Riemann Zeta Function

New Orleans LA

In graduate school, I spent a fair amount of time getting familiar^[z] with the Riemann zeta function. Courses in complex analysis and special functions along with reading a couple of pop math books^[e] increased my familiarity with zeta as well as the Riemann hypothesis. In answer to the gentleman’s question in New Orleans, “It is kind of a big deal, in the math world at least.”

Let’s get you familiar with the Riemann zeta function. Here it is as an infinite series.

That is function notation on the left. Instead of f of x, we have ζ of s, where ζ is the Greek letter zeta and s is our independent variable, our input to the function. The big ∑ is the Greek letter sigma. Sigma indicates “sum.” (Also, sigma is an “s,” not an “e.”)

How does this function work? Choose something for s, such as 2:

Remember, sometimes we can add infinitely many terms of a sequence.

It is almost troubling how lovely that is. But, how did that π get in there? (A common comment you make during the holidays?) Euler sorted out an explanation of this in 1735 in his solution to the Basel problem.^[t]

The Riemann zeta function appeared in Riemann’s “On the Number of Primes Less Than a Given Magnitude.” The manuscript of this paper from 1859 (in Riemann’s own hand, scanned, and output to PDF) is available from the Clay Mathematics Institute.

Why would the Clay Mathematics Institute make the manuscript available? They want someone to prove the Riemann hypothesis. This hypothesis is about the location of the zeros of the zeta function,^[a] which seem to be connected to the distribution of the primes. The folks at Clay are offering a million dollar prize for the proof. It is kind of a big deal.

Thanks to Jaclyn for taking photos during our visit to the French Quarter.

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z.↑ While I was at the University of Tennessee Space Institute, Dr. Kupershmidt altered my thinking about understanding. With his guidance, I started focusing on growing more familiar, rather than understanding. This is a good rule to follow when interacting with people, too. Curiosity yields familiarity no matter what it does to cats.

e.↑ Derbyshire’s Prime Obsession is a good one about the Riemann zeta function and the Riemann hypothesis. There are more.

t.↑ The Basel problem is named for Basel, Switzerland, hometown to Euler and the Bernoulli family. (We met the Bernoullis in the Taylor Series post.) The problem was to find the sum of the reciprocals of the perfect squares. In the 1700s, Euler worked with what Riemann would latter denote as ζ(s). Euler’s life and work inspired many, including Riemann. Here is the beginning of the second (translated) paragraph of Riemann’s 1859 paper:

For this investigation my point of departure is provided by the observation of Euler that the product

if one substitutes for p all prime numbers, and for n all whole numbers.

Right out of the gate, zeta is connected with the primes—all the primes. Let s equal 2 and you have the series of the Basel problem.

a.↑ The zeros of a function are the input values that result in an output of zero. The function f(x) = x² has one zero; it’s zero. That is, f(0) = 0. The function f(x) = x² – 4 has two zeros. Do you know what they are? For what values of x does x² – 4 = 0?

Now, as for our friend ζ(s), things are bit more complicated. First, s is not limited to real numbers. The input s can be a complex number of the form s = σ + it, where i is the imaginary unit, the square root of –1. The real part of s, ℜ(s), is σ, and the imaginary part is t. In summation form, ζ(s) only works for σ = ℜ(s) > 1. However, in Riemann’s 1859 paper, he was able to analytically continue, i.e. expand the domain (i.e. extend the possible values of s), to the strip where 0 < ℜ(s) < 1. This work yields a functional equation for ζ(s).

where Γ refers to the gamma function, an extension of factorial. (Remember? 4! = 4·3·2·1) The “sin” in the equation is good old fashioned sine. Now, ζ(s) in this form has trivial zeros for s = –2, –4, –6, … due to the sine function. Zeta also has zeros of the form s = 1/2 + it. These zeros are called the non-trivial zeros. The Riemann hypothesis says that all of the non-trivial zeros of zeta have real part equal to 1/2. It is a hypothesis. It is yet to be proven.

Thanks to CodeCogs for providing an Online LaTeX Equation Editor for turning my LaTeX into GIFs.