An Improved Asymptotic on the Representations of Integers as Sums of Products

Abstract

In this paper, we improve the error terms of Chace’s results in the study by Chace (1994) on the number of ways of writing an integer as a sum of products of factors, valid for and , 3. More precisely, for , 3, we improve the upper bound , for the error term, to when and when .

1. Introduction

In this paper, we study the number of representations of a natural number as a sum of terms, each being a product of factors. Let denote this number. This problem was studied by Estermann [1, 2] in the case or 3 and by using some properties of Dirichlet -function. His method is not easy to be generalized. Later, Chace [3] generalized Estermann’s result to and . For such and , he gotwhereand

Here, is defined in (72). The way he studied the problem is different from Estermann. The main tools he used are the Hardy–Littlewood method and some results from the divisor problem in arithmetic progressions. He got the main term which is a sum of terms of the form , where are the “singular series” and are the “singular integrals.” They occur in the applications of the Hardy–Littlewood method. In this paper, we improve Chace’s result in the cases and . We get the following result.

Theorem 1. Suppose and . Then,where defined by (72) satisfies (2) and the error term satisfies

We can compare the exponents in (3) with our results (5). For , we see that when and when . Our error terms are better than Chace’s.

The proof of the theorem is an application of the Hardy–Littlewood method (cf, Chapter 3 of [4]), the Voronoi summation formula, and some results from the Kloosterman sums. The estimates on the minor arc were studied by Chace, and his result is sufficient for us. Hence, we will not focus on the minor arc in this paper. The main difficulty arises in treating the error term of the major arcs. In Section 2, we make some preparations for our proof. We state the Voronoi summation formula and give some lemmas related to the Kloosterman sums in this section. In Sections 3.1 and 3.2, we obtain a bound for the contribution from the minor arcs. In , we prove our theorem.

2. Applications of the Voronoi Summation Formula

Let . Suppose that is a smooth function compactly supported in with , satisfying in and

Letbe the Mellin transformation of . To state Voronoi’s summation formula, we introduce the notations below. For , letwhere

We definewherefor and . From Theorem 2 in [5], we know the following lemma.

Lemma 1 (Voronoi’s summation formula). With the above notations, we have, for ,To simplify the integrals in (15), we definewhere . By (12), it is clear thatTo use Lemma 1 in practice, we need to estimate the residue in (15), and . We first compute the residue. For , we definewhere is an integer, and the coefficients are sums of terms of the formfor some function . The coefficients are given explicitly in equation (2.13) in [6]. It has been shown in [6] that for , is independent of .

Lemma 2. Suppose , . We have

Proof. By (9), we can rewrite aswherefor . Therefore,Chace (Theorem 1 and (2.1) in [6]) gave thatwhereSubstituting this into (23), we haveThe proof of the lemma is complete.
Our next lemma, proved by Jiang and Lü (Lemma 2.7 in [7]), is to evaluate the integrals in (15).

Lemma 3. Let be defined as in (16). Then, we have, for ,

Proof. This can be proved by taking with in Lemma 2.7 of [7].
The following two lemmas give estimates for . This two lemmas play an important role in our proof. We use some results for the Kloosterman sum to obtain the power saving in the aspect.

Lemma 4. Suppose , , and let be defined as in (11). We have

Proof. By (13), we obtainwherefor . Here, the notation means that divides each component of . Then, we obtain that the left hand side of the desired equation in our lemma equals toNow, we definefor and . Here, denotes the multiplicative inverse of modulo . In Section 6 in [8], Smith gave thatwhere for all . Therefore, (31) can be written asTaking the change of variables that , we getFrom Theorem 6 in [9], we know thatThe lemma follows immediately.

Lemma 5. Suppose , , and . Let be defined as before. Then, we have

Proof. By (13), for , we haveFor , one hasNow, if we use Weil’s classical boundthen it follows from (39) thatFor and , noting that by (11), we have . Therefore,Hence, we havefor . The proof of the lemma is complete.

3. Proof of Theorem 1

In this section, we prove Theorem 1. The main difficulty arises in treating the major arcs. The results from Kloosterman’s sums will play a role in our proof. Throughout this section, . We choose a smooth function compactly supported in with , satisfying in and (6) and (7). To apply the circle method, we choose the parameters and such that

By Dirichlet’s lemma on rational approximations, each may be written in the form as follows:for some integers and with and . We denote by the set of satisfying (45) and define the major arc by

It follows from that the major arcs are disjoint. The union of minor arcs is just the complement of in . We denote

Then, we have

The upper bound of the integral of on is given by Chace [3].

Lemma 6. Suppose and are integers. Then,

This estimate on the minor arc is sufficient for us. It remains to consider the integral of on the major arc. Let

Now, for , we havewhere and

It is clear that is a smooth function compactly supported in . Moreover, it satisfiesfor any , andfor any . Now, by Lemma 1, we getwhereis the Mellin transformation of and

It follows from Lemma 2 that

From Equation (4.8) in [3], we know that

Then, we havewhere

Now, for , by (51), (55), and (60), we writewhere

Hence, by dyadic analysis, for , formally, we writewherewithfor . Therefore, we havewhere

It has been proved by Chace (Section 5 in [3]) thatwheresatisfying

In the following subsections, we will estimate for .

3.1. The Estimate of

In this section, we give the upper bound of .

Lemma 7. Let . We have

Proof. By our definition of major arc (46), we haveBecause for , is independent of , and we can interchange the order of summation over and the integral and then take the summation of over first. Therefore,By (59) and (68), it is clear thatBy the definition of in (66), we haveWe deduce thatIt is obvious from (61) thatIt remains to consider the summation of . We write be with replaced by in (16). Similar to the assertions of Lemma 3, we haveNotice thatThen, by Lemma 4, we haveTherefore, we getHence, by (79), (80), and (84), for , one hasSubstituting this and (77) into (76), we obtainBy some elementary calculations, we get the desired result.

3.2. Integral of , , on the Major Arcs

The way we treat for for is different from that of . We get the following lemma.

Lemma 8. Let and be defined as before. We have, for ,

Proof. The strategy is similar to that of Section 5 in [3]. Note thatWe obtainSimilar to (83), by (81) and Lemma 5, we have, for and ,Hence, by (52), (61), (64), (90), and (66), we haveNow, for , by Cauchy’s inequality, the definition of major arc (46), and Parseval’s identity, we obtainFor , we haveby the fact that . One then shows thatby using the definition of the major arc and (77). Combining these results, we complete the proof of the lemma.

3.3. Proof of Theorem 1

By Lemma 6 and (69)–(73), we have, for , ,wheresatisfying

Now, by Lemmas 7 and 8, takingwe have, for ,

We complete the proof of Theorem 1.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that there are no conflicts of interest.