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What is Riemann zeta function.

  1. Feb 18, 2010 #1
    Could anyone tell me what is the Riemann zeta function. On Wikipedia , the definition has been given for values with real part > 1 , as :
    Sum ( 1 / ( n^-s) ) as n varies from 1 to infinity.

    but what is the definition for other values of s ? It is mentioned that the zeta function is the analytic continuation of the above definition for other values of s.

    But what exactly do we mean by Analytic continuation ?

    Please explain in simple terms as I am not accustomed to mathematical language
     
  2. jcsd
  3. Feb 18, 2010 #2

    CRGreathouse

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    It can't really be explained in layman's terms. It takes complex analysis, usually a fourth-year college course for students in physics or mathematics.
     
  4. Feb 19, 2010 #3
    Well , after looking at a couple of places , I came to know that the analytic continuation is a function that has the same value as the given function within the given functions domain , but is defined at points in a larger superset of the original domain too . Also that the analytic function is unique.

    It comes as a surprise to me that the continuation of a function that is differentiable at every point within the larger domain, is unique !!

    Does this not imply that it is sufficient to define a differentiable function in a very small domain , and from this we can get the value of the function over as large as possible a domain as it can be defined ( assuming its differentiable at each point within this larger domain) .

    Is this why we use the same formula ( that has been derived from the 1st principle for real numbers ) to compute the derivative / integral of a complex function ?

    Can somebody give some insight on to why this continuation is unique ?
     
  5. Feb 19, 2010 #4
    By the way can we not represent the Riemann zeta function over the larger domain with the help of some power series ?
     
  6. Feb 20, 2010 #5
    Well , I think I have understood something wrong , because I can think of many examples of functions that are completely differential in a domain , but there exist more than 1 continuation of that function over a larger domain, which is still differentiable.


    eg. f (x) = 1/ x^3 and f(x) = |1/ x^3 | , both have same value for x > 0 ( and x is real ) , but they are both differentiable continuations in the domain (- infinity , 0 ]
     
  7. Feb 20, 2010 #6
    Analytic continuation is a very restricted form of exanding the definition of a function. It is used to define a function g of the complex number z = x+iy, given a function f on the real axis x. If you had a function f(x) = x^2, the analytic continuation of it would be g(z) = z^2. g(x) = f(x) for x on the real axis, but g is also defined on the rest of the complex plane. So you see that you are not allowed to use any combination of the x and y variables. They must appear in the linear combination z = x+iy. It is this restriction that makes the complex continuation unique.

    So the definition of the Riemann Zeta function on the positive real axis is used to define a unique function on the complex plane.

    Torquil
     
  8. Feb 20, 2010 #7
    Thanks very much for the explanation. But there are some doubts which persist for me . You say that I should use the definition of Riemann function on the positive real axis , and extentd the definition for complex numbers . But the thing is that I was only able to find the definition for numbers greater than 1 on the real number line . How do I extend this definition for numbers less that 1 . ( For numbers < -1 I could use the functional equation , though I have no idea where that came from either) . But in any case what is the value of the Riemann zeta function when the real part of s lies in [-1,1] ??
     
  9. Feb 20, 2010 #8

    CRGreathouse

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    It doesn't matter, you don't need that little part. It's sufficient to define it for, say, all real numbers greater than N for some N.
     
  10. Feb 27, 2010 #9
    Analytic continuation is for a function of a complex variable. So examples in the real line like |1/x^3| say nothing about it.
     
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