# Definition talk:Riemann Zeta Function

## Refactoring: impossible with all those definitions

This approach doesn't work. The equivalence proof itself uses theorems and definitions about $\zeta$ already. In fact it doesn't look like it was ever the intention to do it this way, a page that should've been called Analytic Continuations of Riemann Zeta Function became Equivalence of Definitions of Riemann Zeta Function, and then those formulas were given the status of definitions, but shouldn't have been.

Clean proposal:

• Define $\zeta$ for $\sigma=\Re (s)>1$ only, series + product.
• Construct analytic continuations, some for $\sigma>0$ etc, others for $\sigma>-\infty$. Proofs on separate pages. As many as you want.
• Overview page Analytic Continuations of Riemann Zeta Function, with one-line equivalence proof using Uniqueness of Analytic Continuation.
• Add a section "Analytic continuation" here (like it was long ago), extending the definition $\zeta$ by any (don't specify) of the analytic continuations, at that point already proved to be equivalent, link to overview page.

--barto (talk) (contribs) 17:10, 14 November 2017 (EST)

Agree, this is quite a mess. In my opinion definition 1 should remain here, (in my opinion def. 2 would better serve as a theorem to def. 1) and the others should be extracted as theorems thereof, linked on this page as you suggest. Caliburn (talk) 09:40, 15 November 2017 (EST)
Done. The structure is there, only the proofs are still incomplete (mainly because general complex analysis has not yet been covered enough) --barto (talk) (contribs) 02:53, 16 November 2017 (EST)

What's important is that the link to Sum of Reciprocals of Powers as Euler Product is an also see. --barto (talk) (contribs) 07:27, 5 January 2018 (EST)