  We present a new and simple method for the determination of the pointwise
H\"{o}lder exponent of Riemann's function $\sum_{n=1}^{\infty} n^{-2}\sin(\pi
n^{2} x)$ at every point of the real line. In contrast to earlier approaches,
where wavelet analysis and the theta modular group were needed for the analysis
of irrational points, our method is direct and elementary, being only based on
the following tools from number theory and complex analysis: the evaluation of
quadratic Gauss sums, the Poisson summation formula, and Cauchy's theorem.
