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Eisenstein series and their applications to some arithmetic identities and congruences
Advances in Difference Equations volume 2013, Article number: 84 (2013)
Utilizing the theory of elliptic curves over ℂ to the normalized lattice , its connection to the Weierstrass ℘-functions and to the Eisenstein series and , we establish some arithmetic identities involving certain arithmetic functions and convolution sums of restricted divisor functions. We also prove some congruence relations involving certain divisor functions and restricted divisor functions.
MSC:11A25, 11A07, 11G99.
The study of arithmetical identities and congruences is classical in number theory and such investigations have been carried out by several mathematicians including the legend Srinivasa Ramanujan. This study constitutes an important and significant part of the subject number theory.
For with and , we define some divisor functions for our use in the sequel. Let
and let us define the restricted divisor function
For , let us define the convolution sum
Ramanujan showed that the sum can be evaluated in terms of for the nine pairs satisfying , , . For example, explicitly, we know (see ) that
and (see [, p.35])
From , we note that for any integer , we have
For an elementary proof of (1) and (2), we refer to . An another interesting arithmetical identity (which was stated by Ramanujan, see [, p.146], for some analytical proofs of this identity, one may refer to [, p.329], [, p.136] and , also ) is for , we have
There are some nice arithmetical identities connecting the divisor functions along with Ramanujan’s τ-function. For instance, we know (see ) that
and from , we observe that for ,
where denotes the number of representations of n as a sum of k triangular numbers, i.e. (with ),
and is the coefficient defined by the expression:
For any integer , let
An another identity worth mentioning (see ) is
where and if is not an integer.
From (3), (4) and (5), it is immediate (and interesting) to note that, for ,
The proofs of all these identities and congruences heavily depend upon the theory of modular functions and the properties of Eisenstein series. Later some of these identities have been proved using only elementary techniques.
Define the integers (for ) by
Also define the integers (for ) by
where is as mentioned before and for ,
It should be mentioned that Bernoulli identities associated with the Weierstrass ℘-function have been studied by Chang and Srivastava in . Families of Weierstrass type functions and their applications have been investigated by Chang, Srivastava and Wu in . It is also interesting to note that the families of Weierstrass type functions, Weber type functions and their applications have been studied by Aygunes and Simsek in . A few more related references are [15, 16] and .
Though there are plenty of identities and congruences involving various arithmetic functions available in the literature, practically nothing seriously known involving restricted divisor functions.
For any integer with , we define .
Throughout the paper, where unless otherwise specified hereafter. The aim of this article is to prove some arithmetical identities involving certain arithmetic functions and convolution sums of restricted divisor functions. We also establish some congruence relations similar to (8), (9) and (10). More precisely, we prove the following theorems.
For any integer , we have
Moreover, if M is odd or is odd for an odd prime p, then
Theorem 1.2 For any integer , we have
Theorem 1.3 Let . Then, we have
Corollary 1.4 For , we have
In particular, if , then we have
Theorem 1.5 Let . Then we have
Remark The main idea in proving Theorems 1.1 and 1.2 is to obtain q-series expansions for and with coefficients being restricted divisor functions and their convolution sums. Then we have to compare these expressions with the already known q-series expressions of and .
The paper is organized as follows. In Section 2, we express and in terms of q-products. In Section 3, and are transformed into expressions involving and where and are q-series expressions with coefficients as restricted divisor functions. Then we obtain q-series expressions for and with coefficients being restricted divisor functions and their convolution sums. Then we compare these expressions of and with already known expressions. Section 4 concludes the proofs of Theorems 1.1 and 1.2. In proving arithmetical identities involving the coefficient (where is defined as in (11)), we need to study the quantities and and in turn the convolution sums of restricted divisor functions along with come out very naturally.
2 q-product expressions for the Eisenstein series and
Let ( the complex upper half plane) be a lattice and . The Weierstrass ℘ function relative to is defined by the series
and the Eisenstein series of weight 2k for with is the series
We shall use the notations and instead of and , respectively, when the lattice has been fixed. Then the Laurent series for about is given by
As is customary, by setting
the algebraic relation between and becomes
We use the following q-product expressions:
Theorem 2.1 We have
To prove Theorem 2.1, we need the following lemma.
Lemma 2.2 Let , and .
Proof See . □
Proof of Theorem 2.1 From [, p.63], we observe that , and are the roots of the equation
Therefore, we have
By the above equations and Lemma 2.2, we deduce that
and the following three identities:
Using (14), (15) and (16), we obtain the identities for and , namely
This proves the theorem. □
3 Eisenstein series and divisor functions
We use the q-series and q-products notions
in the following. Some identities of the basic hypergeometric series type are quoted by Fine (see ). Some of these identities (in a similar form) can also be found in  and . It should be mentioned that some generalizations and basic q-extensions of Bernoulli, Euler and Genocchi polynomials have been studied recently by Srivastava (see ). We also refer to  in which zeta and q-zeta function, associated series and integrals have been investigated by Srivastava and Choi. We mention below two identities (see [, p.78, p.79]) for our further use. These are
Using (19), (20) and the facts,
our aim here is first to prove the following lemma.
Lemma 3.1 Let , . Then we have
From (14), we have(21)
From (15), we have(22)
From (16), we have(23)
This proves the lemma. □
Using the fact that , and are the roots of the equation , indeed we can express and in terms of and . More precisely, we have the following lemma.
Lemma 3.2 We have
Proof Note that , and hence
Also note that , and hence
This completes the proof of the lemma. □
In the next theorem, we give q-series expressions for and with the coefficients involving restricted divisor functions and its convolution sums. Precisely, we prove the following theorem.
Theorem 3.3 We have
From Lemma 3.2, we observe that
Now replacing into q in the above expressions for and , we obtain
This completes the proof of the theorem. □
It should be noted that and themselves have coefficients and in their q-series expansions. However, the aim of the next theorem is to express and in terms of convolution sums involving restricted divisor functions .
Theorem 3.4 (a) If is an integer, then
In particular, if is odd, then
If is an integer, then
In particular, if is odd, then
Proof From [, p.59], we know that
So comparing (24) and (26), we find that
On the other hand, Liouville  proved
for odd M. From (28) and (29), we note that we reprove a result in [, p.300], namely,
for odd M. By the same way, comparing (25) and (27), we deduce
where . Note that for odd , we get
where we used Equation (29). This completes the proof of the theorem. □
4 Proof of the theorems
Proof of Theorem 1.1 Glaisher proved that (see [, p.300])
It follows directly from (28) that
Thus, we have proved the identity
Thus, for any integer , we have
This proves (i).
We note that we can write (34) as
If M is odd, then (35) is equivalent to
Let . Therefore, if is odd for an odd prime p, then we note that
and so we can write for some . Therefore, from (35), we obtain
This means that
This proves (ii). □
For example, and . For any prime , we note that
We observe that
and . Thus, and this implies that the largest prime factor of is strictly <p.
From the fact
and from known results, it is possible to establish the following identity (in an elementary way, without using the Eisenstein series), namely
One can also use this identity to obtain
Proof of Theorem 1.2 Inserting (33) and (34) into (31), we obtain
Thus, we deduce the congruence relation
This completes the proof of Theorem 1.2. □
Proof of Theorem 1.3 We expand as
Then we notice that
by [, Lemma 3.1(b)] and . Since , the right-hand side of (36) is
respectively. This completes the proof. □
Proof of Corollary 1.4
We note that
Therefore, from Theorem 1.3, we obtain
Thus, the first assertion
We note that , and thus whenever and is odd, we have . Thus,
whenever . This proves the second assertion. □
Proof of Theorem 1.5
We note that
Thus, by and [, (11)], we obtain that
We observe that
Thus, the right-hand side of (39) is
Now from (37), (38) and (see [, (4.4)]), we have
and from (see [, Theorem 4]), we have
Thus, the theorem follows. □
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Dedicated to Professor Hari M Srivastava.
The author, Ayyadurai Sankaranarayanan, wishes to thank the National Institute for Mathematical Sciences (NIMS), Daejeon, Republic of Korea, for its warm hospitality and generous support. This research was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean government (B21303).
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read and approved the final manuscript.
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Kim, D., Kim, A. & Sankaranarayanan, A. Eisenstein series and their applications to some arithmetic identities and congruences. Adv Differ Equ 2013, 84 (2013). https://doi.org/10.1186/1687-1847-2013-84
- divisor functions
- restricted divisor functions
- Eisenstein series
- convolution sums
- congruence relations