A remark on algebraic curves derived from convolution sums

Daeyeoul Kim; Aeran Kim; Min-Soo Kim

doi:10.1186/1029-242X-2013-58

Research
Open Access

A remark on algebraic curves derived from convolution sums

Daeyeoul Kim¹,
Aeran Kim² and
Min-Soo Kim³Email author

Journal of Inequalities and Applications20132013:58

https://doi.org/10.1186/1029-242X-2013-58

Received: 5 December 2012
Accepted: 4 February 2013
Published: 19 February 2013

Abstract

Hahn (Rocky Mt. J. Math. 37:1593-1622, 2007) established three differential equations according to $P (q)$ , $E (q)$ , and $Q (q)$ , which allows us to obtain the values of the formulas for

\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n), \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n),

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

\begin{matrix} σ_{s} (N) = \sum_{d | N} d^{s}, σ (N) : = σ_{1} (N) = \sum_{d | N} d, \\ {\tilde{σ}}_{s} (n) = \sum_{d | n} {(- 1)}^{d - 1} d^{s}, {\hat{σ}}_{s} (n) = \sum_{d | n} {(- 1)}^{(n / d) - 1} d^{s} \end{matrix}

for

N, s, d \in N

. 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,

\sum_{m = 1}^{n - 1} σ (m) σ (n - m) = \frac{1}{12} (5 σ_{3} (n) + (1 - 6 n) σ (n))

(1)

is the result of Ramanujan and also Huard, Ou, Spearman, and Williams [[1], (3.10)]. We can see

\sum_{m = 1}^{n - 1} m σ (m) σ (n - m) = \frac{n}{24} (5 σ_{3} (n) + (1 - 6 n) σ (n)),

(2)

\sum_{m = 1}^{n - 1} σ (m) σ_{3} (n - m) = \frac{1}{240} (21 σ_{5} (n) + (10 - 30 n) σ_{3} (n) - σ (n))

(3)

in [[1], (3.11), (3.12)], respectively. Moreover, there are

\sum_{m < n / 2} σ (m) σ (n - 2 m) = \frac{1}{24} (2 σ_{3} (n) + (1 - 3 n) σ (n) + 8 σ_{3} (\frac{n}{2}) + (1 - 6 n) σ (\frac{n}{2}))

(4)

in [[1], (4.4)] and

\begin{aligned} \sum_{m < n / 2} σ_{3} (m) σ (n - 2 m) = \frac{1}{240} (σ_{5} (n) - σ (n) + 20 σ_{5} (\frac{n}{2}) + (10 - 30 n) σ_{3} (\frac{n}{2})), \\ \sum_{m < n / 2} σ (m) σ_{3} (n - 2 m) = \frac{1}{240} (5 σ_{5} (n) + (10 - 15 n) σ_{3} (n) + 16 σ_{5} (\frac{n}{2}) - σ (\frac{n}{2})) \end{aligned}

(5)

in [[1], Theorem 6]. In addition, Lahiri [2] gave the value of the sum

\sum_{m_{1} + \dots + m_{r} = n} m_{1}^{a_{1}} \dots m_{r}^{a_{r}} σ_{b_{1}} (m_{1}) \dots σ_{b_{r}} (m_{r}) (r \geq 3),

where the sum is over all positive integers

m_{1}, \dots, m_{r}

satisfying

m_{1} + \dots + m_{r} = n

,

a_{i} \in N \cup {0}

, and

b_{i} \in N

. In this paper we substitute

{\hat{σ}}_{b_{i}} (m_{i})

for

σ_{b_{i}} (m_{i})

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

| q | < 1

,

P (q) : = 1 + 8 \sum_{n = 1}^{\infty} \tilde{σ} (n) q^{n},

(6)

E (q) : = 1 + 24 \sum_{n = 1}^{\infty} \hat{σ} (n) q^{n},

(7)

Q (q) : = 1 - 16 \sum_{n = 1}^{\infty} {\tilde{σ}}_{3} (n) q^{n} .

(8)

Three functions (6), (7), and (8) satisfy the following differential equations:

q \frac{d P (q)}{d q} = \frac{P^{2} (q) - Q (q)}{4},

(9)

q \frac{d E (q)}{d q} = \frac{E (q) P (q) - Q (q)}{2},

(10)

q \frac{d Q (q)}{d q} = P (q) Q (q) - E (q) Q (q) .

(11)

The paper is organized as follows. In Section 2, we obtain the values of the formulas for

\begin{matrix} \sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n), \\ \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n), \end{matrix}

etc. and insist on the following. Let

\begin{aligned} f (N) \in & {\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n), \sum_{m = 1}^{N - 1} \hat{σ} (m) \hat{σ} (N - m), \\ \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n), \sum_{m = 1}^{N - 1} \hat{σ} (m) {\tilde{σ}}_{3} (N - m), \\ \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n), \sum_{m = 1}^{N - 1} m \hat{σ} (m) \hat{σ} (N - m)} . \end{aligned}

If N is an odd integer and

n \in N \cup {0}

, then there exist

u, a, b, c, d, e, g \in Z

satisfying

f (N) = \frac{1}{u} [a σ_{5} (N) + (b N + c) σ_{3} (N) + (d N^{2} + e N + g) σ (N)]

with $a + b + c + d + e + g = 0$ (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 $\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n)$

Theorem 2.1 Let N (≥3) be any positive integer. Then we have

\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n) = \frac{1}{64} {{\tilde{σ}}_{5} (N) + 6 (1 - N) {\tilde{σ}}_{3} (N) + (8 N^{2} - 12 N + 3) \tilde{σ} (N)} .

Proof Multiplying

P (q)

on both sides in (9), we obtain

P^{3} (q) = 4 P (q) \cdot q \frac{d P (q)}{d q} + P (q) Q (q) .

(12)

Employing the definition of

P (q)

and

Q (q)

, we can rewrite (12) as

\begin{aligned} {(1 + 8 \sum_{n = 1}^{\infty} \tilde{σ} (n) q^{n})}^{3} = & 4 (1 + 8 \sum_{n = 1}^{\infty} \tilde{σ} (n) q^{n}) q (8 \sum_{m = 1}^{\infty} m \tilde{σ} (m) q^{m - 1}) \\ + (1 + 8 \sum_{n = 1}^{\infty} \tilde{σ} (n) q^{n}) (1 - 16 \sum_{m = 1}^{\infty} {\tilde{σ}}_{3} (m) q^{m}) . \end{aligned}

Thus, we have

\begin{matrix} 512 \sum_{N = 3}^{\infty} \sum_{L = 2}^{N - 1} \sum_{l = 1}^{L - 1} \tilde{σ} (N - L) \tilde{σ} (L - l) \tilde{σ} (l) q^{N} \\ = - 16 \sum_{N = 1}^{\infty} {\tilde{σ} (N) + {\tilde{σ}}_{3} (N) - 2 N \tilde{σ} (N)} q^{N} - 192 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} \tilde{σ} (l) \tilde{σ} (N - l) q^{N} \\ - 128 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} \tilde{σ} (l) {\tilde{σ}}_{3} (N - l) q^{N} + 256 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} l \tilde{σ} (l) \tilde{σ} (N - l) q^{N} . \end{matrix}

Then we refer to

4 \sum_{m < n} \tilde{σ} (m) \tilde{σ} (n - m) = - {\tilde{σ}}_{3} (n) + (2 n - 1) \tilde{σ} (n)

in [[3], (4.4)],

16 \sum_{m < n} \tilde{σ} (m) {\tilde{σ}}_{3} (n - m) = - {\tilde{σ}}_{5} (n) + 2 (n - 1) {\tilde{σ}}_{3} (n) + \tilde{σ} (n)

in [[3], (4.9)] and by direct calculation we get

\sum_{l = 1}^{N - 1} l \tilde{σ} (l) \tilde{σ} (N - l) = \frac{N}{2} \sum_{l = 1}^{N - 1} \tilde{σ} (l) \tilde{σ} (N - l) .

This completes the proof of the theorem. □

Corollary 2.2 We obtain

ln | Q (q) | = 8 \sum_{n = 1}^{\infty} \frac{\tilde{σ} (n) - 3 \hat{σ} (n)}{n} q^{n},

where $Q (q) = 1 - 16 \sum_{n = 1}^{\infty} {\tilde{σ}}_{3} (n) q^{n}$ as (8).

Proof By separating the variables in (11), we can have

- \int_{0}^{q} \frac{d Q (x)}{Q (x)} = \int_{0}^{q} \frac{E (x) - P (x)}{x} d x .

Thus,

- ln | Q (q) | = \int_{0}^{q} 8 \sum_{n = 1}^{\infty} (3 \hat{σ} (n) - \tilde{σ} (n)) x^{n - 1} d x .

We note that

3 \hat{σ} (n) - \tilde{σ} (n)

is positive because

3 \hat{σ} (n) - \tilde{σ} (n) = 2 (σ (n) - σ (n / 2))

according to

{\tilde{σ}}_{s} (n) = σ_{s} (n) - 2^{s + 1} σ_{s} (n / 2) and {\hat{σ}}_{s} (n) = σ_{s} (n) - 2 σ_{s} (n / 2)

in [[3], (1.12), (1.13)]. Therefore, we can exchange the summation and the integral. This completes the proof of the corollary. □

We introduce the following Lemma 2.3 to deduce Theorem 2.6.

Lemma 2.3 Let N (≥2) be any positive integer. Then we have

(a)

$\sum_{m = 1}^{N - 1} \hat{σ} (m) \hat{σ} (N - m) = \frac{1}{12} (σ_{3} (N) + 4 σ_{3} (\frac{N}{2}) - \hat{σ} (N))$ .
(b)

$\sum_{m = 1}^{N - 1} \hat{σ} (m) {\tilde{σ}}_{3} (N - m) = - \frac{1}{48} ({\tilde{σ}}_{5} (N) + 2 {\tilde{σ}}_{3} (N) - 3 \hat{σ} (N))$ .
(c)

$\sum_{m = 1}^{N - 1} m \hat{σ} (m) \hat{σ} (N - m) = \frac{N}{24} (σ_{3} (N) + 4 σ_{3} (\frac{N}{2}) - \hat{σ} (N))$ .

Proof (a) The proof is the same as Remark in [[3], p.13]. Also, Hahn showed that

36 \sum_{m < n} \hat{σ} (m) \hat{σ} (n - m) = {\begin{matrix} - 3 \hat{σ} (n) + 3 {\tilde{σ}}_{3} (n), & if n is odd, \\ - 3 \hat{σ} (n) - 5 {\tilde{σ}}_{3} (n) + 4 {\tilde{σ}}_{3} (n / 2), & if n is even \end{matrix}

in [[3], Theorem 4.2] derived by the identity

E^{2} (q) = z^{4} {(1 + x)}^{2}

, which is the same result (a).

(b)

The convolution sum can be written as
$\begin{aligned} \sum_{m = 1}^{N - 1} \hat{σ} (m) {\tilde{σ}}_{3} (N - m) = & \sum_{m = 1}^{N - 1} {σ (m) - 2 σ (\frac{m}{2})} {σ_{3} (N - m) - 16 σ_{3} (\frac{N - m}{2})} \\ = & \sum_{m = 1}^{N - 1} σ (m) σ_{3} (N - m) - 16 \sum_{m = 1}^{N - 1} σ (m) σ_{3} (\frac{N - m}{2}) \\ - 2 \sum_{m = 1}^{N - 1} σ (\frac{m}{2}) σ_{3} (N - m) + 32 \sum_{m = 1}^{N - 1} σ (\frac{m}{2}) σ_{3} (\frac{N - m}{2}) . \end{aligned}$

Then we obtain

\begin{aligned} \sum_{m = 1}^{N - 1} σ (m) σ_{3} (\frac{N - m}{2}) = \sum_{t < N / 2} σ (N - 2 t) σ_{3} (t), \\ \sum_{m = 1}^{N - 1} σ (\frac{m}{2}) σ_{3} (N - m) = \sum_{t < N / 2} σ (t) σ_{3} (N - 2 t), \\ \sum_{m = 1}^{N - 1} σ (\frac{m}{2}) σ_{3} (\frac{N - m}{2}) = \sum_{t < N / 2} σ (t) σ_{3} (\frac{N}{2} - t) . \end{aligned}

Therefore, we have proved (b) by using (3) and (5).

(c)

We obtain the proof by direct calculation of the index m. □

Corollary 2.4 Let

N = 2 q + 1

be an odd prime in Lemma 2.3 and let

T_{i} (n) = \sum_{k = 1}^{n} k^{i}

(

i \geq 1

). Then we have

(a)

$\sum_{m = 1}^{2 q} \hat{σ} (m) \hat{σ} (2 q + 1 - m) = 2 T_{2} (q)$ .
(b)

$\sum_{m = 1}^{2 q} \hat{σ} (m) {\tilde{σ}}_{3} (2 q + 1 - m) = - 2 (q^{2} + q + 1) T_{2} (q)$ .
(c)

$\sum_{m = 1}^{2 q} m \hat{σ} (m) \hat{σ} (2 q + 1 - m) = N T_{2} (q)$ .

Proof

From Lemma 2.3(a), (b), and (c), we obtain

\begin{aligned} \sum_{m = 1}^{2 q} \hat{σ} (m) \hat{σ} (2 q + 1 - m) = \frac{1}{12} {{(2 q + 1)}^{3} - (2 q + 1)}, \\ \sum_{m = 1}^{2 q} \hat{σ} (m) {\tilde{σ}}_{3} (2 q + 1 - m) = - \frac{1}{48} {{(2 q + 1)}^{5} + 2 {(2 q + 1)}^{3} - 3 (2 q + 1)}, \\ \sum_{m = 1}^{2 q} m \hat{σ} (m) \hat{σ} (2 q + 1 - m) = \frac{2 q + 1}{24} {{(2 q + 1)}^{3} - (2 q + 1)} . \end{aligned}

This completes the proof of the corollary. □

Example 2.5 The first eleven values of

\sum_{m = 1}^{2 q} \hat{σ} (m) \hat{σ} (2 q + 1 - m)

and

2 T_{2} (q)

are listed in Table 1.

Table 1

Examples for $\sum_{m = 1}^{2 q} \hat{σ} (m) \hat{σ} (2 q + 1 - m)$ and $2 T_{2} (q)$ ( $1 \leq q \leq 11$ )

q	1	2	3	4	5	6	7	8	9	10	11
$\sum_{m = 1}^{2 q} \hat{σ} (m) \hat{σ} (2 q + 1 - m)$	2	10	28	62	110	182	292	408	570	800	$1, 012$
$2 T_{2} (q)$	2	10	28	60	110	182	280	408	570	770	$1, 012$

Theorem 2.6 Let N (≥3) be any positive integer. Then we have

\begin{aligned} \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n) = & \frac{1}{576} {{\tilde{σ}}_{5} (N) + 4 {\tilde{σ}}_{3} (N) + \tilde{σ} (N) - 6 (2 N - 1) \hat{σ} (N) \\ + 6 (N - 1) (σ_{3} (N) + 4 σ_{3} (\frac{N}{2}))} . \end{aligned}

Proof Multiplying

E (q)

in (10), we obtain

E^{2} (q) P (q) = E (q) Q (q) + 2 E (q) \cdot q \frac{d E (q)}{d q} .

So, we have

\begin{matrix} {(1 + 24 \sum_{n = 1}^{\infty} \hat{σ} (n) q^{n})}^{2} (1 + 8 \sum_{m = 1}^{\infty} \tilde{σ} (m) q^{m}) \\ = (1 + 24 \sum_{n = 1}^{\infty} \hat{σ} (n) q^{n}) (1 - 16 \sum_{m = 1}^{\infty} {\tilde{σ}}_{3} (m) q^{m}) \\ + 2 (1 + 24 \sum_{n = 1}^{\infty} \hat{σ} (n) q^{n}) q (24 \sum_{m = 1}^{\infty} m \hat{σ} (m) q^{m - 1}) . \end{matrix}

Therefore

\begin{matrix} 4, 608 \sum_{N = 3}^{\infty} \sum_{L = 2}^{N - 1} \sum_{l = 1}^{L - 1} \hat{σ} (N - L) \hat{σ} (L - l) \tilde{σ} (l) q^{N} \\ = - 8 \sum_{N = 1}^{\infty} {\tilde{σ} (N) + 2 {\tilde{σ}}_{3} (N) + 3 \hat{σ} (N) - 6 N \hat{σ} (N)} q^{N} \\ - 384 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} \hat{σ} (l) \tilde{σ} (N - l) q^{N} - 384 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} \hat{σ} (l) {\tilde{σ}}_{3} (N - l) q^{N} \\ - 576 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} \hat{σ} (l) \hat{σ} (N - l) q^{N} + 1, 152 \sum_{N = 2}^{\infty} \sum_{l = 1}^{N - 1} l \hat{σ} (l) \hat{σ} (N - l) q^{N} . \end{matrix}

Lastly, we use Lemma 2.3. □

Theorem 2.7 Let N (≥3) be any positive integer. Then we have

\sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n) = \frac{1}{192} {σ_{5} (N) - 8 σ_{5} (\frac{N}{2}) - 2 σ_{3} (N) - 8 σ_{3} (\frac{N}{2}) + \hat{σ} (N)} .

Proof

Let us consider

\begin{aligned} \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n) \\ = \sum_{l = 1}^{N - 2} \hat{σ} (l) \sum_{m + n = N - l} \hat{σ} (m) \hat{σ} (n) \\ = \sum_{l = 1}^{N - 2} \hat{σ} (l) \frac{1}{12} {σ_{3} (N - l) + 4 σ_{3} (\frac{N - l}{2}) - \hat{σ} (N - l)} \\ = \sum_{l = 1}^{N - 1} \hat{σ} (l) \frac{1}{12} {σ_{3} (N - l) + 4 σ_{3} (\frac{N - l}{2}) - \hat{σ} (N - l)} \end{aligned}

(13)

by Lemma 2.3(a). Then (13) becomes

\begin{aligned} \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n) = & \frac{1}{12} {\sum_{l = 1}^{N - 1} σ (l) σ_{3} (N - l) - 2 \sum_{l < N / 2} σ (l) σ_{3} (N - 2 l) \\ + 4 \sum_{t < N / 2} σ_{3} (t) σ (N - 2 t) - 8 \sum_{t < N / 2} σ (t) σ_{3} (\frac{N}{2} - t) \\ - \sum_{l = 1}^{N - 1} \hat{σ} (l) \hat{σ} (N - l)} \end{aligned}

by $\hat{σ} (n) = σ (n) - 2 σ (\frac{n}{2})$ . □

Remark 2.8 Let

N = 2 q + 1

be an odd prime in Theorem 2.7. Then we have

\sum_{l + m + n = 2 q + 1} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n) = T_{1} (q) \cdot T_{2} (q) .

Proof It is obvious. □

Importantly, we propose the following theorem.

Theorem 2.9 Let

\begin{aligned} f \in & {\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n), \sum_{m = 1}^{N - 1} \hat{σ} (m) \hat{σ} (N - m), \\ \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n), \sum_{m = 1}^{N - 1} \hat{σ} (m) {\tilde{σ}}_{3} (N - m), \\ \sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n), \sum_{m = 1}^{N - 1} m \hat{σ} (m) \hat{σ} (N - m)} . \end{aligned}

If N is an odd integer and

n \in N \cup {0}

, then there exist

u, a, b, c, d, e, g \in Z

satisfying

f (N) = \frac{1}{u} [a σ_{5} (N) + (b N + c) σ_{3} (N) + (d N^{2} + e N + g) σ (N)]

with $a + b + c + d + e + g = 0$ .

Proof It is satisfied by Theorem 2.1, Lemma 2.3, Theorem 2.6, and Theorem 2.7. □

The expressions are shown in Table 2.

Table 2

Formulas for $f (N)$ with odd N

	f(N)
$\sum_{l + m + n = N} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (n)$	$\frac{1}{64} (σ_{5} (N) + 6 (1 - N) σ_{3} (N) + (8 N^{2} - 12 N + 3) σ (N))$
$\sum_{m = 1}^{N - 1} \hat{σ} (m) \hat{σ} (N - m)$	$\frac{1}{12} (σ_{3} (N) - σ (N))$
$\sum_{m = 1}^{N - 1} \hat{σ} (m) {\tilde{σ}}_{3} (N - m)$	$- \frac{1}{48} (σ_{5} (N) + 2 σ_{3} (N) - 3 σ (N))$
$\sum_{m = 1}^{N - 1} m \hat{σ} (m) \hat{σ} (N - m)$	$\frac{N}{24} (σ_{3} (N) - σ (N))$
$\sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (n)$	$\frac{1}{576} (σ_{5} (N) + (6 N - 2) σ_{3} (N) + (7 - 12 N) σ (N))$
$\sum_{l + m + n = N} \hat{σ} (l) \hat{σ} (m) \hat{σ} (n)$	$\frac{1}{192} (σ_{5} (N) - 2 σ_{3} (N) + σ (N))$

3 Algebraic curves derived from convolution sums

To obtain the result of this section, we need a general theory and it is this that we describe. Suppose that the two polynomials

\begin{matrix} f_{1} (x) = c_{0} x^{n} + \dots + c_{n - 1} x + c_{n}, \\ f_{2} (x) = d_{0} x^{m} + \dots + d_{m - 1} x + d_{m} \end{matrix}

have common zero, say

x_{0}

. Then each of the equations

f_{1} (x) = x f_{1} (x) = \dots = x^{m - 1} f_{1} (x) = 0 = f_{2} (x) = x f_{2} (x) = \dots = x^{n - 1} f_{2} (x)

is of the form

p (x) = 0

, where p is a polynomial of degree at most

m + n - 1

, and as each of these equations is satisfied when

x = x_{0}

, the determinant of the coefficients must vanish. This

(m + n) \times (m + n)

determinant is the resultant

R (f_{1}, f_{2})

of

f_{1}

and

g_{1}

and

R (f_{1}, f_{2}) = | \begin{array}{c} c_{0} & \dots & \dots & c_{n} \\ ⋱ & ⋱ \\ c_{0} & \dots & \dots & c_{n} \\ d_{0} & \dots & \dots & d_{m} \\ ⋱ & ⋱ \\ d_{0} & \dots & \dots & d_{m} \end{array} |,

where the omitted elements are zero, and the diagonal of

R (f_{1}, f_{2})

contains m occurrences of

c_{0}

and n of

d_{m}

[[4], p.206]. We can obtain Table 3 from the results of Section 2.

Table 3

Formulas for $a_{1} \sim a_{12}$

	x = n	x = p
$\sum_{m = 1}^{x - 1} σ (m) σ (x - m)$	$\frac{1}{12} (5 σ_{3} (n) + (1 - 6 n) σ (n))$	$\frac{1}{12} (p - 1) (p + 1) (5 p - 6)$
$\sum_{m = 1}^{x - 1} m σ (m) σ (x - m)$	$\frac{n}{24} (5 σ_{3} (n) + (1 - 6 n) σ (n))$	$\frac{1}{24} p (p - 1) (p + 1) (5 p - 6)$
$\sum_{m = 1}^{x - 1} σ (m) σ_{3} (n - m)$	$\begin{array}{l} \frac{1}{240} (21 σ_{5} (n) + (10 - 30 n) σ_{3} (n) \\ - σ (n)) \end{array}$	$\begin{array}{l} \frac{1}{240} (p - 1) (p + 1) \\ \times (21 p^{3} - 30 p^{2} + 31 p - 30) \end{array}$
$\sum_{m < x / 2} σ (m) σ (x - 2 m)$	$\begin{array}{l} \frac{1}{24} (2 σ_{3} (n) + (1 - 3 n) σ (n) \\ + 8 σ_{3} (n / 2) + (1 - 6 n) σ (n / 2)) \end{array}$	$\frac{1}{24} (p - 1) (p + 1) (2 p - 3)$
$\sum_{m < x / 2} σ_{3} (m) σ (x - 2 m)$	$\begin{array}{l} \frac{1}{240} (σ_{5} (n) - σ (n) + 20 σ_{5} (n / 2) \\ + (10 - 30 n) σ_{3} (n / 2)) \end{array}$	$\frac{1}{240} (p - 1) p (p + 1) (p^{2} + 1)$
$\sum_{m < x / 2} σ (m) σ_{3} (x - 2 m)$	$\begin{array}{l} \frac{1}{240} (5 σ_{5} (n) + (10 - 15 n) σ_{3} (n) \\ + 16 σ_{5} (n / 2) - σ (n / 2)) \end{array}$	$\frac{1}{48} (p - 1) (p + 1) (p^{3} - 3 p^{2} + 3 p - 3)$
$\sum_{l + m + s = x} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (s)$	$\begin{array}{l} \frac{1}{64} {{\tilde{σ}}_{5} (n) + 6 (1 - n) {\tilde{σ}}_{3} (n) \\ + (8 n^{2} - 12 n + 3) \tilde{σ} (n)} \end{array}$	$\frac{1}{64} (p + 1) {(p - 1)}^{2} (p^{2} - 5 p + 10)$
$\sum_{m = 1}^{x - 1} \hat{σ} (m) \hat{σ} (x - m)$	$\frac{1}{12} (σ_{3} (n) + 4 σ_{3} (\frac{n}{2}) - \hat{σ} (n))$	$\frac{1}{12} (p - 1) p (p + 1)$
$\sum_{m = 1}^{x - 1} \hat{σ} (m) {\tilde{σ}}_{3} (x - m)$	$- \frac{1}{48} ({\tilde{σ}}_{5} (n) + 2 {\tilde{σ}}_{3} (n) - 3 \hat{σ} (n))$	$- \frac{1}{48} (p - 1) p (p + 1) (p^{2} + 3)$
$\sum_{m = 1}^{x - 1} m \hat{σ} (m) \hat{σ} (x - m)$	$\frac{n}{24} (σ_{3} (n) + 4 σ_{3} (\frac{n}{2}) - \hat{σ} (n))$	$\frac{1}{24} (p - 1) p^{2} (p + 1)$
$\sum_{l + m + s = x} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (s)$	$\begin{array}{l} \frac{1}{576} {{\tilde{σ}}_{5} (n) + 4 {\tilde{σ}}_{3} (n) + \tilde{σ} (n) \\ - 6 (2 n - 1) \hat{σ} (n) \\ + 6 (n - 1) (σ_{3} (n) + 4 σ_{3} (\frac{n}{2}))} \end{array}$	$\frac{1}{576} {(p - 1)}^{2} {(p + 1)}^{2} (p + 6)$
$\sum_{l + m + s = x} \hat{σ} (l) \hat{σ} (m) \hat{σ} (s)$	$\begin{array}{l} \frac{1}{192} (σ_{5} (n) - 8 σ_{5} (\frac{n}{2}) - 2 σ_{3} (n) \\ - 8 σ_{3} (\frac{n}{2}) + \hat{σ} (n)) \end{array}$	$\frac{1}{192} {(p - 1)}^{2} p {(p + 1)}^{2}$

Corollary 3.1 Let

\begin{matrix} a_{1} (n) : = \sum_{m = 1}^{n - 1} σ (m) σ (n - m), a_{2} (n) : = \sum_{m = 1}^{n - 1} m σ (m) σ (n - m), \\ a_{3} (n) : = \sum_{m = 1}^{n - 1} σ (m) σ_{3} (n - m), a_{4} (n) : = \sum_{m < n / 2} σ (m) σ (n - 2 m), \\ a_{5} (n) : = \sum_{m < n / 2} σ_{3} (m) σ (n - 2 m), a_{6} (n) : = \sum_{m < n / 2} σ (m) σ_{3} (n - 2 m), \\ a_{7} (n) : = \sum_{l + m + s = n} \tilde{σ} (l) \tilde{σ} (m) \tilde{σ} (s), a_{8} (n) : = \sum_{m = 1}^{n - 1} \hat{σ} (m) \hat{σ} (n - m), \\ a_{9} (n) : = \sum_{m = 1}^{n - 1} \hat{σ} (m) {\tilde{σ}}_{3} (n - m), a_{10} (n) : = \sum_{m = 1}^{n - 1} m \hat{σ} (m) \hat{σ} (n - m), \\ a_{11} (n) : = \sum_{l + m + s = n} \hat{σ} (l) \hat{σ} (m) \tilde{σ} (s), a_{12} (n) : = \sum_{l + m + s = n} \hat{σ} (l) \hat{σ} (m) \hat{σ} (s) . \end{matrix}

There exists a polynomial

T (x, y) \in Z [x, y]

such that

T (a_{i} (p), a_{j} (p)) = 0

with

i, j \in {1, 2, \dots, 12}

where p is an odd prime. We abbreviate

a_{1} (n)

to

a_{1}

and it is also applied to the other values. Then we get Table 4.

Table 4

Formulas of $T (x (p), y (p)) = 0$

(x,y)	T(x,y)
$(a_{1}, a_{2})$	$- 3 a_{1}^{3} + 6 a_{1}^{4} + 5 a_{1}^{2} a_{2} + 12 a_{1} a_{2}^{2} - 20 a_{2}^{3}$
$(a_{1}, a_{3})$	$\begin{array}{l} 168 a_{1}^{2} + 162, 358 a_{1}^{3} - 143, 775 a_{1}^{4} + 83, 349 a_{1}^{5} - 750 a_{1} a_{3} - 1, 045, 050 a_{1}^{2} a_{3} \\ + 283, 500 a_{1}^{3} a_{3} + 825 a_{3}^{2} + 2, 221, 875 a_{1} a_{3}^{2} - 1, 562, 500 a_{3}^{3} \end{array}$
$(a_{1}, a_{4})$	$- 5 a_{1}^{2} + 32 a_{1}^{3} + 32 a_{1} a_{4} - 480 a_{1}^{2} a_{4} - 44 a_{4}^{2} + 2, 400 a_{1} a_{4}^{2} - 4, 000 a_{4}^{3}$
$(a_{1}, a_{5})$	$\begin{array}{l} - 61 a_{1}^{2} + 194 a_{1}^{3} - 180 a_{1}^{4} + 72 a_{1}^{5} + 6, 100 a_{1} a_{5} - 28, 000 a_{1}^{2} a_{5} + 18, 000 a_{1}^{3} a_{5} \\ + 67, 100 a_{5}^{2} + 1, 525, 000 a_{1} a_{5}^{2} - 12, 500, 000 a_{5}^{3} \end{array}$
$(a_{1}, a_{6})$	$\begin{array}{l} - 415 a_{1}^{2} + 408 a_{1}^{3} - 360 a_{1}^{4} + 144 a_{1}^{5} + 2, 656 a_{1} a_{6} + 304 a_{1}^{2} a_{6} \\ - 3, 600 a_{1}^{3} a_{6} - 3, 652 a_{6}^{2} + 38, 000 a_{1} a_{6}^{2} - 200, 000 a_{6}^{3} \end{array}$
$(a_{1}, a_{7})$	$\begin{array}{l} 22, 032 a_{1}^{3} + 3, 402 a_{1}^{4} + 243 a_{1}^{5} - 3, 264 a_{1} a_{7} - 217, 152 a_{1}^{2} a_{7} \\ - 32, 400 a_{1}^{3} a_{7} + 5, 984 a_{7}^{2} + 762, 000 a_{1} a_{7}^{2} - 800, 000 a_{7}^{3} \end{array}$
$(a_{1}, a_{8})$	$- a_{1}^{2} + 2 a_{1}^{3} + 10 a_{1} a_{8} - 30 a_{1}^{2} a_{8} + 11 a_{8}^{2} + 150 a_{1} a_{8}^{2} - 250 a_{8}^{3}$
$(a_{1}, a_{9})$	$\begin{array}{l} - 111 a_{1}^{2} + 258 a_{1}^{3} - 81 a_{1}^{4} + 18 a_{1}^{5} - 1, 110 a_{1} a_{9} + 3, 940 a_{1}^{2} a_{9} \\ - 900 a_{1}^{3} a_{9} + 1, 221 a_{9}^{2} + 22, 750 a_{1} a_{9}^{2} + 25, 000 a_{9}^{3} \end{array}$
$(a_{1}, a_{10})$	$\begin{array}{l} 3 a_{1}^{2} - 12 a_{1}^{3} + 12 a_{1}^{4} + 72 a_{1} a_{10} - 94 a_{1}^{2} a_{10} + 132 a_{10}^{2} \\ + 2, 400 a_{1} a_{10}^{2} - 5, 000 a_{10}^{3} \end{array}$
$(a_{1}, a_{11})$	$105 a_{1}^{4} + a_{1}^{5} - 10, 224 a_{1}^{2} a_{11} + 2, 400 a_{1}^{3} a_{11} + 5, 808 a_{11}^{2} + 350, 000 a_{1} a_{11}^{2} - 2, 400, 000 a_{11}^{3}$
$(a_{1}, a_{12})$	$- 9 a_{1}^{4} + 18 a_{1}^{5} - 1, 440 a_{1}^{2} a_{12} + 3, 600 a_{1}^{3} a_{12} + 1, 936 a_{12}^{2} + 124, 000 a_{1} a_{12}^{2} - 1, 600, 000 a_{12}^{3}$
$(a_{2}, a_{3})$	$\begin{array}{l} 10, 080 a_{2}^{2} + 16, 307, 536 a_{2}^{3} - 36, 244, 800 a_{2}^{4} + 28, 005, 264 a_{2}^{5} - 2, 700 a_{2} a_{3} \\ - 23, 500, 980 a_{2}^{2} a_{3} - 37, 184, 400 a_{2}^{3} a_{3} - 12, 375 a_{3}^{2} - 14, 849, 125 a_{2} a_{3}^{2} \\ + 208, 935, 000 a_{2}^{2} a_{3}^{2} + 23, 536, 500 a_{3}^{3} + 22, 500, 000 a_{2} a_{3}^{3} - 187, 500, 000 a_{3}^{4} \end{array}$
$(a_{2}, a_{4})$	$- 15 a_{2}^{2} + 128 a_{2}^{3} - 18 a_{2} a_{4} - 768 a_{2}^{2} a_{4} + 33 a_{4}^{2} - 944 a_{2} a_{4}^{2} + 2, 736 a_{4}^{3} - 24, 000 a_{4}^{4}$
$(a_{2}, a_{5})$	$\begin{array}{l} 61 a_{2}^{3} - 120 a_{2}^{4} + 144 a_{2}^{5} + 7, 320 a_{2}^{2} a_{5} - 18, 000 a_{2}^{3} a_{5} + 236, 375 a_{2} a_{5}^{2} \\ - 165, 000 a_{2}^{2} a_{5}^{2} + 1, 006, 500 a_{5}^{3} + 22, 500, 000 a_{2} a_{5}^{3} - 187, 500, 000 a_{5}^{4} \end{array}$
$(a_{2}, a_{6})$	$\begin{array}{l} - 415 a_{2}^{2} + 1, 344 a_{2}^{3} - 1, 344 a_{2}^{4} + 384 a_{2}^{5} - 498 a_{2} a_{6} + 5, 192 a_{2}^{2} a_{6} - 6, 528 a_{2}^{3} a_{6} \\ + 913 a_{6}^{2} + 21, 816 a_{2} a_{6}^{2} - 68, 800 a_{2}^{2} a_{6}^{2} + 35, 392 a_{6}^{3} - 480, 000 a_{2} a_{6}^{3} - 800, 000 a_{6}^{4} \end{array}$
$(a_{2}, a_{7})$	$\begin{array}{l} 55, 080 a_{2}^{3} + 7, 290 a_{2}^{4} + 243 a_{2}^{5} - 2, 040 a_{2} a_{7} - 121, 140 a_{2}^{2} a_{7} \\ - 16, 956 a_{2}^{3} a_{7} - 1, 870 a_{7}^{2} + 127, 401 a_{2} a_{7}^{2} - 747, 000 a_{2}^{2} a_{7}^{2} \\ + 261, 968 a_{7}^{3} - 2, 160, 000 a_{2} a_{7}^{3} - 1, 600, 000 a_{7}^{4} \end{array}$
$(a_{2}, a_{8})$	$4 a_{2}^{3} + 36 a_{2}^{2} a_{8} + 83 a_{2} a_{8}^{2} + 33 a_{8}^{3} - 750 a_{8}^{4}$
$(a_{2}, a_{9})$	$\begin{array}{l} 444 a_{2}^{3} - 240 a_{2}^{4} + 48 a_{2}^{5} - 4, 440 a_{2}^{2} a_{9} + 2, 880 a_{2}^{3} a_{9} + 11, 877 a_{2} a_{9}^{2} \\ + 5, 800 a_{2}^{2} a_{9}^{2} - 4, 884 a_{9}^{3} - 60, 000 a_{2} a_{9}^{3} - 100, 000 a_{9}^{4} \end{array}$
$(a_{2}, a_{10})$	$\begin{array}{l} 2 a_{2}^{4} + 3 a_{2}^{2} a_{10} - 40 a_{2}^{3} a_{10} - 30 a_{2} a_{10}^{2} + 300 a_{2}^{2} a_{10}^{2} - 33 a_{10}^{3} \\ - 1, 000 a_{2} a_{10}^{3} + 1, 250 a_{10}^{4} \end{array}$
$(a_{2}, a_{11})$	$\begin{array}{l} - 315 a_{2}^{4} + a_{2}^{5} + 7, 668 a_{2}^{2} a_{11} + 30, 228 a_{2}^{3} a_{11} - 1, 089 a_{11}^{2} - 110, 101 a_{2} a_{11}^{2} \\ - 916, 200 a_{2}^{2} a_{11}^{2} + 554, 544 a_{11}^{3} + 10, 800, 000 a_{2} a_{11}^{3} \\ - 43, 200, 000 a_{11}^{4} \end{array}$
$(a_{2}, a_{12})$	$\begin{array}{l} 9 a_{2}^{5} + 360 a_{2}^{3} a_{12} - 121 a_{2} a_{12}^{2} - 30, 600 a_{2}^{2} a_{12}^{2} + 5, 808 a_{12}^{3} \\ + 720, 000 a_{2} a_{12}^{3} - 4, 800, 000 a_{12}^{4} \end{array}$
$(a_{3}, a_{4})$	$\begin{array}{l} - 6, 625 a_{3}^{2} + 64, 000 a_{3}^{3} + 20, 140 a_{3} a_{4} - 612, 000 a_{3}^{2} a_{4} - 11, 872 a_{4}^{2} \\ + 1, 492, 080 a_{3} a_{4}^{2} - 1, 040, 816 a_{4}^{3} - 2, 721, 600 a_{3} a_{4}^{3} \\ + 4, 723, 920 a_{4}^{4} - 10, 668, 672 a_{4}^{5} \end{array}$
$(a_{3}, a_{5})$	$\begin{array}{l} - a_{3}^{2} + 8 a_{3}^{3} - 144 a_{3}^{4} + 1, 152 a_{3}^{5} + 52 a_{3} a_{5} + 56 a_{3}^{2} a_{5} \\ + 4, 896 a_{3}^{3} a_{5} - 120, 960 a_{3}^{4} a_{5} + 224 a_{5}^{2} - 22, 736 a_{3} a_{5}^{2} + 144, 576 a_{3}^{2} a_{5}^{2} \\ + 5, 080, 320 a_{3}^{3} a_{5}^{2} - 181, 328 a_{5}^{3} - 735, 264 a_{3} a_{5}^{3} - 106, 686, 720 a_{3}^{2} a_{5}^{3} \\ + 50, 985, 936 a_{5}^{4} + 1, 120, 210, 560 a_{3} a_{5}^{4} - 4, 704, 884, 352 a_{5}^{5} \end{array}$
$(a_{3}, a_{6})$	$\begin{array}{l} - 341, 375 a_{3}^{2} + 4, 749, 000 a_{3}^{3} - 18, 000, 000 a_{3}^{4} + 7, 200, 000 a_{3}^{5} + 1, 037, 780 a_{3} a_{6} \\ - 21, 435, 400 a_{3}^{2} a_{6} + 133, 848, 000 a_{3}^{3} a_{6} - 151, 200, 000 a_{3}^{4} a_{6} - 611, 744 a_{6}^{2} \\ + 43, 392, 400 a_{3} a_{6}^{2} - 587, 858, 400 a_{3}^{2} a_{6}^{2} + 1, 270, 080, 000 a_{3}^{3} a_{6}^{2} \\ - 43, 483, 216 a_{6}^{3} + 1, 153, 638, 720 a_{3} a_{6}^{3} - 5, 334, 336, 000 a_{3}^{2} a_{6}^{3} \\ - 669, 437, 136 a_{6}^{4} + 11, 202, 105, 600 a_{3} a_{6}^{4} - 9, 409, 768, 704 a_{6}^{5} \end{array}$
$(a_{3}, a_{7})$	$\begin{array}{l} 1, 155, 060, 000 a_{3}^{3} + 104, 034, 375 a_{3}^{4} + 3, 037, 500 a_{3}^{5} - 27, 379, 200 a_{3} a_{7} \\ - 4, 160, 682, 000 a_{3}^{2} a_{7} - 566, 676, 000 a_{3}^{3} a_{7} - 85, 050, 000 a_{3}^{4} a_{7} + 25, 553, 920 a_{7}^{2} \\ + 6, 748, 222, 080 a_{3} a_{7}^{2} - 20, 313, 741, 600 a_{3}^{2} a_{7}^{2} + 952, 560, 000 a_{3}^{3} a_{7}^{2} - 3, 624, 210, 688 a_{7}^{3} \\ + 20, 040, 791, 040 a_{3} a_{7}^{3} - 5, 334, 336, 000 a_{3}^{2} a_{7}^{3} + 176, 788, 224 a_{7}^{4} \\ + 14, 936, 140, 800 a_{3} a_{7}^{4} - 16, 728, 477, 696 a_{7}^{5} \end{array}$
$(a_{3}, a_{8})$	$\begin{array}{l} - 125 a_{3}^{2} + 1, 000 a_{3}^{3} + 650 a_{3} a_{8} - 6, 750 a_{3}^{2} a_{8} + 280 a_{8}^{2} + 5, 820 a_{3} a_{8}^{2} \\ - 5, 978 a_{8}^{3} + 56, 700 a_{3} a_{8}^{3} - 7, 695 a_{8}^{4} - 166, 698 a_{8}^{5} \end{array}$
$(a_{3}, a_{9})$	$\begin{array}{l} - 52, 125 a_{3}^{2} + 360, 750 a_{3}^{3} + 421, 875 a_{3}^{4} + 225, 000 a_{3}^{5} - 271, 050 a_{3} a_{9} \\ + 2, 621, 450 a_{3}^{2} a_{9} + 3, 667, 500 a_{3}^{3} a_{9} + 4, 725, 000 a_{3}^{4} a_{9} + 116, 760 a_{9}^{2} + 4, 007, 030 a_{3} a_{9}^{2} \\ - 780, 750 a_{3}^{2} a_{9}^{2} + 39, 690, 000 a_{3}^{3} a_{9}^{3} + 2, 270, 618 a_{9}^{3} - 43, 218, 900 a_{3} a_{9}^{3} + 166, 698, 000 a_{3}^{2} a_{9}^{3} \\ + 9, 146, 115 a_{9}^{4} + 350, 065, 800 a_{3} a_{9}^{4} + 294, 055, 272 a_{9}^{5} \end{array}$
$(a_{3}, a_{10})$	$\begin{array}{l} 1, 875 a_{3}^{2} - 30, 000 a_{3}^{3} + 120, 000 a_{3}^{4} + 22, 500 a_{3} a_{10} - 311, 950 a_{3}^{2} a_{10} \\ + 1, 440, 000 a_{3}^{3} a_{10} + 16, 800 a_{10}^{2} + 37, 800 a_{3} a_{10}^{2} + 2, 826, 000 a_{3}^{2} a_{10}^{2} \\ - 955, 472 a_{10}^{3} - 2, 613, 600 a_{3} a_{10}^{3} + 12, 946, 608 a_{10}^{4} - 56, 010, 528 a_{10}^{5} \end{array}$
$(a_{3}, a_{11})$	$\begin{array}{l} 5, 315, 625 a_{3}^{4} + 6, 250 a_{3}^{5} - 133, 844, 400 a_{3}^{2} a_{11} + 358, 344, 000 a_{3}^{3} a_{11} \\ - 1, 575, 000 a_{3}^{4} a_{11} + 12, 192, 768 a_{11}^{2} + 1, 542, 181, 760 a_{3} a_{11}^{2} + 607, 773, 600 a_{3}^{2} a_{11}^{2} \\ + 158, 760, 000 a_{3}^{3} a_{11}^{2} - 5, 851, 104, 000 a_{11}^{3} - 41, 822, 645, 760 a_{3} a_{11}^{3} - 8, 001, 504, 000 a_{3}^{2} a_{11}^{3} \\ + 213, 353, 968, 896 a_{11}^{4} + 201, 637, 900, 800 a_{3} a_{11}^{4} - 2, 032, 510, 040, 064 a_{11}^{5} \end{array}$
$(a_{3}, a_{12})$	$\begin{array}{l} - 28, 125 a_{3}^{4} + 225, 000 a_{3}^{5} - 1, 170, 000 a_{3}^{2} a_{12} + 12, 420, 000 a_{3}^{3} a_{12} \\ - 18, 900, 000 a_{3}^{4} a_{12} + 250, 880 a_{12}^{2} + 33, 272, 960 a_{3} a_{12}^{2} - 205, 380, 000 a_{3}^{2} a_{12}^{2} \\ + 635, 040, 000 a_{3}^{3} a_{12}^{2} - 228, 392, 192 a_{12}^{3} + 287, 539, 200 a_{3} a_{12}^{3} - 10, 668, 672, 000 a_{3}^{2} a_{12}^{3} \\ + 5, 816, 344, 320 a_{12}^{4} + 89, 616, 844, 800 a_{3} a_{12}^{4} - 301, 112, 598, 528 a_{12}^{5} \end{array}$
$(a_{4}, a_{5})$	$\begin{array}{l} - 13 a_{4}^{2} + 176 a_{4}^{3} - 720 a_{4}^{4} + 1, 152 a_{4}^{5} + 260 a_{4} a_{5} - 4, 880 a_{4}^{2} a_{5} + 14, 400 a_{4}^{3} a_{5} \\ + 1, 625 a_{5}^{2} + 52, 000 a_{4} a_{5}^{2} - 64, 000 a_{5}^{3} \end{array}$
$(a_{4}, a_{6})$	$\begin{array}{l} - 25 a_{4}^{2} + 168 a_{4}^{3} - 576 a_{4}^{4} + 1, 152 a_{4}^{5} + 50 a_{4} a_{6} - 200 a_{4}^{2} a_{6} - 576 a_{4}^{3} a_{6} \\ - 25 a_{6}^{2} + 544 a_{4} a_{6}^{2} - 512 a_{6}^{3} \end{array}$
$(a_{4}, a_{7})$	$\begin{array}{l} 6, 156 a_{4}^{3} + 5, 589 a_{4}^{4} + 1, 944 a_{4}^{5} - 228 a_{4} a_{7} - 10, 980 a_{4}^{2} a_{7} - 9, 072 a_{4}^{3} a_{7} \\ + 95 a_{7}^{2} + 8, 544 a_{4} a_{7}^{2} - 2, 048 a_{7}^{3} \end{array}$
$(a_{4}, a_{8})$	$- 4 a_{4}^{2} + 32 a_{4}^{3} + 8 a_{4} a_{8} - 96 a_{4}^{2} a_{8} + 5 a_{8}^{2} + 96 a_{4} a_{8}^{2} - 32 a_{8}^{3}$
$(a_{4}, a_{9})$	$\begin{array}{l} - 84 a_{4}^{2} + 816 a_{4}^{3} - 1, 296 a_{4}^{4} + 1, 152 a_{4}^{5} - 168 a_{4} a_{9} + 2, 432 a_{4}^{2} a_{9} \\ - 2, 880 a_{4}^{3} a_{9} + 105 a_{9}^{2} + 2, 848 a_{4} a_{9}^{2} + 512 a_{9}^{3} \end{array}$
$(a_{4}, a_{10})$	$3 a_{4}^{2} - 48 a_{4}^{3} + 192 a_{4}^{4} + 18 a_{4} a_{10} - 112 a_{4}^{2} a_{10} + 15 a_{10}^{2} + 384 a_{4} a_{10}^{2} - 128 a_{10}^{3}$
$(a_{4}, a_{11})$	$175 a_{4}^{4} + 8 a_{4}^{5} - 900 a_{4}^{2} a_{11} + 816 a_{4}^{3} a_{11} + 125 a_{11}^{2} + 5, 920 a_{4} a_{11}^{2} - 6, 144 a_{11}^{3}$
$(a_{4}, a_{12})$	$- 9 a_{4}^{4} + 72 a_{4}^{5} - 72 a_{4}^{2} a_{12} + 720 a_{4}^{3} a_{12} + 25 a_{12}^{2} + 1, 312 a_{4} a_{12}^{2} - 2, 048 a_{12}^{3}$
$(a_{5}, a_{6})$	$\begin{array}{l} - 125 a_{5}^{2} + 24, 000 a_{5}^{3} - 1, 800, 000 a_{5}^{4} + 7, 200, 000 a_{5}^{5} - 20 a_{5} a_{6} + 2, 600 a_{5}^{2} a_{6} \\ - 432, 000 a_{5}^{3} a_{6} - 7, 200, 000 a_{5}^{4} a_{6} + a_{6}^{2} + 200 a_{5} a_{6}^{2} - 158, 400 a_{5}^{2} a_{6}^{2} \\ + 2, 880, 000 a_{5}^{3} a_{6}^{2} - 16 a_{6}^{3} + 4, 320 a_{5} a_{6}^{3} - 576, 000 a_{5}^{2} a_{6}^{3} + 144 a_{6}^{4} \\ + 57, 600 a_{5} a_{6}^{4} - 2, 304 a_{6}^{5} \end{array}$
$(a_{5}, a_{7})$	$\begin{array}{l} 10, 732, 500 a_{5}^{3} + 15, 946, 875 a_{5}^{4} + 6, 075, 000 a_{5}^{5} - 15, 900 a_{5} a_{7} - 4, 972, 500 a_{5}^{2} a_{7} \\ - 10, 827, 000 a_{5}^{3} a_{7} - 8, 100, 000 a_{5}^{4} a_{7} + 265 a_{7}^{2} + 183, 960 a_{5} a_{7}^{2} - 12, 355, 200 a_{5}^{2} a_{7}^{2} \\ + 4, 320, 000 a_{5}^{3} a_{7}^{2} - 2, 016 a_{7}^{3} + 17, 280 a_{5} a_{7}^{3} - 1, 152, 000 a_{5}^{2} a_{7}^{3} + 3, 328 a_{7}^{4} \\ + 153, 600 a_{5} a_{7}^{4} - 8, 192 a_{7}^{5} \end{array}$
$(a_{5}, a_{8})$	$2, 000 a_{5}^{3} - 500 a_{5}^{2} a_{8} + 40 a_{5} a_{8}^{2} - a_{8}^{3} - 36 a_{8}^{5}$
$(a_{5}, a_{9})$	$\begin{array}{l} 1, 500 a_{5}^{3} + 450, 000 a_{5}^{5} + 400 a_{5}^{2} a_{9} + 450, 000 a_{5}^{4} a_{9} + 35 a_{5} a_{9}^{2} + 180, 000 a_{5}^{3} a_{9}^{2} \\ + a_{9}^{3} + 36, 000 a_{5}^{2} a_{9}^{3} + 3, 600 a_{5} a_{9}^{4} + 144 a_{9}^{5} \end{array}$
$(a_{5}, a_{10})$	$60, 000 a_{5}^{4} + 25 a_{5}^{2} a_{10} - 3, 000 a_{5}^{2} a_{10}^{2} - a_{10}^{3} + 24 a_{10}^{4} - 144 a_{10}^{5}$
$(a_{5}, a_{11})$	$\begin{array}{l} 809, 375 a_{5}^{4} + 25, 000 a_{5}^{5} - 11, 100 a_{5}^{2} a_{11} + 561, 000 a_{5}^{3} a_{11} - 300, 000 a_{5}^{4} a_{11} \\ + 37 a_{11}^{2} - 760 a_{5} a_{11}^{2} - 1, 353, 600 a_{5}^{2} a_{11}^{2} + 1, 440, 000 a_{5}^{3} a_{11}^{2} - 5, 280 a_{11}^{3} \\ - 51, 840 a_{5} a_{11}^{3} - 3, 456, 000 a_{5}^{2} a_{11}^{3} + 186, 624 a_{11}^{4} + 4, 147, 200 a_{5} a_{11}^{4} - 1, 990, 656 a_{11}^{5} \end{array}$
$(a_{5}, a_{12})$	$\begin{array}{l} 28, 125 a_{5}^{5} - 112, 500 a_{5}^{4} a_{12} - 5 a_{5} a_{12}^{2} + 180, 000 a_{5}^{3} a_{12}^{2} \\ - 4 a_{12}^{3} - 144, 000 a_{5}^{2} a_{12}^{3} + 57, 600 a_{5} a_{12}^{4} - 9, 216 a_{12}^{5} \end{array}$
$(a_{6}, a_{7})$	$\begin{array}{l} 44, 388 a_{6}^{3} + 12, 879 a_{6}^{4} + 3, 888 a_{6}^{5} - 1, 644 a_{6} a_{7} + 756 a_{6}^{2} a_{7} + 7, 776 a_{6}^{3} a_{7} \\ - 25, 920 a_{6}^{4} a_{7} + 685 a_{7}^{2} - 6, 840 a_{6} a_{7}^{2} - 101, 088 a_{6}^{2} a_{7}^{2} + 69, 120 a_{6}^{3} a_{7}^{2} \\ - 672 a_{7}^{3} + 43, 776 a_{6} a_{7}^{3} - 92, 160 a_{6}^{2} a_{7}^{3} + 61, 440 a_{6} a_{7}^{4} - 16, 384 a_{7}^{5} \end{array}$
$(a_{6}, a_{8})$	$\begin{array}{l} - 4 a_{6}^{2} + 64 a_{6}^{3} + 8 a_{6} a_{8} - 176 a_{6}^{2} a_{8} + 5 a_{8}^{2} + 16 a_{6} a_{8}^{2} + 24 a_{8}^{3} \\ + 432 a_{6} a_{8}^{3} + 72 a_{8}^{4} - 144 a_{8}^{5} \end{array}$
$(a_{6}, a_{9})$	$\begin{array}{l} - 4 a_{6}^{2} + 48 a_{6}^{3} + 240 a_{6}^{4} + 256 a_{6}^{5} - 8 a_{6} a_{9} + 144 a_{6}^{2} a_{9} + 960 a_{6}^{3} a_{9} \\ + 1, 280 a_{6}^{4} a_{9} + 5 a_{9}^{2} + 72 a_{6} a_{9}^{2} + 288 a_{6}^{2} a_{9}^{2} + 2, 560 a_{6}^{3} a_{9}^{2} \\ - 24 a_{9}^{3} - 1, 344 a_{6} a_{9}^{3} + 2, 560 a_{6}^{2} a_{9}^{3} + 384 a_{9}^{4} + 1, 280 a_{6} a_{9}^{4} + 256 a_{9}^{5} \end{array}$
$(a_{6}, a_{10})$	$\begin{array}{l} a_{6}^{2} - 32 a_{6}^{3} + 256 a_{6}^{4} + 6 a_{6} a_{10} - 168 a_{6}^{2} a_{10} + 1, 536 a_{6}^{3} a_{10} + 5 a_{10}^{2} \\ - 88 a_{6} a_{10}^{2} + 2, 624 a_{6}^{2} a_{10}^{2} - 48 a_{10}^{3} + 1, 344 a_{6} a_{10}^{3} + 384 a_{10}^{4} - 384 a_{10}^{5} \end{array}$
$(a_{6}, a_{11})$	$\begin{array}{l} 4, 025 a_{6}^{4} + 16 a_{6}^{5} - 20, 700 a_{6}^{2} a_{11} + 124, 512 a_{6}^{3} a_{11} - 960 a_{6}^{4} a_{11} \\ + 2, 875 a_{11}^{2} - 13, 000 a_{6} a_{11}^{2} + 705, 888 a_{6}^{2} a_{11}^{2} + 23, 040 a_{6}^{3} a_{11}^{2} - 99, 936 a_{11}^{3} \\ + 546, 048 a_{6} a_{11}^{3} - 276, 480 a_{6}^{2} a_{11}^{3} + 1, 327, 104 a_{11}^{4} + 1, 658, 880 a_{6} a_{11}^{4} - 3, 981, 312 a_{11}^{5} \end{array}$
$(a_{6}, a_{12})$	$\begin{array}{l} - 9 a_{6}^{4} + 144 a_{6}^{5} - 72 a_{6}^{2} a_{12} + 1, 440 a_{6}^{3} a_{12} - 2, 880 a_{6}^{4} a_{12} + 25 a_{12}^{2} - 184 a_{6} a_{12}^{2} \\ + 5, 472 a_{6}^{2} a_{12}^{2} + 23, 040 a_{6}^{3} a_{12}^{2} - 1, 632 a_{12}^{3} - 6, 912 a_{6} a_{12}^{3} - 92, 160 a_{6}^{2} a_{12}^{3} \\ + 36, 864 a_{12}^{4} + 184, 320 a_{6} a_{12}^{4} - 147, 456 a_{12}^{5} \end{array}$
$(a_{7}, a_{8})$	$\begin{array}{l} - 80 a_{7}^{2} + 512 a_{7}^{3} + 480 a_{7} a_{8} - 4, 512 a_{7}^{2} a_{8} + 9, 360 a_{7} a_{8}^{2} - 3, 240 a_{8}^{3} \\ + 3, 888 a_{7} a_{8}^{3} - 1, 215 a_{8}^{4} - 486 a_{8}^{5} \end{array}$
$(a_{7}, a_{9})$	$\begin{array}{l} - 2, 480 a_{7}^{2} + 14, 592 a_{7}^{3} + 6, 912 a_{7}^{4} + 8, 192 a_{7}^{5} - 14, 880 a_{7} a_{9} + 139, 392 a_{7}^{2} a_{9} \\ + 129, 024 a_{7}^{3} a_{9} + 30, 720 a_{7}^{4} a_{9} + 378, 000 a_{7} a_{9}^{2} + 505, 440 a_{7}^{2} a_{9}^{2} + 46, 080 a_{7}^{3} a_{9}^{2} \\ + 100, 440 a_{9}^{3} - 147, 744 a_{7} a_{9}^{3} + 34, 560 a_{7}^{2} a_{9}^{3} - 27, 945 a_{9}^{4} + 12, 960 a_{7} a_{9}^{4} + 1, 944 a_{9}^{5} \end{array}$
$(a_{7}, a_{10})$	$\begin{array}{l} 25 a_{7}^{2} - 320 a_{7}^{3} + 1, 024 a_{7}^{4} + 300 a_{7} a_{10} - 2, 010 a_{7}^{2} a_{10} + 9, 216 a_{7}^{3} a_{10} + 17, 460 a_{7} a_{10}^{2} \\ + 22, 320 a_{7}^{2} a_{10}^{2} - 8, 100 a_{10}^{3} + 9, 288 a_{7} a_{10}^{3} - 3, 645 a_{10}^{4} - 486 a_{10}^{5} \end{array}$
$(a_{7}, a_{11})$	$\begin{array}{l} 4, 655 a_{7}^{4} + 16 a_{7}^{5} - 134, 064 a_{7}^{2} a_{11} + 241, 812 a_{7}^{3} a_{11} - 720 a_{7}^{4} a_{11} + 1, 034, 208 a_{7} a_{11}^{2} \\ + 2, 320, 650 a_{7}^{2} a_{11}^{2} + 12, 960 a_{7}^{3} a_{11}^{2} - 1, 994, 544 a_{11}^{3} + 1, 787, 508 a_{7} a_{11}^{3} - 116, 640 a_{7}^{2} a_{11}^{3} \\ - 2, 735, 937 a_{11}^{4} + 524, 880 a_{7} a_{11}^{4} - 944, 784 a_{11}^{5} \end{array}$
$(a_{7}, a_{12})$	$\begin{array}{l} - 5 a_{7}^{4} + 32 a_{7}^{5} - 240 a_{7}^{2} a_{12} + 1, 836 a_{7}^{3} a_{12} - 480 a_{7}^{4} a_{12} + 4, 320 a_{7} a_{12}^{2} \\ + 29, 322 a_{7}^{2} a_{12}^{2} + 2, 880 a_{7}^{3} a_{12}^{2} - 19, 440 a_{12}^{3} + 8, 316 a_{7} a_{12}^{3} - 8, 640 a_{7}^{2} a_{12}^{3} \\ - 23, 085 a_{12}^{4} + 12, 960 a_{7} a_{12}^{4} - 7, 776 a_{12}^{5} \end{array}$
$(a_{8}, a_{9})$	$3 a_{8}^{3} + 9 a_{8}^{5} + 10 a_{8}^{2} a_{9} + 11 a_{8} a_{9}^{2} + 4 a_{9}^{3}$
$(a_{8}, a_{10})$	$6 a_{8}^{4} + a_{8}^{2} a_{10} - 4 a_{10}^{3}$
$(a_{8}, a_{11})$	$35 a_{8}^{4} + 2 a_{8}^{5} - 48 a_{8}^{2} a_{11} + 144 a_{8}^{3} a_{11} + 16 a_{11}^{2} - 32 a_{8} a_{11}^{2} - 1, 536 a_{11}^{3}$
$(a_{8}, a_{12})$	$9 a_{8}^{5} - 16 a_{8} a_{12}^{2} - 256 a_{12}^{3}$
$(a_{9}, a_{10})$	$32 a_{9}^{4} + 3 a_{9}^{2} a_{10} - 104 a_{9}^{2} a_{10}^{2} - 12 a_{10}^{3} + 48 a_{10}^{4} - 48 a_{10}^{5}$
$(a_{9}, a_{11})$	$\begin{array}{l} - 1, 365 a_{9}^{4} + 8 a_{9}^{5} + 1, 872 a_{9}^{2} a_{11} + 7, 584 a_{9}^{3} a_{11} + 480 a_{9}^{4} a_{11} - 624 a_{11}^{2} \\ - 1, 280 a_{9} a_{11}^{2} + 103, 392 a_{9}^{2} a_{11}^{2} + 11, 520 a_{9}^{3} a_{11}^{2} + 66, 816 a_{11}^{3} - 27, 648 a_{9} a_{11}^{3} \\ + 138, 240 a_{9}^{2} a_{11}^{3} - 684, 288 a_{11}^{4} + 829, 440 a_{9} a_{11}^{4} + 1, 990, 656 a_{11}^{5} \end{array}$
$(a_{9}, a_{12})$	$\begin{array}{l} 9 a_{9}^{5} + 180 a_{9}^{4} a_{12} - 16 a_{9} a_{12}^{2} + 1, 440 a_{9}^{3} a_{12}^{2} + 192 a_{12}^{3} \\ + 5, 760 a_{9}^{2} a_{12}^{3} + 11, 520 a_{9} a_{12}^{4} + 9, 216 a_{12}^{5} \end{array}$
$(a_{10}, a_{11})$	$\begin{array}{l} - 105 a_{10}^{4} + 2 a_{10}^{5} + 36 a_{10}^{2} a_{11} + 1, 752 a_{10}^{3} a_{11} - 3 a_{11}^{2} - 290 a_{10} a_{11}^{2} \\ - 10, 512 a_{10}^{2} a_{11}^{2} + 576 a_{11}^{3} + 27, 648 a_{10} a_{11}^{3} - 27, 648 a_{11}^{4} \end{array}$
$(a_{10}, a_{12})$	$9 a_{10}^{5} - a_{10} a_{12}^{2} - 72 a_{10}^{2} a_{12}^{2} - 1, 536 a_{12}^{4}$
$(a_{11}, a_{12})$	$\begin{array}{l} - 81 a_{11}^{4} + 7, 776 a_{11}^{5} + 108 a_{11}^{3} a_{12} - 12, 960 a_{11}^{4} a_{12} + 594 a_{11}^{2} a_{12}^{2} + 8, 640 a_{11}^{3} a_{12}^{2} \\ - 420 a_{11} a_{12}^{3} - 2, 880 a_{11}^{2} a_{12}^{3} - 1, 225 a_{12}^{4} + 480 a_{11} a_{12}^{4} - 32 a_{12}^{5} \end{array}$

Proof We illustrate the proof for the first

T (x (p), y (p)) = 0

in Table 4. In Table 3 we consider

\sum_{m = 1}^{x - 1} σ (m) σ (x - m)

and

\sum_{m = 1}^{x - 1} m σ (m) σ (x - m)

with an odd prime x put by

2 q + 1

. As

\begin{matrix} \sum_{m = 1}^{2 q} σ (m) σ (2 q + 1 - m) = \frac{1}{3} (10 q^{3} + 9 q^{2} - q), \\ \sum_{m = 1}^{2 q} m σ (m) σ (2 q + 1 - m) = \frac{1}{6} (20 q^{4} + 28 q^{3} + 7 q^{2} - q), \end{matrix}

the polynomials

\begin{matrix} f_{1} (X) = 10 X^{3} + 9 X^{2} - X - 3 a_{1} (2 q + 1), \\ f_{2} (X) = 20 X^{4} + 28 X^{3} + 7 X^{2} - X - 6 a_{2} (2 q + 1) \end{matrix}

have common zero, namely,

X = q

. We deduce that for each odd prime p (

= 2 q + 1

),

| \begin{array}{c} 10 & 9 & - 1 & - 3 a_{1} & 0 & 0 & 0 \\ 0 & 10 & 9 & - 1 & - 3 a_{1} & 0 & 0 \\ 0 & 0 & 10 & 9 & - 1 & - 3 a_{1} & 0 \\ 0 & 0 & 0 & 10 & 9 & - 1 & - 3 a_{1} \\ 20 & 28 & 7 & - 1 & - 6 a_{2} & 0 & 0 \\ 0 & 20 & 28 & 7 & - 1 & - 6 a_{2} & 0 \\ 0 & 0 & 20 & 28 & 7 & - 1 & - 6 a_{2} \end{array} | = 0

and this simplifies to give $R (f_{1}, f_{2}) = 108, 000 (- 3 a_{1}^{3} + 6 a_{1}^{4} + 5 a_{1}^{2} a_{2} + 12 a_{1} a_{2}^{2} - 20 a_{2}^{3}) = 0$ , so we can find the irreducible polynomial $T (a_{1}, a_{2}) = - 3 a_{1}^{3} + 6 a_{1}^{4} + 5 a_{1}^{2} a_{2} + 12 a_{1} a_{2}^{2} - 20 a_{2}^{3} = 0$ . Other results in Table 4 can also be obtained by using the resultant. □

Remark 3.2 The plane curves $T (x, y) = 0$ in Corollary 3.1 all have zero-genus since x, y are polynomials of p, which leads to a morphism from the projective line $P^{1}$ to plane curves $T (x, y) = 0$ . 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

Z_{i, j} = T (a_{i}, a_{j})

and

T (a_{i}, a_{j}) = 0

.

Figure 1
$Z_{1, 2}$ **and** $T (a_{1}, a_{2}) = 0$ .

Figure 2
$Z_{1, 3}$ **and** $T (a_{1}, a_{3}) = 0$ .

Figure 3
$Z_{1, 4}$ **and** $T (a_{1}, a_{4}) = 0$ .

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This work was supported by the Kyungnam University Foundation Grant, 2013.

Authors' original submitted files for images

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13660_2012_502_MOESM1_ESM.eps Authors’ original file for figure 1

13660_2012_502_MOESM2_ESM.eps Authors’ original file for figure 2

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13660_2012_502_MOESM4_ESM.png Authors’ original file for figure 4

13660_2012_502_MOESM5_ESM.png Authors’ original file for figure 5

13660_2012_502_MOESM6_ESM.png Authors’ original file for figure 6

Authors’ Affiliations

(1)

National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon, 305-340, Republic of Korea

(2)

Department of Mathematics and Institute of Pure and Applied Mathematics, Chonbuk National University, Chonju, Chonbuk, 561-756, Republic of Korea

(3)

Division of Cultural Education, Kyungnam University, 7(Woryeong-dong) kyungnamdaehak-ro, Masanhappo-gu, Changwon-si, Gyeongsangnam-do, 631-701, Republic of Korea

References

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.Google Scholar
Lahiri DB:On Ramanujan’s function $τ (n)$ and the divisor function $σ (n)$ , I. Bull. Calcutta Math. Soc. 1946, 38: 193–206.MathSciNetGoogle Scholar
Hahn H: Convolution sums of some functions on divisors. Rocky Mt. J. Math. 2007, 37: 1593–1622. 10.1216/rmjm/1194275937View ArticleGoogle Scholar
Beardon AF: Sums of powers of integers. Am. Math. Mon. 1996, 103: 201–213. 10.2307/2975368MathSciNetView ArticleGoogle Scholar
Rosen M Graduate Texts in Mathematics 210. In Number Theory in Function Fields. Springer, Berlin; 2002.View ArticleGoogle Scholar

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