in [, Theorem 6]. In addition, Lahiri  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 . Officially, we look into Hanh’s definition of three functions for ,
Three functions (6), (7), and (8) satisfy the following differential equations:
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
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 [, p.206]. We can obtain Table 3 from the results of Section 2.
There exists a polynomialsuch thatwithwherepis an odd prime. We abbreviatetoand it is also applied to the other values. Then we get Table 4.
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
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 ).
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.
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