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We construct normal numbers in base q by concatenating q-ary expansions of pseudo-polynomials evaluated at primes. This extends a recent result by Tichy and the author.

The present paper deals with the summatory function of functions acting on the digits of an $q$-ary expansion. In particular let $n$ be a positive integer, then we call $$\begin{array}{c}n=\sum _{r=0}^{\ell }{d}_r\left(n\right)q^r\text{with}{d}_r\left(n\right)\in \{0,...,q-1\}\end{array}$$ its $q$-ary expansion. We call a function $f$ strictly $q$-additive, if for a given value, it acts only on the digits of its representation, i.e., $$f\left(n\right)=\sum _{r=0}^{\ell }f\left(d_r\left(n\right)\right).$$ Let $p\left(x\right)={\alpha }_0{x}^{{\beta }_0}+\cdots +{\alpha }_d{x}^{{\beta }_d}$ with ${\alpha }_0,{\alpha }_1,...,{\alpha }_d,\in ℝ$, ${\alpha }_0\>0$, ${\beta }_0\>\cdots \>{\beta }_d\ge 1$ and at least one ${\beta }_i\notin \mathbb{Z}$. Then we call $p$ a pseudo-polynomial. ...