We prove the existence of functions $f\in A\left(𝔻\right)$, the Fourier series of which being universally divergent on countable subsets of $𝕋=\partial 𝔻$. The proof is based on a uniform estimate of the Taylor polynomials of Landau’s extremal functions on $𝕋\setminus \left\{1\right\}$.

We establish upper bounds for certain trigonometric sums involving cosine powers. Part of these results extend previous ones valid for the sum ${\sum }_{m=1}^{k-1}\left|sin(\pi rm/k)\right|/sin(\pi m/k)$. We apply our results to estimate character sums in an explicit and elementary way.