[FOM] (no subject)
kohlenb at brics.dk
Mon Sep 9 06:17:40 EDT 2002
Adam Epstein wrote
> The same person also told me that there is a theorem in number
> theory whose proof goes as follows:
> "If the Riemann Hypothesis then ... while if the Riemann Hypothesis fails
> then ...' "
> Can anyone supply details about this one?
Probably he referred to Littlewood's theorem which states that
the sign of pi(n)-li(n) changes infinitely often, where pi(n)
is the number of primes <=n and li(n) the logarithmic integral.
Littlewood's theorem can be written as a Pi-0-2 statement
Already in the 50's, G. Kreisel indicated the possibility to
extract bounds on the theorem from Littlewood's proof.
For detailed historical discussions see the articles:
"H. Luckhardt: Bounds extracted by Kreisel from ineffective proofs"
and "S. Feferman: Kreisel's `Unwinding' Program" which are both
in "P. Oddifreddi (ed.), Kreiseliana, A.K.Peters. 1996.
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