In this talk, we present two topics. One is an elucidation of the Richaud-Degert fundamental units in the real quadratic field $\mathbb{Q}(\sqrt{D})$, where $D=c^2+r$, $c$ is a positive integer and $r=\pm 1, \pm 4$. We reveal the underlying structure in terms of Chebyshev polynomials (standard and of hyperbolic type) in relation to Pisot numbers.

The other is about an asymptotic formula for the convolution of the PNT-related function and the Piltz divisor function. This is a good example of the hyperbola method and reveals the contribution of the error terms $\sqrt{x}\delta(x)$ of PNT and $\sqrt{x}^{\alpha_\kappa+\varepsilon}$, $\alpha_\kappa\le \frac{\kappa-1}{2\kappa}￥of the Piltz divisor problem. These are cowork with R.–Y. Li.