Abstract

We examine the average order of some arithmetic functions written as sums over Euler function in arithmetic progression and in general over such that is a prime number, an integer and is a polynomial function with integer coefficients and a degree that is not constant modulo Our results are based on various estimates of rational exponential sums with the Euler Function in arithmetic progression which are due to William Banks and Igor E. Shparlinski.

1991 Mathematics Subject Classification. 11N37, 11T23.

1. Introduction

In this paper, always denotes a prime number fixed throughout, the Euler function and the greatest common divisor. If is a real number, we have We recall that the notation is equivalent to the statement for positive functions and where it means that there exists a constant such that and the implied constants in the symbols are absolute. We also use the expression that is equivalent to Furthermore, we have the abbreviation for any real number and we will note by .

While the behavior of a number theoretic function for large is often difficult to determine because the function values can fluctuate considerably as n increases, it is more fruitful to study partial sums and seek asymptotic formulas of the form

where is a known function of and represents the error, a function of smaller order than for all x in some prescribed range. Some of these arithmetic functions are called multiplicative when they satisfy and whenever and are coprimes. In this work, we will focus on the following arithmetic functions:

Ÿ : The number of non-negative divisors function.

Ÿ : The sum of the -th powers of all the non-negative divisors function, for . In particular,

Ÿ The Euler function.

Ÿ The Jordan totient function counting the number of tuples. In particular,

Ÿ : The Pillai function known also as the sum function.

Problems involving exponential sums over some arithmetic functions have been considered in many papers as in ^{2, 3, 7} where they obtained non trivial bounds in certain ranges for rational exponential sums of the form such that . In particular, Banks and Shparlinski ² gave upper bounds for the number of solutions to congruences with the Euler functions and the Carmichael function and also non trivial bounds for the exponential sums involving They expected that their methods could be also applied to exponential sums with replaced by an arbitrary positive integer although certain arguments would be more complicated. Analogous results for similar arithmetic functions such as the aliquot divisor function in ³. Later on, Kerr ⁵ solved the problem given by Shparlinski in [ ⁸, Problem 27] about bounding the rational exponential sums over the divisor function for specifically odd integer and coprime with a.

In this work, we inquire about calculating the mean values of the already listed arithmetic functions for large values of N where they will be written as Dirichlet convolutions under some constraints involving the Euler function. Indeed, we will be focusing on the following

The main goal of this paper is to give an average order of some chosen arithmetic functions under constraints involving the Euler function. In particular our work is organized as follows: Section 2 is devoted to give some auxiliary lemmas that are necessary for the proofs of our theorems. Section 3 and Section 4 establish the mean values of the defined arithmetic functions in the above first while satisfying the conditions then

2. Technical Lemmas

We start by providing a rigorous definition of the zeta function for

The Euler-Mascheroni's constant is usually known as

In order to be able to state our main results, we need the following mean values of the form

Lemma 2.1 ([ ¹, Theorem 3.2]). If we have

(1)

(2) if

(3) if

(4) if

A crucial part in our proofs is the estimation of exponential sums. One plain formula that helps to detect the congruences is the following simple but important observation

(2.1)

The key input to the present paper is the application of the following bounds of the exponential sums given by Banks and Shparlinski ².

Theoerem A [ ², Theorem 5.1]: The following bound holds, for

(2.2)

It is to be noted that Banks and Shparlinski bounds are non trivial for a wide range of values of starting with They remarked that although it might be possible to improve on this power of , for very small values of relative to , it is simply not possible to obtain nontrivial bounds.

Now, we define the following set of polynomials with integers coefficients of degree whose leading coefficient is relatively prime to that is

Theoerem B [ ², Theorem 5.2]: For any and , the following bound holds for

(2.3)

3. Sums with Congruences with φ

Theorem 3.1. Let and . Then, for a sufficiently large , we have

Proof. Given large enough, we have

Next, we use Dirichlet's hyperbola formula and we get

Each summand will be treated separately in the sequel.

First, we write thanks to the orthogonality formula (2.1)

(3.1)

The first sum in the right hand side of (3.1) can be estimated using Lemma 2.1 giving

(3.2)

Afterwards, we use Abel's summation formula and we get

(3.3)

Due to Theorem A, we have for all while we mean by

(3.4)

Hence, we obtain

(3.5)

Now, we apply Theorem A again and we get

(3.6)

Summing up the identities (3.5}) and (3.6) lead to

(3.7)

Taking the identities (3.2) and (3.7) and going back to (3.1) therefore yield

(3.8)

It is to be said that this choice of is to balance the first term inside the parentheses in the term.

Next, we turn our attention to bounding the . We write again due to the orthogonality formula,

When using Lemma 2.1, it is easy to see that

(3.9)

While calling Theorem A and denoting , we get for every

Afterwards by writing as and since , then necessarily we have .

Consequently, the upper bound becomes while using Lemma 2.1

(3.10)

Moreover, by choosing it follows that .

Consequently, by putting (3.9) and (3.10) jointly gives

(3.11)

Finally, for the last sum, we obtain in view of (2.1) and the first assertion of Lemma 2.1,

(3.12)

We reach the desired result by assembling (3.8), (3.11) and (3.12).

3.2. Average Order of the Function

The case was already studied in the last paragraph since .

Theorem 3.2. Let and . For a real number and for a sufficiently large , we state that

For

such that

For ,

Proof. We shall treat the subcases , and separately.

At first, we might write due to Lemma 2.1 and Abel’s summation formula

We will consider separately the last equation in the two positive subcases.

Ÿ If s = 1: we get by using the orthogonality relation and Lemma 2.1

(3.13)

Thus, by Lemma 2.1, the first sum of (3.13) becomes

Consequently, we obtain

(3.14)

Ÿ If We set and we write thanks to Lemma

(3.15)

Thus, we summarize both cases (3.14}) and (3.15) by writing as follows

(3.16)

while denotes

Next, we apply Abel's formula to for every so we obtain

For all thanks to Theorem A, the latter integral is absolutely convergent and

Therefore, we have

(3.17)

such that for every

We call Theorem A once again, so that we have

(3.18)

Joining (3.17) to (3.18), we find that

Afterwards, we go back to

(3.19)

with

Hence, taking (3.19) and putting it in (3.16) lead us to the desired asymptotic formula.

Now in order to study the case we shall set where

Then, it follows from the orthogonality formula that

Afterwards, it follows that Lemma 2.1,

(3.20)

Whence, we have

(3.21)

Furthermore, for we denote . Then, Theorem A and Lemma \ref{lemma} lead for

(3.22)

It is to be noted that we used and that with the choice of it is easily seen that .

Finally, by combining (3.21) and (3.22), the proof is completed.

Theorem 3.3. Let and . We assert, for and a sufficiently large ,

where is defined as

.

Proof. Due to the orthogonality formula and Lemma 2.2, we have

(3.23)

such that B denotes

(3.24)

Knowing that the last step in (3.23) follows from the fact that the Möbius function is the inverse convolution of 1. Indeed, we formally have for such that

Each of the two series being absolutely convergent leads to

(3.25)

Next, we apply Abel's summation again to in order to obtain

One can easily see that the integral is convergent. Subsequently, we get

(3.26)

with

Clearly, we have

(3.27)

Considering the identities (3.26) and (3.27) jointly, we find that

(3.28)

Hence, by going back to (3.26) and putting (3.27) in ,

(3.29)

with

Finally, gathering (3.23) and (3.29), we reach the desired expression.

One can easily see that the Euler function is the Jordan totient function for It follows that from Theorem 3.3.

Corollary 3.4. Let and , then for a sufficiently large , we have

where defined as in Theorem 3.3 for

Theorem 3.5. Let and . For a sufficiently large , we state

such that is defined as in Theorem 3.3 for

Proof.

(3.30)

We need to evaluate the second sum in the product in (3.30). Thanks to Abel's formula, we have

Observing that

we get

(3.31)

Therefore, by putting the last equation in (3.30) for , we obtain

Now, we use (2.1) and we get

(3.32)

So on one hand, we have

such that the last equation follows from Lemma 2.1.

Going back to (3.25), it gives by differentiation for

(3.33)

Afterwards, again by (3.25) and using (3.33), we write

(3.34)

such that where is the Glaischer-Kinkelin constant (for further details about this see ^{4, 6}).

On the other hand, we have while calling (3.31),

(3.35)

where the error term is equal to thanks to Lemma 2.1.

For the first sum, all we need to do is to appeal the property (3.28) and apply it for so that we get

Now, for the second sum in (3.35), we use Abel's summation and we obtain

Since

(3.36)

it follows that

(3.37)

Appealing the term (3.26}) for allows us to find that the first integral in (3.37) is only equal to

As for the second integral in (3.37), we have from (3.36) and since the integral is trivially convergent, it follows that

Thus,

(3.38)

Collecting (3.37) and (3.38) gives us

(3.39)

Hence, we insert (3.39) in (3.35) in order to get

So that we obtain

(3.40)

At the end, by inserting (3.34) and (3.40) in (3.32), we achieve the desired asymptotic formula.

4. Sums with Congruences with f(φ)

In this section, we turn our attention to the mean value of the listed functions before with restrictions including the polynomials from the class given at the beginning with In fact, our methods can be used without any further modifications to estimate the similar sums when . One only needs to use appropriately Theorem B instead of Theorem A when needed.

Theorem 4.1. Let and . Then we have for a sufficiently large

(1)

(2)

For , we have

(3)

(4)

(5) For

such that the mentioned constants are defined as follows

Acknowledgements

The authors would like to thank the referee for comments about an earlier version of this paper.

References