On a Sum Involving the Sum-of-Divisors Function

Abstract

Let be the sum of all divisors of and let be the integral part of . In this paper, we shall prove that for , and that the error term of this asymptotic formula is .

1. Introduction

As usual, denote by the Euler function and by the integral part of real , respectively. Recently, Bordellès et al. [1] studied the asymptotic behaviour of the quantityfor . By exponential sum technique, they proved thatand conjectured that

Very recently, Wu [2] improved (2) and Zhai [3] resolved conjecture (3) by showingand also proved that the error term in (4) is , where denotes the iterated logarithm. Some related works can be found in [4, 5]. Since the sum-of-divisors function has similar properties as the Euler function in many cases, it seems natural and interesting to consider its analogy of (3).

Our result is as follows.

Theorem 1. (i)For , we have(ii)Let be the error term in (5). Then, for , we have

Let be the Möbius function and define and for all integers . Then, and . In Zhai’s approach proving (4), the inequalityplays a key role, where is a positive constant. Clearly, such a bound is not true for . By refining Zhai’s approach, we shall prove our result.

2. Preliminary Lemmas

As in [3], we need some bounds on exponential sums of the type where . For large values of , Zhai used the theory of exponent pair, and for smaller ones the Vinogradov method. Both estimates are contained in the following general theorem of Karatsuba [6, Theorem 1], which will be a key tool for proving Theorem 1.

Lemma 1. Let and and be integers, being positive. Let . Suppose that there exist positive absolute constants , and such that , and ; an integer such that ; and distinct numbers not exceeding , such that for the following inequalities are satisfied:(i).(ii).Then, for each positive integer not exceeding , we havewhere and , are absolute constants.

The next two lemmas are essentially a special case of [7, Lemmas 2.5 and 2.6] with . The only difference is that the ranges of and here are slightly larger than those of [7, Lemmas 2.5 and 2.6] ( in place of and in place of , respectively). Although the proof is completely similar, for the convenience of readers, we still reproduce a proof here.

Lemma 2. Let and . Then, there exists an absolute positive constant such thatwhere the implied constant is absolute.

Proof. We apply Lemma 1 to with . For this, we chooseand take the to be all integers such thatObviously the number of is between and . Next we shall verify that satisfies the conditions (i) and (ii) of Lemma 1 with the parameters chosen above.
For , we havewhereSimilarly for , we find the inequality , whereFor the lower bound of (ii), we havewhereFrom Lemma 1, there exist two positive constants and such thatwith . This completes the proof of Lemma 2.

Lemma 3. Define . Let be the constant defined by Lemma 2 and . Then, we haveuniformly for , and .

Proof. By invoking a classical result on (see 8, page 39]) we can write, for any ,An application of Lemma 2 with yieldsTaking , we easily deduce thatThe first term can be absorbed by the second, since can be chosen small enough to ensure that and since implies . Hence,and an Abel summation produces the required result.

Lemma 4. Let and . Denote by the total variation of on . Then,where the implied constant is absolute.

Proof. If is a partition of the interval , thenSince for all , we haveOn the other hand, since is of period 1, we haveInserting these two bounds into (24), we obtain the required result.

3. Proof of Theorem 1

3.1. A Formula on the Mean Value of

Lemma 5. (i)For and , we have where(ii)For , we have

Proof. Using , the hyperbole principle of Dirichlet allows us to writewhereFirstly we haveSecondly we can writewhere is as in (28). Inserting (32), (33), and (34) into (30) and using , we get (27).
Taking in (27) and noticing thatwe obtain the required bound. This completes the proof.

3.2. Estimates of Error Terms

Lemma 6. Let , where is given as in Lemma 3. Let be defined by (28). Then, for and , we have

Proof. Denote by and two sums on the left-hand side of (36), respectively. By (28) of Lemma 5, we can writewhereFor , let and defineNoticing that , we can apply Lemma 3 to derive thatwith . It is clear that is increasing on . On the other hand, for and , we haveThus,which implies that . Inserting this into the expression of , we getNext we bound . Let be a function of bounded variation on for each integer and let be the total variation of on . Integrating by parts, we haveFrom this, we can derive thatfor . Summing over , we find thatWe apply this formula toAccording to Lemma 4, we have , and thus by putting , we obtain, with the notation and ,where we have used the fact that and and the boundUsing (48), a simple partial integration allows us to derive thatCombining (43) and (50), it follows thatSimilarly, we can prove the same bound for . This completes the proof.

3.3. End of the Proof of Theorem 1

Let be the constant given as in Lemma 3 and . Let be a parameter to be chosen later.

Putting , we have and . We have, with the convention ,

By the hyperbole principle of Dirichlet, we can writewhere

With the help of the bound , we can derive that

For evaluating , we write

With the help of Lemma 5 (ii), a simple partial integration gives us

Inserting these estimates into (56), we find that

Finally, we evaluate . For this, we writewhere

By the bound that , we have

On the other hand, (27) of Lemma 5 allows us to derive thatwhere is given by (28). Thus,wherethanks to Lemma 6. Inserting these estimates into (59), we find that

Now (5) follows from (53), (55), (58), and (66) with the choice of .

3.4. Proof of Theorem 1

(ii) For any odd prime , (52) allows us to writewhere . On the other hand, we have

Thus, for all odd primes.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported in part by the National Natural Science Foundation of China (grant nos. 11871193, 11971370, and 12071375), Natural Science Foundation of Chongqing (grant no. cstc2019jcyj-msxm1651), National Natural Science Foundation of Henan Province (grant no. 202300410274), and Research Projects for Overseas Researchers of Department of Human Resources and Social Security of Henan Province 2020.