Hahn (Rocky Mt. J. Math. 37:1593-1622, 2007) established three differential equations according to , , and , which allows us to obtain the values of the formulas for
etc. Finally, by using the above equations, we derive the algebraic curves.
MSC:11A67.
1 Introduction
Let ℕ denote the sets of positive integers. In number theory, divisor functions are defined as
for . The convolution sum is special one of divisor functions begun by Ramanujan’s effort and expanded by many authors; e.g., see [1]. For example,
(1)
is the result of Ramanujan and also Huard, Ou, Spearman, and Williams [[1], (3.10)]. We can see
(2)
(3)
in [[1], (3.11), (3.12)], respectively. Moreover, there are
in [[1], Theorem 6]. In addition, Lahiri [2] gave the value of the sum
where the sum is over all positive integers satisfying , , and . In this paper we substitute for and obtain the formulas as Lahiri’s evaluation. So, we refer to Hahn’s paper [3]. Officially, we look into Hanh’s definition of three functions for ,
(6)
(7)
(8)
Three functions (6), (7), and (8) satisfy the following differential equations:
(9)
(10)
(11)
The paper is organized as follows. In Section 2, we obtain the values of the formulas for
etc. and insist on the following. Let
If N is an odd integer and , then there exist satisfying
with (see Theorem 2.9).
In Section 2, we derive the convolution sums of the restricted divisor functions. In Section 3, we obtain the algebraic curves by using the convolution sums in Section 2.
2 Convolution sum
Theorem 2.1LetN (≥3) be any positive integer. Then we have
Proof Multiplying on both sides in (9), we obtain
(12)
Employing the definition of and , we can rewrite (12) as
To obtain the result of this section, we need a general theory and it is this that we describe. Suppose that the two polynomials
have common zero, say . Then each of the equations
is of the form , where p is a polynomial of degree at most , and as each of these equations is satisfied when , the determinant of the coefficients must vanish. This determinant is the resultant of and and
where the omitted elements are zero, and the diagonal of contains m occurrences of and n of [[4], p.206]. We can obtain Table 3 from the results of Section 2.
Proof We illustrate the proof for the first in Table 4. In Table 3 we consider and with an odd prime x put by . As
the polynomials
have common zero, namely, . We deduce that for each odd prime p (),
and this simplifies to give , so we can find the irreducible polynomial . Other results in Table 4 can also be obtained by using the resultant. □
Remark 3.2 The plane curves in Corollary 3.1 all have zero-genus since x, y are polynomials of p, which leads to a morphism from the projective line to plane curves . Then it follows easily from the Riemann-Hurwitz theorem (e.g., see Corollary 1 on page 91 of [5]).
Example 3.3 We suggest Figure 1, Figure 2 and Figure 3 for results of and .
Huard JG, Ou ZM, Spearman BK, Williams KS: Elementary evaluation of certain convolution sums involving divisor functions. II. Number Theory for the Millennium 2002, 229–274.
This work was supported by the Kyungnam University Foundation Grant, 2013.
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Authors and Affiliations
National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon, 305-340, Republic of Korea
Daeyeoul Kim
Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Chonju, Chonbuk, 561-756, Republic of Korea
Aeran Kim
Division of Cultural Education, Kyungnam University, 7(Woryeong-dong) kyungnamdaehak-ro, Masanhappo-gu, Changwon-si, Gyeongsangnam-do, 631-701, Republic of Korea
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kim, D., Kim, A. & Kim, MS. A remark on algebraic curves derived from convolution sums.
J Inequal Appl2013, 58 (2013). https://doi.org/10.1186/1029-242X-2013-58