What is the Riemann zeta function used for?

What is the Riemann zeta function used for?

Physics Nobel Prize Winner MIT Prof Frank Wilczek

The Riemann zeta function encodes information about the prime numbers. Moreover, the atoms of arithmetic and critical to modern cryptography for which e-commerce sits on. Finding a proof has been the holy grail of number theory since Riemann first published his hypothesis.


Now, let’s discuss the Riemann-zeta function and its analytic continuation.

The Riemann-zeta function, widely used in statistics and physics, however, most importantly, in analytic number theory. This function provides so much valuable information linking the distribution of prime numbers. However, in order to understand this function beyond some domain, we need to learn about the analytic continuation of this function, which requires implementation of complex analysis.

The tools we learned during the course, such as integral over complex contour. Residue Theorem, Liouville’s Theorem, etc. were essential in understanding how the analytic continuation is performed. And what kind of characters related functions have. 

(a) ζ(s) for Re(s) > 1 (b) ζ(s) after analytic continuation 
Figure 1. Visualization of Riemann-zeta function 
Riemann Zeta Function and Gamma Function 

Since Re(s) > 1, this function converges. However, we want to expand the definition of Riemann zeta function into the whole complex plane. This cannot occur by letting us take any complex value since if Re(s) 1. We know that the series diverges, which means that the function would have no meaning. Thus, we need to use analytic continuation to extend the definition. Before delving closely into analytic continuation, let’s first consider the gamma function Γ(s) and the integral J(s), along with their connection to the Riemann zeta function. 

One of the properties of this gamma function comes from the Euler’s reflection formula. 

Here, C is a Euler-Mascheroni constant. And this form, often called the Weierstrass form of gamma function. This form is important in that it lets us examine the analyticity of gamma function. Since the gamma function is the inverse of the right-hand side, the function has singularity when any of the terms in the product is 0 or s itself is 0. This is equivalent to having 1 + sn = 0 or s = 0, which means that the gamma function has singularity when s is a negative integer or 0. 

Moreover, if some factor of the product is zero, any other factors will be nonzero. Thus, we have simple poles at s = 0, −1, −2, · · ·. It is found that gamma function is actually analytic everywhere else except for s ∈ {0, −1, −2, · · ·}

This gamma function itself has several other interesting properties.

So we can write it simply as ζ(s) = Γ(1 − s)I(s).

We can also find that I(s) is analytic by using L-M estimates in showing the uniform convergence over the contour C.   

Although the relationship ζ(s) = Γ(1 − s)I(s) established for Re(s) > 1. The components constituting the Riemann zeta function defined for any complex number s (Γ(s) defined for any complex number in Weierstrass form). Therefore, we extend the definition of Riemann zeta function to the whole complex plane by defining ζ(s) = Γ(1 − s)I(s) for Re(s) 1. Thus, because we know that ζ(s) = Γ(1 − s)I(s) holds for Re(s) > 1, ζ(s) = Γ(1 − s)I(s) now holds for all complex numbers. 

The Riemann zeta function over the whole complex plane is actually analytic for all s except for possibly at simple pole s = 1. This is because I(s) is analytic for all s, and Γ(1 − s) is analytic except for 1 − s = 0, −1, −2, · · ·. These points are s = 1, 2, 3, · · ·. Thus, possible points where ζ(s) is not analytic would be the points s = 1, 2, 3, · · ·. However, we know that ζ(s) is defined as Pn=11nsfor Re(s) > 1. Which converges uniformly for Re(s) > 1. Thus, ζ(s) is actually analytic for s = 2, 3, 4, · · ·. Therefore, ζ(s) is analytic everywhere except for possibly at s = 1. 

We will use Cauchy’s Residue Theorem and L’hospital’s rule.

Putting things together, we have extended the Riemann zeta function into the whole complex plane and showed that the function defined is now analytic at all complex planes except at simple poles = 1 with residue 1. This shows how analytic continuation is so powerful in extending the domain for a function with restricted domain. 

Some other studies of this Riemann zeta function is on finding the roots for ζ(s). There are trivial roots s = 2n (n ∈ N), but the more interesting roots are the ones that Riemann Hypothesized: the non-trivial root of ζ(s) has Re(s) = 12. This hypothesis still needs to be proven. And is one of the Millennium Prize Problems by the Clay Mathematics Institute. This Riemann zeta function is important in the distribution of prime numbers as it has close relationship with prime distribution function Chebyshev’s function. 

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What is the Riemann zeta function used for? References 

1. Apostol, T., Introduction To Analytic Number Theory. Undergraduate Texts in Mathematics. Springer, 2013, pp.249-265. 

2. Whittaker, E. T., and G. N. Watson. A Course of Modern Analysis. Cambridge University Press, 1946, pp.265-272.