Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Zeros of the Zeta Function


Date: 08/09/99 at 15:36:04
From: Xi Wang
Subject: Zeros of the Zeta Function

Hello Dr. Math,

I recently read in a book that the only real zeros to the zeta 
function are -2, -4, -6, -8, etc. However, since:

     zeta(x) = 1 + 1/2^x + 1/3^x + 1/4^x + ...

How can any of the zeros work? It doesn't matter which one of them you 
plug in, the answer is obviously not zero.

Thanks in advance for your help, 
Xi Wang


Date: 08/09/99 at 16:10:33
From: Doctor Tom
Subject: Re: Zeros of the Zeta Function

To understand what's going on, you have to understand enough of 
complex analysis to know what "analytic continuation" means.

Unlike functions of a real variable, differentiable functions of a 
complex variable are not flexible - once you know the values of the 
function on any infinite set of points, that function can be extended 
in a unique way to a certain function.

The expression for zeta(x) that you gave above converges for positive 
x, so it does define an infinite set of values of the function. This 
function defined on the positive reals can be extended in a unique way 
to most of the complex plane, and that extended function has a set of 
zeroes outside the range of convergence of the series you gave above.

Here's an example that's very easy to understand. You've probably 
learned that:

     f(x) = 1 + x + x^2 + x^3 + ... = 1/(1-x)

But if x is greater than 1 (say, for example, x = 2), the series above 
doesn't converge, it gives:

     f(2) = 1 + 2 + 4 + 8 + ...

But the expression 1/(1-x) makes perfect sense for x = 2. The function 
1/(1-x) is the analytic continuation of the series:

     1 + x + x^2 + x^3 + ...

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
College Analysis

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/