From: "G. A. Edgar" Subject: Re: could the Riemann hypothesis be neither true nor false? Date: Mon, 24 Jan 2000 12:26:23 -0500 Newsgroups: sci.math Summary: [missing] In article , Markus Redeker wrote: > G. A. Edgar wrote: > > If the Riemann zeta function has a zero in the critical strip with > > real part not 1/2, then there is a finite computation that will verify > > that fact. (For example, doing a numerical integration to sufficiently > > high precision around a certain contour surrounding the zero.) > > Can you tell a bit more about that calculation? I always that it is > impossible to prove numerically that there is a zero at a certain point > because there is always an error left. How do you want to circumvent that? > > The trick seems to be that you do a contour integration, only asking > whether there is a zero somewhere inside. But which kind of numeric > integration tells you that with certainity? Suppose there is a zero p of the Riemann zeta function with 1/2 < Re(p) < 1. Then there is a small rectangle surrounding that zero and no other, strictly contained in the strip 1/2 < Re(z) < 1. Then a certain contour integral around that rectangle is nonzero. [It's (2 pi i) times the number of zeros inside... see a complex variables text.] Say the value is b. Now all we need to do is a numerical integration to approximate the value of our integral correct within |b|/2, and see that the result has modulus > |b|/2. If we can do this rigorously, then it will prove the existence of the zero somewhere inside the contour. Numerical integrals can (theoretically at least) be done as accurately as we wish using only rational arithmetic, and the zeta function at the rational points we need can be evaluated as accurately as we wish. So if there is such an anomolous zero, then a computation with rational numbers will verify that fact. -- Gerald A. Edgar edgar@math.ohio-state.edu