Congruences of the Weierstrass $\wp(x)$ and $\wp''(x)(x= \frac{1}{2},\frac{\tau}{2}, \frac{\tau+1}{2})$-functions on divisors

Daeyeoul Kim; Aeran Kim; Hwasin Park

doi:10.4134/BKMS.2013.50.1.241

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2013; 50(1): 241-261

Printed January 31, 2013

https://doi.org/10.4134/BKMS.2013.50.1.241

Congruences of the Weierstrass $\wp(x)$ and $\wp''(x)(x= \frac{1}{2},\frac{\tau}{2}, \frac{\tau+1}{2})$-functions on divisors

Daeyeoul Kim, Aeran Kim, and Hwasin Park

National Institute for Mathematical Sciences, Chonbuk National University, Chonbuk National University

Abstract

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Abstract

In this paper, we find the coefficients for the Weierstrass $\wp(x)$ and $\wp''(x)(x= \frac{1}{2},\frac{\tau}{2}, \frac{\tau+1}{2})$-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan (\cite{S. Ram}). Finally, we reprove congruences for the functions $\mu(n)$ and $\nu(n)$ in Hahn's article \cite[Theorems 6.1 and 6.2]{Hahn}.

Keywords: Weierstrass $\wp(x)$ functions, convolution sums

MSC numbers: 11A67