We characterize nonempty open subsets of the complex plane where the sum $\zeta (s, \alpha )+ {e}^{\pm i\pi s} \hspace{0.167em} \zeta (s, 1- \alpha )$ of Hurwitz zeta functions has no zeros in $s$ for all $0\leq \alpha \leq 1$ . This problem is motivated by the construction of fundamental cardinal splines of complex order $s$ .

In this paper we investigate the zeros of the Estermann zeta function $E(s; k/ \ell , \alpha )= { \mathop{\sum }\nolimits}_{n= 1}^{\infty } {\sigma }_{\alpha } (n) \exp (2\pi ink/ \ell ){n}^{- s} $ as a function of a complex variable $s$, where $k$ and $\ell $ are coprime integers and ${\sigma }_{\alpha } (n)= {\mathop{\sum }\nolimits}_{d\vert n} {d}^{\alpha } $ is the generalized divisor function with a fixed complex number $\alpha $. In particular, we study the question on how the zeros of $E(s; k/ \ell , \alpha )$ depend on the parameters $k/ \ell $ and $\alpha $. It turns out that for some zeros there is a continuous dependency whereas for other zeros there is not.

We have investigated AlxGa1−xN /GaN heterostructures (0<x<0.22) grown by metal organic vapor phase epitaxy on sapphire with photoluminescence (PL), reflexion and cathodo-luminescence experiments. The energetic positions of the free A-exciton in GaN and AlGaN as a function of the alloy compositions are deduced from temperature dependent PL and from reflexion measurements. We obtain a small bowing parameter and no evidence for a Stokes shift between absorption and emission. The AlxGa1−xN films induce additional compressive strain on the underlying GaN film. Compositional inhomogeneities are present, but the fluctuations are too small to be important for carrier localisation. The broadening of the luminescence line width in the alloy can be described by statistical disorder of a random alloy.

We have investigated AlxGal-xN/GaN heterostructures (0<x<0.22) grown by metal organic vapor phase epitaxy on sapphire with photoluminescence (PL), reflexion and cathodo-luminescence experiments. The energetic positions of the free A-exciton in GaN and AIGaN as a function of the alloy compositions are deduced from temperature dependent PL and from reflexion measurements. We obtain a small bowing parameter and no evidence for a Stokes shift between absorption and emission. The AlxGal-xN films induce additional compressive strain on the underlying GaN film. Compositional inhomogeneities are present, but the fluctuations are too small to be important for carrier localisation. The broadening of the luminescence line width in the alloy can be described by statistical disorder of a random alloy.