What Is the Analytic Continuation of the Riemann Zeta Function?

[srijithju]

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

 

[CRGreathouse]

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.

 

[srijithju]

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 ?

 

[srijithju]

By the way can we not represent the Riemann zeta function over the larger domain with the help of some power series ?

 

[srijithju]

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 ]

 

[torquil]

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 ?

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

 

[srijithju]

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] ??

 

[CRGreathouse]

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.

 

[g_edgar]

Analytic continuation is for a function of a complex variable. So examples in the real line like |1/x^3| say nothing about it.