Note on a theorem of Farkas and Kra
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Note on a theorem of Farkas and Kra Kazuhide Matsuda1 Received: 5 September 2018 / Accepted: 19 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we focus on applications of high-level versions of Jacobi’s derivative formula to number theory, such as quaternary quadratic forms and convolution sums of some arithmetical functions. Keywords Theta function · Theta constant · Rational characteristics · Hecke congruence subgroup Mathematics Subject Classification 14K25 · 11E25
1 Introduction Theta functions and theta constants are powerful tools for the study of function theory. This theory also yields deep applications for number theory and combinatorics. For example, Rademacher [9, pp. 197–208] applied the theory to derive the formula for the number of representations of the positive integer n as the sums of squares. In particular, Farkas and Kra [5, p. 318] treated some modular forms on Hecke congruence subgroup 0 (k) for each odd prime k and evaluated the derivative of eta quotient of weight 0 plus the sum of squares of logarithmic theta derivatives. The aim of this paper is to consider analogues of the result of Farkas and Kra for k = 4, 6, 8 and their applications to number theory, such as quaternary quadratic forms and convolution sums of some arithmetical functions. √ √ For √ this purpose, we first use the Hecke theta functions of Q( −1), Q( −3), and Q( −2), namely,
This work was supported by JSPS KAKENHI Grant Number JP17K14213.
B 1
Kazuhide Matsuda [email protected] Faculty of Fundamental Science, National Institute of Technology, Niihama College, 7-1 Yagumo-chou, Niihama, Ehime 792-8580, Japan
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K. Matsuda
m,n∈Z
qm
2 +n 2
,
m,n∈Z
qm
2 +mn+n 2
, and
qm
2 +2n 2
.
(1.1)
m,n∈Z
We then use the results from our previous papers [6,7], where we considered highlevel versions of Jacobi’s derivative formula. We note that Zemel [10] considered the relationship between the Hecke theta functions (1.1) and derivative formulas. Throughout this paper, following Farkas and Kra, we relate theta functions and eta quotients in weight 2. Lemke Oliver [8] treated the case with weights 1/2 and 3/2. The remainder of this paper is organized as follows. In Sect. 2 we fix notations of some arithmetical functions. In Sect. 3 we review the basic properties of the theta functions. In Sect. 4 we consider derivative formulas of the theta functions. In Sect. 5 we deal with 0 (4) and prove the 4 squares theorem and the 4 triangular numbers theorem. In Sect. 6 we treat 0 (6) and apply the results to additional quadratic forms, some of which are not diagonalizable over Z. In Sect. 7 we deal with 0 (8) and apply the results to quadratic forms and mixed sums of squares and triangular numbers.
2 Notations 2 = {τ ∈ C | τ > Throughout this paper, the upper half-plane is defined by H ∞ 1 (1 − q n ), q = 0}, and the Dedekind eta function is defined by η(τ ) = q 24 n=1
exp(2πiτ ) for τ ∈ H2 . For each positive integer k we define the Hecke congruence s
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