(adsbygoogle = window.adsbygoogle || []).push({}); From a post by Mathwonk in the recent thread about randomness in prime numbers:Barry Mazur's article on "Error Terms" in the current issue of the Bulliten of the AMS (BAMS) uses the "Prime Number Races" that Mathwonk is talking about as a "jumping off point." mathwonk said: ↑OK, here you go, an article in the math monthly, written for anyone.

http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf

The argument there is that IF we had the naively expected degree of randomness of prime distribution, THEN for all large x, the number of primes less than x, and congruent to 1 mod4, should have a 50-50 chance of being greater than the number of such primes congruent to 3 mod4;

but, it isn't true. The primes congruent to 3 mod4, exceed the others for almost all x.

This a good readable article.

In general granville excels at being able to make number theory interesting and understandable to a broad audience, when that is his goal.

Has anybody read Barry Mazur's article in the current issue of the Bulliten of the AMS?

If so, please let me know what you think about it.

Here are my thoughts:

1) Mazur references the excellent Prime Race article quoted by Mathwonk above. This raises the following questions.

A) Did Mathwonk get the reference to the monthly article from Mazur?

B) Did Mazur get the reference from Mathwonk?

C) Is Mathwonk really Barry Mazur?

2) Mazur's article gives a really good introduction to modern number theory for non-number theory specialists like me. For example, he has an elementary introduction to Galois deformations, one of the primary highly specialized techniques that Wiles used to prove FLT.

3) Mazur gives really understandable explanations of the links between

A) Prime Races

B) the (yet to be found) proof of the Riemann hypothesis

C) elliptic modular functions

D) L-series

E) Wiles proof of FLT

4) Mazur makes it seem to me (although he does not say this) that it is highly likely that the same body of knowledge that allowed Wiles to prove FLT will allow some genius to prove the Riemann conjecture in the near future (or, at least in this century).

5) Before reading Mazur's article, I was slowly becoming convincedc that the Langlands program was key in Wiles proof and will be even more key in future breakthroughs. After reading Mazur's article, I feel even more strongly this way.

6) The equivalence of demonstrating suspected magnitudes of various error terms to the Riemann hypothesis sure looks like a promising approach to resolving the Riemann hypothesis.

7) Note that I said "resolving" and not "proving" the RH. Even the experts admit that, at this point in time, it's an even bet. This is not true of the FLT before Wiles proved it. After Ribet's work, everybody was pretty sure it was true. Everybody who followed Ribet's work, that is.

DJ

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# Barry Mazur's current BAMS article

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