What Is the Analytic Continuation of the Riemann Zeta Function?

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Discussion Overview

The discussion centers on the Riemann zeta function and its analytic continuation, exploring its definition beyond the region where the real part of s is greater than 1. Participants seek clarification on the concept of analytic continuation and its implications for defining functions in complex analysis.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants inquire about the definition of the Riemann zeta function for values of s outside the region where the real part is greater than 1, referencing its analytic continuation.
  • There is a discussion on the nature of analytic continuation, with some participants suggesting it allows for a function to be defined on a larger domain while maintaining the same values as the original function within its initial domain.
  • One participant expresses surprise at the uniqueness of analytic continuation, questioning whether it is sufficient to define a differentiable function in a small domain to extend it to a larger one.
  • Another participant proposes that power series might be used to represent the Riemann zeta function over a larger domain.
  • Concerns are raised about the existence of multiple differentiable continuations for certain functions, challenging the uniqueness aspect of analytic continuation.
  • Some participants clarify that analytic continuation is a restricted form of extending a function, emphasizing the importance of the complex variable representation.
  • There is a request for clarification on how to extend the definition of the Riemann zeta function for values less than 1, with mention of using a functional equation for values less than -1.
  • One participant suggests that defining the zeta function for all real numbers greater than a certain threshold may be sufficient.
  • A participant notes that examples from the real line, such as |1/x^3|, do not directly apply to the concept of analytic continuation.

Areas of Agreement / Disagreement

Participants express varying levels of understanding regarding analytic continuation, with some agreeing on its uniqueness while others challenge this notion. The discussion remains unresolved regarding the specific values of the Riemann zeta function for real parts of s in the range [-1, 1].

Contextual Notes

Limitations include the lack of consensus on the uniqueness of analytic continuation and the specific methods for extending the Riemann zeta function's definition. Participants also express uncertainty about the functional equation and its application.

srijithju
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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
 
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srijithju said:
But what exactly do we mean by Analytic continuation ?

Please explain in simple terms as I am not accustomed to mathematical language

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.
 
CRGreathouse said:
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.

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 ?
 
By the way can we not represent the Riemann zeta function over the larger domain with the help of some power series ?
 
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 ]
 
srijithju said:
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 !
...
Can somebody give some insight on to why this continuation is unique ?

srijithju said:
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 ]

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
 
torquil said:
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

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] ??
 
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.
 
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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