From: bumby@lagrange.rutgers.edu (Richard Bumby)
Newsgroups: sci.math
Subject: Re: Number theory question
Date: 10 Apr 1998 19:13:39 -0400

"Dan Kucerovsky" <kucerov@uwovax.uwo.ca> writes:

>Hi,

>I am trying to remember something that I think I saw a few years ago. If I
>remember right, the pi function (number of primes less than n) can be
>approximated by an infinite sum over the complex zeros of the zeta
>function, and the approximation can be derived in a fairly elementary way,
>using the Euler-Maclaurin series. Is this correct, and if yes, where do I
>find it? 
>I'm not a number theorist, but a I have a student who wants to know
>something about the Riemann hypothesis.
>Thanks,
>Dan Kucerovsky
>dkucerov@fields.utoronto.ca

I have something like that in front of me right now.  Chapter 17 of
Davenport's "Multiplicative Number Theory" (I have the 1967 Markham
edition, but I'm sure it has been reprinted) is devoted to such an
"explicit formula".  You get a more precise formula by counting the
prime $p$ with the weight log $p$, and even better results by counting
prime-powers $p^n$ with weight log $p$.  This is the way it is done in
this reference,but converting such results to information about the
number of primes is straightforward.

-- 
R. T. Bumby **  Rutgers Math || Amer. Math. Monthly Problems Editor 1992--1996
bumby@math.rutgers.edu       ||   
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