The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that are supposed to verify the analogue of the Riemann Hypothesis?
Thanks in advance.
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asked Aug 22, 2013 at 9:40
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1$\begingroup$ Davenport-Heilbronn= Hurwitz zeta with irrational $q$? $\endgroup$– Marc PalmCommented Aug 22, 2013 at 11:10
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1 Answer
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http://siauliaims.su.lt/pdfai/2008/laurincikas-08.pdf
Abstract. In the paper, the joint universality in the Voronin sense for Hurwitz zeta-functions with parameters $a_1; \dots ; a_r$ such that the system $\{ \log(m + a_j) : m = 0;1;2; \dots ; j = 1; \dots ; r \}$ is linearly independent over the field of rational numbers is obtained.
For irrational $q$, the requirements are true (see Corollary 2).
Interesting side remark:
Quote: If q is algebraic irrational, then J. W. S. Cassels proved that at least 51 percent of elements of the set $\{ log(m + q) : m \in \mathbb{N}_0 \}$ are linearly independent over $\mathbb{Q}$.
answered Aug 22, 2013 at 11:17
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$\begingroup$ +1, You really are an expert on universality! $\endgroup$ Commented Aug 22, 2013 at 13:30
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1$\begingroup$ Thanks for the flowers. To be honest, this did require only googling for "universality of Hurwitz zeta" and some cross-reading though;) $\endgroup$ Commented Aug 22, 2013 at 14:09