Functional equations as global formulas

[Tendex]

TL;DR Summary

I've been recently rereading the first pages of Edwards' book on Riemann's zeta function and would like to explore its ideas about global formula or functional equation as opposed to the more usual Weierstrass' local way of continuing functions.

In page 9 it says: "It is interesting to note that Riemann does not speak of the “analytic continuation” of the function beyond the halfplane Re s > 1, but
speaks rather of finding a formula for it which “remains valid for all s.” [...]. The view of analytic continuation in terms of chains of disks and power series convergent in each disk descends from Weierstrass and is quite antithetical to Riemann’s basic philosophy that analytic functions should be dealt with globally, not locally in terms of power series."

So does this suggest that the power series representation of the Riemann's zeta function is of little or no use for its investigation?

Also, when I think about circles of convergence overlapping as they extend a path it is inmediate to visualize this in the complex plane but when using the global functional equation formula and given the rigidity that analiticity demands the setting is in 4 real dimensions, that is the conditions for the continuation to exist being global and rigid it determines the whole graph of the function in 4 dimensions.
Ultimately both the local and global views must coincide of course, but it hints that for certain functions like the zeta function only the global formula way is feasible for investigating them. For others, like for rational functions, with polynomial denominators, the local way seems more natural.

 

[fresh_42]

First of all forget Riemann. The power series $\sum \dfrac{1}{n^s}$ have been studied prior to Riemann by Cotes, de Moivre and Euler - $\zeta(2)$. And its origin is indeed in the realm of power series as representation of the elementary functions $\exp,\log,\sin,\cos$ and later on Bessel to solve differential equations. They regarded those power series - phrased in modern terms - as algebraic elements of the algebra of formal power series rather than functions. The radius of convergence wasn't introduced prior to Cauchy. Next stops have been at Legendre, Dirichlet, Chebyshev, and Weierstraß's preparation theorem ($1860$), which you probably refer to. Most goals, including Weierstraß's, was to calculate, to find algorithms! So the functional character of those elementary functions wasn't at the center of investigation. The major problem without computers is to actually compute!

In $1859$, Bernhard Riemann decisively elaborated the relationship between the zeta function and the prime numbers already given by Euler in his publication On the Number of Prime Numbers Under a Given Size. The major achievement was recognizing the relevance of extending the domain to complex numbers. It was only with this approach that it became possible to obtain specific information about prime numbers $2, 3, 5, 7 ...$. This is remarkable in that prime numbers are real numbers. Riemann, who was a student of Carl Friedrich Gauß, wrote in his ten-page work a function-theoretical interpretation and evaluation of the Euler product, which created a connection between prime numbers and the non-trivial zeros of the Zeta function.

So, again, there is less a fundamental difference in concepts - analytical continuation versus locally convergent power series - as it is about an extension of methods!

And this leads to the answer of your question:
"Is only the globale character of a function feasible to investigate it?"

Let's learn from the mathematicians mentioned above:
Choose your screwdriver according to the screw, not according to your personal taste.

 

[Tendex]

First of all forget Riemann. The power series $\sum \dfrac{1}{n^s}$ have been studied prior to Riemann by Cotes, de Moivre and Euler - $\zeta(2)$. And its origin is indeed in the realm of power series as representation of the elementary functions $\exp,\log,\sin,\cos$ and later on Bessel to solve differential equations. They regarded those power series - phrased in modern terms - as algebraic elements of the algebra of formal power series rather than functions. The radius of convergence wasn't introduced prior to Cauchy. Next stops have been at Legendre, Dirichlet, Chebyshev, and Weierstraß's preparation theorem ($1860$), which you probably refer to. Most goals, including Weierstraß's, was to calculate, to find algorithms! So the functional character of those elementary functions wasn't at the center of investigation. The major problem without computers is to actually compute!

So, again, there is less a fundamental difference in concepts - analytical continuation versus locally convergent power series - as it is about an extension of methods!

And this leads to the answer of your question:
"Is only the globale character of a function feasible to investigate it?"

Let's learn from the mathematicians mentioned above:
Choose your screwdriver according to the screw, not according to your personal taste.

I see, so I interpret that in this extension case the screw calls for a functional equation that sets conditions on the four dimensions in which the graph of the function lives rather than for a local continuation method in its investigation, right?

 

[fresh_42]

I see, so I interpret that in this extension case the screw calls for a functional equation that sets conditions on the four dimensions in which the graph of the function lives rather than for a local continuation method in its investigation, right?

No, I just wanted to say, whether a local or a global view is better depends on the goal, not definition or taste. If someone could show by the use of power series - however such a series may look like, that $\zeta(s)\neq 0$ on any open neighborhood at any point $s=0.5 + iy$, who would complain about it? I just mentioned that history was driven by computational needs and thus the power series. The analytical point of view came late.

And power series are still of great importance when it comes to computations and experiments. The question which one is feasible is a question which should read: Which one will lead to the goal?

 

[Tendex]

No, I just wanted to say, whether a local or a global view is better depends on the goal, not definition or taste. If someone could show by the use of power series - however such a series may look like, that $\zeta(s)\neq 0$ on any open neighborhood at any point $s=0.5 + iy$, who would complain about it? I just mentioned that history was driven by computational needs and thus the power series. The analytical point of view came late.

And power series are still of great importance when it comes to computations and experiments. The question which one is feasible is a question which should read: Which one will lead to the goal?

Ok, remember the context is the meromorphic extension of the Riemann's zeta, not just the Euler zeta function only defined for s>1 you seemed to think of at the beguinning of your first answer. And this is a quote from Laugwitz, a mathematician and math history expert from his technical biography of Riemann:

"Restriction of complex analysis to power series has conceptual as well as
practical consequences. Important functions did not even enter the field of
vision of Weierstrass and his students. One example is the zeta function. It
has a pole at z = 1 and an essential singularity at z = infinity and is holomorphic
everywhere else. This means that it has a series representation $f (z)=\frac{1}{(z-1)}+\displaystyle\sum_{0}^{\infty} c_n(z-1)$
that is everywhere uniformly convergent. But there is not a word about any
of this in the whole comprehensive theory of the zeta function. Indeed, this
representation is of no use."

How am I to understand the remark about the uselessness of the power series representation here?

 

[Tendex]

The power series $\sum \dfrac{1}{n^s}$

You certainly meant the Dirichlet series.

 

[martinbn]

...
So does this suggest that the power series representation of the Riemann's zeta function is of little or no use for its investigation?
...

Which power series representation do you have in mind?

 

[Tendex]

Which power series representation do you have in mind?

See post #5

 

[Tendex]

I'm interested in what about the Riemann zeta extension makes the local approach of no use. Is it maybe that it's not a rational function? Maybe @mathwonk might be interested in chiming in.

 

[martinbn]

See post #5

That is a consequence of the analytic extension. You don't know it before you have proven it. My question is: given the zeta function as a Dirichlet series, which power series expansion do you have in mind?

 

[Infrared]

This means that it has a series representation $f (z)=\frac{1}{(z-1)}+\displaystyle\sum_{0}^{\infty} c_n(z-1)$
that is everywhere uniformly convergent. But there is not a word about any
of this in the whole comprehensive theory of the zeta function. Indeed, this
representation is of no use.

This is a strong statement! It's not the most well-known identity, but some searching finds the Laurent expansion $\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\gamma_n(s-1)^n$ where $\gamma_n=\lim_{m\to\infty}\left(\sum_{k=1}^m\frac{\log(k)^n}{k}-\frac{\log(m)^{n+1}}{n+1}\right)$. See http://mathworld.wolfram.com/StieltjesConstants.html

I'm also not sure what you mean by "everywhere uniformly convergent". Do you just mean uniform convergence on compact sets?

 

[Tendex]

That is a consequence of the analytic extension. You don't know it before you have proven it. My question is: given the zeta function as a Dirichlet series, which power series expansion do you have in mind?

I only had in mind the extended function, as I said I was reading Edwards book.

 

[Tendex]

This is a strong statement! It's not the most well-known identity, but some searching finds the Laurent expansion $\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}\gamma_n(s-1)^n$ where $\gamma_n=\lim_{m\to\infty}\left(\sum_{k=1}^m\frac{\log(k)^n}{k}-\frac{\log(m)^{n+1}}{n+1}\right)$. See http://mathworld.wolfram.com/StieltjesConstants.html

I'm also not sure what you mean by "everywhere uniformly convergent". Do you just mean uniform convergence on compact sets?

Those are not my statements, are Laugwitz's.

 

[martinbn]

I only had in mind the extended function, as I said I was reading Edwards book.

The quote from the book talks about the process of analytically extending the function. That is how I understand your question. It contrasts the Weirstrass idea of extending a function using chains of discs on which the function is given by a converging power series. That is not what Riemann does. He uses the integral representations and the transformation properties of the theta functions.

 

[mathwonk]

I am a novice on this topic, but I did enjoy reading cursorily Riemann's paper when I reviewed the Kendrick Press translation of his works for Math Reviews some 15 years ago. Here is what I said, abbreviated for publication:

 

[Tendex]

I am a novice on this topic, but I did enjoy reading cursorily Riemann's paper when I reviewed the Kendrick Press translation of his works for Math Reviews some 15 years ago. Here is what I said, abbreviated for publication:

View attachment 257256

Thanks for that. It helps me to understand the quote by Edwards. I guess it also helps with the strong statement by Laugwitz when having in mind the goal of Riemann of connecting the primes distribution with global properties of meromorphic functions through the Euler product formula, and the power series of local paths extensions seem to have no use here.
In this sense would a meromorphic rational function, whose singularity at infinity would not be an essential one, serve the same goal as a global object?

 

[zinq]

What Riemann understood was that whenever a power series in the complex plane has a positive radius of convergence, then it belongs to a "global analytic function" with a well-defined maximal domain.

That domain does not need to be a subset of the plane! The simplest example is the square root function f(z) = √z. It has a nice convergent power series in a neighborhood of z = 1, taking r exp(iθ) to √r exp(iθ/2). But as this definition is continued around the origin, by the time it gets back to where it started it disagrees with itself. But going twice around the origin it will once again agree with the original definition.

This leads to the "Riemann surface" of this function as the double cover of the punctured plane (punctured at z = 0). Any convergent power series (meaning: it must have a positive radius of convergence) has a uniquely defined maximal domain, its Riemann surface.

So while the choice of the center of a power series for an analytic (also called holomorphic) function is an arbitrary choice, the Riemann surface is a well-defined characteristic of the function.

P.S. A point like the z = 0 for f(z) = √z is called a branch point. If going around a branch point a finite number of times returns the function to its original value (without approaching infinity at the branch point) — as does the function f(z) = z^1/n for any positive integer n — then the Riemann surface of the function can be completed by filling in the branch point by gluing a small disk D over it via the map ξ → ξⁿ.

 

[benorin]

This is what this conversation made me think of: the global analytic continuation of the Riemann zeta function.
Ground work: define a Dirichlet series as $\xi (s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ where $a_n , s\in\mathbb{C}$; if $a_{n}$ is multiplicative, namely if $\forall n,m\in\mathbb{Z}^+, a_{nm}=a_n a_m$, then

$$\xi (s) + \sum_{n=1}^{\infty}(-1)^n\frac{a_n}{n^s}=\sum_{n=1}^{\infty}\frac{a_n}{n^s}+\sum_{n=1}^{\infty}(-1)^n\frac{a_n}{n^s} =2\sum_{n=1}^{\infty}\frac{a_{2n}}{(2n)^s}=2^{1-s}a_{2}\xi (s)$$

Let us denote the series thus obtained by analytic continuation by

$$\hat {\xi} (s) = (1-2^{1-s}a_2)^{-1}\sum_{n=1}^{\infty}(-1)^{n-1}\frac{a_n}{n^s}$$

which is analogous to the typical analytic continuation of the Riemann zeta function to the half-plane $\Re \left[ s\right] >0$.
Let us suppose that the resulting alternating series is convergent. Define the Euler's series transformation by the equality I shall not here prove:

$$\sum_{n=1}^{\infty}(-1)^{n-1}b_n = \sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{m=0}^{n}(-1)^m \cdot \, _n C_m b_{m+1}$$

where $_n C_m$ is a binomial coefficient. Now apply the Euler's series transformation to $\hat {\xi } (s)$ to obtain

$$\xi (s) = (1-2^{1-s}a_2 )^{-1}\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{m=0}^{n}(-1)^{m}\cdot \, _n C_m \frac{ a_{m+1}}{(m+1)^s}$$

which at once gives the global analytic continuation of the Riemann zeta function, namely

$$\forall s\in \mathbb{C}-\left\{ 1\right\} , \, \zeta (s) = (1-2^{1-s} )^{-1}\sum_{n=0}^{\infty}\frac{1}{2^{n+1}}\sum_{m=0}^{n}(-1)^{m}\cdot \, _n C_m (m+1)^{-s}$$

Note: sorry couldn't remember the code for the pretty binomial coefficient.