From: nikl+sm000493@pchelwig1.mathematik.tu-muenchen.de (Gerhard Niklasch) Newsgroups: sci.math Subject: Riemann Hyp., was Re: what has Thomas Hales just ... Date: 31 Aug 1998 13:51:43 GMT In article <2glno6kxxb.fsf@hera.wku.edu>, Allen Adler writes: |> "Arthur L. Rubin" <216-5888@mcimail.com> states the Riemann hypothesis |> as follows: |> > |> > Pi(x) = integral(2 to x)(1 / ln(x)) dx + O(x^(1/2 + eps)) for any eps > 0. |> |> I don't know much about the Riemann hypothesis, so let me just ask |> one naive question about this. Let Z denote the set of real parts of |> zeroes of the Riemann zeta function. Let E denote the set of all real |> numbers u such that Pi(x) = integral(2 to x)(1 / ln(x)) dx + O(x^u). |> Is it true that the inf of E equals the sup of Z and, if so, can one |> prove it without assuming the Riemann hypothesis? Yes and yes. More precisely, if S = sup(Z), you have Pi(x) - li(x) = O(x^S log x), and it is known that the difference is not O(x^s) for any s < 1/2. The proof of the former fact is based on what is known as `Explicit Formulae', expressing Pi(x) in terms of li(x), some fudge terms, and a sum over x^rho where rho ranges over the nonreal zeros of the Riemann zeta function, ordered by increasing absolute value of the imaginary part (or grouped in complex conjugate pairs) to ensure convergence. (My standard reference for these things is E. Trost's booklet `Primzahlen', but it's in German and I'm not aware of any English translation...) Gerhard -- * Gerhard Niklasch * spam totally unwelcome * http://hasse.mathematik.tu-muenchen.de/~nikl/ ******* all browsers welcome * This .signature now fits into 3 lines and 77 columns * newsreaders welcome ============================================================================== From: gerry@mpce.mq.edu.au (Gerry Myerson) Newsgroups: sci.math Subject: Re: Riemann Hyp., was Re: what has Thomas Hales just ... Date: Tue, 01 Sep 1998 11:00:01 +1100 In article <6se9pf$dbs$2@sparcserver.lrz-muenchen.de>, nikl+sm000494@mathematik.tu-muenchen.de wrote: > (My standard reference for these things is E. Trost's booklet > `Primzahlen', but it's in German and I'm not aware of any English > translation...) I'm sure Allen can handle the German but if others want English they may find the relation between the zeros of zeta and the error term in the Prime Number Theorem is discussed in S J Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Chapter 5, starting in Section 8. As to whether this rephrasing of the Riemann Hypothesis makes it more attractive to amateurs, said amateurs still have to know what an integral is, what a (natural) logarithm is, and what big-oh notation means, just to understand the statement. Not a crushing burden, to be sure, but the entry requirement is still considerably higher than it is for Wiles' Theorem, squaring the circle, trisecting the angle, and other historically favored pursuits. Gerry Myerson (gerry@mpce.mq.edu.au)