Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of primitive) then Riemann hypothesis holds for the Zeta function?

  • $\begingroup$ I thought the Dirichlet L-function with character $\chi(1)$ is the Zeta function and isn't $\chi(1)$ primitive? $\endgroup$ – Carlo Beenakker Sep 11 '13 at 6:10
  • $\begingroup$ You are right, I should be more precise; I mean the primitive character NOT principal ($\chi(1)$being the only that can be qualified of principal and primitive) $\endgroup$ – Bertrand Sep 11 '13 at 12:25
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    $\begingroup$ Very roughly speaking, RH for zeta is about the inner product of $\Lambda$ with $1$ (where I am being ambiguous as to what precise inner product is intended here), while RH for other Dirichlet L-functions is about the inner product of $\Lambda$ with $\chi$. As the primitive characters $\chi$ are orthogonal to each other, there is no a priori relation between the various hypotheses; if $\Lambda$ was completely arbitrary, then the inner products with different $\chi$ would be completely independent. But of course, $\Lambda$ is NOT completely arbitrary, so this is not conclusive... $\endgroup$ – Terry Tao Sep 11 '13 at 18:15
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    $\begingroup$ EDIT to previous comment: of course, the Bessel inequality (or the large sieve, etc.) does give some relation between all of these versions of RH, namely that there can't be too many flagrant violations of RH across all characters. This gives rise to various "RH on the average" results such as the Bombieri-Vinogradov inequality or zero density estimates on average. But these sort of arguments do not appear strong enough to prohibit one single version of RH failing, and in particular from zeta RH failing even when the rest of GRH is OK. $\endgroup$ – Terry Tao Sep 11 '13 at 21:04
  • $\begingroup$ Thanks it confirms that there is no obvious relation between the two problems: each one requires a demonstration! $\endgroup$ – Bertrand Sep 12 '13 at 6:10

This is more a suggestion how to interpret the question, then a real answer. But I think it's hard to definitely answer your question, because a "good"proof of RH for any one $L$-function will most likely suggest a proof for all automorphic $L$-functions.

So here is a suggestion: Can we express zeta in terms of other primitive $L$-functions.

I don't think so, because he joint universality theorem for Dirichlet $L$-functions implies that there is not functional relation between the Riemann zeta function and any finite set of primitive Dirichlet $L$-functions, e.g,, take any finite number of primite Dirichlet character $\chi_1, \dots, \chi_n$.

For every continuous function $F: \mathbb{R}^{n+1} \rightarrow \mathbb{R}$ with $$ F(\zeta(\delta_0+ it),L(\delta_1+ it, \chi_1), \dots, L(\delta_n+ it, \chi_n)) = 0$$ for all $t \in \mathbb{R}$ and some values $\delta_j \in \{ \Re s \in (1/2,1)\}$ implies $F=0!$

Ref: http://books.google.de/books/about/The_Riemann_Zeta_Function.html?id=fNontpCu9kQC&redir_esc=y

  • $\begingroup$ Thanks, I think comments above and your answer close the subject. $\endgroup$ – Bertrand Sep 12 '13 at 6:11

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