[FOM] (no subject)

[Ulrich Kohlenbach]

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. 

Ulrich Kohlenbach