Riemann's Fragment on Limit Values of Elliptic Modular Functions

Abstract

We give an interpretation different from that of Dedekind to work n° 28 of the complete works of Riemann ‘Fragmente über die Grenzfälle der Elliptischen Modulfunctionen’.

We prove a theorem of inversion of radial limit and sum in a series of functions. This allows us to justify all of Riemann's reasoning in the fragment to obtain the limit values of modular elliptic functions. In particular we prove the statement of Riemann that for every rational number x we have

$\sum\limits_{n = 1}^\infty {\frac{{\varphi \left( {nx} \right)}}{n}} = \sum\limits_{t = 1}^\infty {\left( {\sum\limits_\theta { - ( - 1)^\theta } } \right)\frac{{\sin 2\pi t x}}{{\pi t}},}$

where ϕ denotes the periodic function with period 1, such that ϕ(x) = x when |x| < 1/2, and ϕ(n + $\frac{1}{2}$) = 0 for every n ∈ Z.

This assertion of Riemann was criticized by Dedekind. We also give the transformation formulae of the logarithms of the classical theta-function ϑ₃(0), giving an alternative form to that obtained by B. C. Berndt [1].