A particular consequence of the famous Carleson–Hunt theorem is that the Taylor series expansion of a bounded holomorphic function on the open unit disk converges almost everywhere on the boundary, whereas on single points the convergence may fail. In contrast, Bayart, Konyagin, and Queffélec constructed an example of an ordinary Dirichlet series ∑ann-s\documentclass[12pt]{minimal}
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\begin{document}$$\sum a_n n^{-s}$$\end{document}, which on the open right half-plane [Re>0]\documentclass[12pt]{minimal}
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\begin{document}$$[{{\,\textrm{Re}\,}}>0]$$\end{document} converges pointwise to a bounded holomorphic function—but diverges at each point of the imaginary line, although its limit function extends continuously to the closed right half plane. Inspired by a result of M. Riesz, we study the boundary behavior of holomorphic functions f on the right half-plane which for some ℓ≥0\documentclass[12pt]{minimal}
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\begin{document}$$\ell \ge 0$$\end{document} satisfy the growth condition |f(s)|=O((1+|s|)ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$|f(s)| = O((1 + |s|)^\ell )$$\end{document} and are generated by some Riesz germ, i.e., there is a frequency λ=(λn)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda = (\lambda _n)$$\end{document} and a λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}-Dirichlet series ∑ane-λns\documentclass[12pt]{minimal}
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\begin{document}$$\sum a_n e^{-\lambda _n s}$$\end{document} such that on some open subset of [Re>0]\documentclass[12pt]{minimal}
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\begin{document}$$[{{\,\textrm{Re}\,}}>0]$$\end{document} and for some m≥0\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 0$$\end{document} the function f coincides with the pointwise limit (as x→∞\documentclass[12pt]{minimal}
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\begin{document}$$x \rightarrow \infty $$\end{document}) of so-called (λ,m)\documentclass[12pt]{minimal}
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\begin{document}$$(\lambda ,m)$$\end{document}-Riesz means ∑λn<xane-λns(1-λnx)m,x>0.\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{\lambda _n < x} a_n e^{-\lambda _n s}\big ( 1-\frac{\lambda _n}{x}\big )^m,\,x >0\,.$$\end{document} Our main results present criteria for pointwise and uniform Riesz summability of such functions on the boundary line [Re=0]\documentclass[12pt]{minimal}
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\begin{document}$$[{{\,\textrm{Re}\,}}=0]$$\end{document}, which include conditions that are motivated by classics like the Dini-test or the principle of localization.
