | n and (d, n/d) = 1. Let σ^{∗}(n) = ∑ _{d|n, (d,n/d)=1} d be the sum of all unitary divisors of n; for example, σ^{∗}(12) = 1 + 12 + 3 + 4 = 20. Robin [8, p. 210] notes that the proof of (1.1) can be adapted to show that

(1.5)  lim sup \frac{σ^{∗}(n)}{n loglogn} = \frac{6e^{γ}}{π^{2}} = 1.08 …,

see also [6, p. 21]. Ivić [6] showed that

σ^{∗}(n) < \frac{28}{15} n loglogn, (n ⩾ 31).

This was improved by Robin who showed that

σ^{∗}(n) < 1.63601n loglogn, (n ⩾ 31),

except for n = 42 when σ^{∗}(n) = 1.7366 …n loglogn. A direct comparison of these results with those in §1.1 compels us to ask the following questions.

(1) Given (1.5) can a Robin-esque criterion for the Riemann hypothesis à la (1.2) be given for σ^{∗}(n)?

(2) Analogous to (1.4) can one obtain a relatively close approximation to (1.5) of the form

σ^{∗}(n) < (1 + ϵ)\frac{6e^{γ}}{π^{2}} n loglogn, (n ⩾ n_{0}),

for reasonably small values of ϵ and n_{0}? Concerning 1, Robin has conjectured [8, Prop. 1(i), p. 210] that there are infin-

itely many n for which

σ^{∗}(n) > \frac{6e^{γ}}{π^{2}} n loglogn.

A related conjecture is given in Proposition 1(ii) in [8], viz. that

(1.6)  \frac{σ(n)}{σ^{∗}(n) loglogn} < e^{γ},

for all n sufficiently large. The interest in this conjecture stems from the limiting relation

limsup \frac{σ(n)}{σ^{∗}(n) loglogn} = e^{γ}.

Derbal [3] proved (1.6) for all n ⩾ 17.

This article answers Question 2 above, at least partially, by proving

Theorem 1.1. For n ⩾ 570 571,

(1.7)  σ^{∗}(n) ⩽ 1.3007 n loglogn.

It takes less than 40 seconds on a 1.8 GHz laptop to compute σ^{∗}(n) for all 1 ⩽ n ⩽ 570 570. One may therefore justify the number 570 571 appearing in Theorem 1.1 as being “reasonably small", as stipulated in Question 2, as least in regards to computational resources.

It would be of interest to address the following problem. Fix an ϵ > 0 and determine the least value of n_{0} such that σ^{∗}(n) < (1 + ϵ)\frac{6e^{γ}}{π^{2}} n loglogn for all n ⩾ n_{0}. The method used to prove Theorem 1.1 is incapable of reducing the right-hand side of (1.7) to anything less than 1.29887n loglogn.

Theorem 1.1 is proved in §2. An application is given in §3. Two concluding questions are raised in §4.

2. Proof of Theorem 1.1

We proceed as in Robin [8, p. 211]. It is sufficient to verify the inequality on numbers N_{k} = ∏ _{i=1}^{k} p_{i}, where k ⩾ 2, since, for N_{k} ⩽ n < N_{k+1} , we have σ^{∗}(n)/n ⩽ σ^{∗}(N_{k})/N_{k}, whence

(2.1)  \frac{σ^{∗}(n)}{n loglogn} ⩽ \frac{σ^{∗}(N_{k})}{N_{k} loglogN_{k}} .

Since σ^{∗}(p^{α}) = 1 + p^{α} and σ^{∗}(n) is a multiplicative function, the right-hand side of (2.1) is

(2.2)  \frac{∏ _{i⩽k} (1 + p_{i}^{−1} )}{logθ(p_{k})} ,

where θ(x) = ∑ _{p⩽x} logp. To bound the numerator in (2.2) we use

∑_{p⩽x} \frac{1}{p} ⩽ loglogx + B + \frac{1}{10 log^{2} x} + \frac{4}{15 log^{3} x}, (x ⩾ 10 372),

where

B = γ + ∑_{p⩾2} \{ log( 1 − \frac{1}{p}) + \frac{1}{p}\} = 0.26149 …,

see Dusart [4]. To bound the denominator in (2.2) we use

θ(x) ⩾ x( 1 − \frac{0.006788}{logx}) , (x ⩾ 10 544 111),

which is also found in [4]. Therefore, since e^{x} ⩾ x + 1 we have

∏_{i⩽k} ( 1 + \frac{1}{p_{i}} ) ⩽ exp( ∑_{i⩽k} \frac{1}{p_{i}} ) ⩽ A_{1}(p_{k}) logp_{k},

where

A_{1}(x) = exp( B + \frac{1}{10 log^{2} x} + \frac{4}{15 log^{2} x}) , (x ⩾ 10 372).

Also

logθ(p_{k}) ⩾ A_{2}(p_{k}) logp_{k},

where

A_{2}(x) = 1 + \frac{log(1 − 0.006788/ logx)}{logx} , (x ⩾ 10 544 111).

It is clear that

(2.3)  A_{2}(x) < 1 < e^{B} = 1.29887 … < A_{1}(x).

We choose a suitably large lower bound on k in order to make A_{1}(x) and A_{2}(x) sufficiently close to e^{B} and 1 respectively. Indeed, we shall bound (2.2) for p_{k} ⩾ 15 485 863, which is equivalent to k ⩾ 1 000 000. Therefore

(2.4)  \frac{∏ _{i⩽k} (1 + p_{i}^{−1} )}{logθ(p_{k})} ⩽ \frac{A_{1}(p_{k})}{A_{2}(p_{k})} ⩽ 1.3007,

whence

(2.5)  \frac{σ^{∗}(n)}{n loglogn} ⩽ \frac{σ^{∗}(N_{k})}{N_{k} loglogN_{k}} ⩽ 1.3007,

for all k ⩾ 10^{6}. One may check that (2.5) also holds for 8 ⩽ k ⩽ 10^{6}, On a single core PC with 32 GB of RAM, this calculation took less than a minute using Magma. All that remains are the numbers 3 ⩽ n ⩽ p_{1} ⋯ p_{8} = 9 699 690. A quick computational check shows that

\frac{σ^{∗}(570 570)}{570 570 loglog570 570} ⩾ 1.3125,

and that, for all n > 570 570, the inequality (1.7) holds, which proves Theorem 1.1. Were this lower bound on n too large for one’s tastes, one could also show

σ^{∗}(n) ⩽ 1.3007 n loglogn,

for all n ⩾ 53 131 with only two exceptions, namely

σ^{∗}(510 510) = (1.3245 …)510 510 loglog510 510, and
σ^{∗}(570 570) = (1.3125 …)570 570 loglog570 570.

Our bounds for σ^{∗}(n) depend on an upper bound for A_{1}(p_{k})/A_{2}(p_{k}) in (2.4). We see at once from (2.3) that our method is incapable of reducing the bound 1.3007 in Theorem 1.1 to anything below 1.29887.

3. Application to exponential divisors

Given an n = p_{1}^{a_{1}} ⋯ p_{s}^{a_{s}} the integer d = p_{1}^{b_{1}} ⋯ p_{s}^{b_{s}} is an exponential divisor of n if b_{j} | a_{j} for every 1 ⩽ j ⩽ s. Define the functions d^{(e)}(n) and σ^{(e)}(n) to be the number of exponential divisors of n and the sum of the exponential divisors of n, respectively. Since these functions are multiplicative we have

d^{(e)}(n) = ∏_{j=1}^{r} d(a_{j}), σ^{(e)}(n) = ∏_{j=1}^{r} ( ∑_{b_{j} |a_{j}} p_{j}^{b_{j}} ) ,

where d(n) is the number of divisors of n. Minculete [7, Thm. 2.1 and Cor. 2.5] has given the following bounds for σ^{(e)}(n) and d(n) d^{(e)}(n)

σ^{(e)}(n) ⩽ \frac{28}{15} n loglogn, (n ⩾ 6),
d^{(e)}(n) d(n) ⩽ \frac{28}{15} n loglogn, (n ⩾ 5).

An application of the proof of Theorem 1.1 improves these bounds.

Corollary 3.1. For n ⩾ 37,

(3.1)  σ^{(e)}(n) ⩽ 1.3007 n loglogn.

For n ⩾ 8,

(3.2)  d^{(e)}(n) d(n) ⩽ 1.3007 n loglogn.

Proof. The displayed formula halfway down page 1529 in [7] gives

σ^{(e)} ⩽ n ∏_{p|n} ( 1 + \frac{1}{p}) ,

so that

(3.3)  \frac{σ^{(e)}(n)}{n loglogn} ⩽ \frac{∏ _{p|n} (1 + \frac{1}{p})}{loglogn} .

As before, we need only consider (3.3) on N_{k} ⩽ n < N_{k+1} . Using (2.4) and the calculations in §2 we have

\frac{σ^{(e)}(n)}{n loglogn} ⩽ 1.3007, (n ⩾ 9 699 691).

Checking the range 37 ⩽ n ⩽ 9 699 691 establishes (3.1). Minculete [7, Eq. (12)] showed that d(n) d^{(e)}(n) ⩽ σ^{(e)}(n) for all n ⩾ 1. Using this, (3.1), and a simple computer check for 8 ⩽ n ⩽ 36, establishes (3.2). □

4. Conclusion

Both of the functions σ^{∗}(n) and σ^{(e)}(n) are multiplicative. We have

σ^{∗}(p) = 1 + p > σ^{(e)}(p) = p,

and, for a ⩾ 2,

σ^{∗}(p^{a}) = 1 + p^{a} < p + p^{a} ⩽ σ^{(e)}(p^{a}),

since a = a · 1, where a and 1 are distinct. Therefore, on square-free numbers σ^{∗}(n) > σ^{(e)}(n). We conclude this section by raising two questions.

(1) What is the proportion of n for which σ^{∗}(n) > σ^{(e)}(n)?

(2) Are there infinitely many values of n for which σ^{∗}(n) = σ^{(e)}(n)? The proportion in Question 1 must be at least that of the square-free numbers, viz. 6/π^{2} ≈ 0.607. A computation shows the proportion of 1 ⩽ n ⩽ 10^{9} to be approximately 0.778307. It follows from the Erdős–Wintner theorem (see, e.g., [9, III.4]) that the density of n for which σ^{∗}(n) > σ^{(e)}(n) is well defined. In [2] the density of the set of integers n for which σ(n)/n ⩾ 2 was estimated. It seems possible that similar methods may be brought to bear on Question 1.

As for Question 2, only five values of n were found in the range 1 ⩽ n ⩽ 10^{9} for which σ^{∗}(n) = σ^{(e)}(n), namely

n = 20, 45, 320, 6615, 382 200.

Andrew Lelechenko has also found

n = 680 890 228 200,

which is the next smallest n after 382 200. He has also communicated to me that σ^{∗}(n) = σ^{(e)}(n) also for

n = 2 456 687 209 744 634 987 008 753 664 = 2^{49} × 4 363 953 127 297.

Acknowledgements. I am grateful to Danesh Jogia who verified (2.5) for 8 ⩽ k ⩽ 10^{6}, Scott Morrison who provided a much-needed tutorial on programming, Greg Martin for a discussion on limiting distributions, and Andrew Lelechenko for providing the last examples in §4.

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Mathematical Sciences Institute (Received 03 06 2014) The Australian National University Canberra Australia

    timothy.trudgian@anu.edu.au