The Bernoulli numbers are a sequence of signed rational numbers that can be defined by the exponential generating function
These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis.
The Bernoulli number ￼ can be defined by the contour integral
where the contour encloses the origin, has radius less than (to avoid the poles at ), and is traversed in a counterclockwise direction.
The numbers of digits in the numerator of for the , 4, … are 1, 1, 1, 1, 1, 3, 1, 4, 5, 6, 6, 9, 7, 11, … , while the numbers of digits in the corresponding denominators are 1, 2, 2, 2, 2, 4, 1, 3, 3, 3, 3, 4, 1, 3, 5, 3, …. Both of these are plotted above.
The denominator of is given by
where the product is taken over the primes , a result which is related to the von Staudt-Clausen theorem.
In 1859 Riemann published a paper giving an explicit formula for the number of primes up to any preassigned limit—a decided improvement over the approximate value given by the prime number theorem. However, Riemann’s formula depended on knowing the values at which a generalized version of the zeta function equals zero. (The Riemann zeta function is defined for all complex numbers—numbers of the form x + iy, where i = √(−1), except for the line x = 1.) Riemann knew that the function equals zero for all negative even integers −2, −4, −6, … (so-called trivial zeros), and that it has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1, and he also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1/2. Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis. In 1900 the German mathematician David Hilbert called the Riemann hypothesis one of the most important questions in all of mathematics, as indicated by its inclusion in his influential list of 23 unsolved problems with which he challenged 20th-century mathematicians. In 1915 the English mathematician Godfrey Hardy proved that an infinite number of zeros occur on the critical line, and by 1986 the first 1,500,000,001 nontrivial zeros were all shown to be on the critical line. Although the hypothesis may yet turn out to be false, investigations of this difficult problem have enriched the understanding of complex numbers.
Suppose you want to put a probability distribution on the natural numbers for the purpose of doing number theory. What properties might you want such a distribution to have? Well, if you’re doing number theory then you want to think of the prime numbers as acting “independently”: knowing that a number is divisible by p should give you no information about whether it’s divisible by q.
That quickly leads you to the following realization: you should choose the exponent of each prime in the prime factorization independently. So how should you choose these? It turns out that the probability distribution on the non-negative integers with maximum entropy and a given mean is a geometric distribution. So let’s take the probability that the exponent of p is k to be equal to (1−rp)rpk for some constant rp.
This gives the probability that a positive integer n = p1e1…pkek occurs as
C ∏ki=1 rpei
C = ∏p(1-rp)
So we need to choose rp such that this product converges. Now, we’d like the probability that n occurs to be monotonically decreasing as a function of n. It turns out that this is true iff rp = p−s for some s > 1 (since C has to converge), which gives the probability that n occurs as
ζ(s) is the zeta function.
The Riemann-Zeta function is a complex function that tells us many things about the theory of numbers. Its mystery is increased by the fact it has no closed form – i.e. it can’t be expressed a single formula that contains other standard (elementary) functions.
The plot above shows the “ridges” of ￼ for ￼ 0 < x < 1 and 0 < y < 100￼. The fact that the ridges appear to decrease monotonically for ￼0 ≤ x ≤ 1/2 is not a coincidence since it turns out that monotonic decrease implies the Riemann hypothesis.
On the real line with , the Riemann-Zeta function can be defined by the integral
The Riemann zeta function can also be defined in the complex plane by the contour integral
∀ ￼, where the contour is illustrated below
Zeros of ￼ come in (at least) two different types. So-called “trivial zeros” occur at all negative even integers ￼, ￼, ￼, …, and “nontrivial zeros” at certain
for in the “critical strip” . The Riemann hypothesis asserts that the nontrivial Riemann zeta function zeros of all have real part , a line called the “critical line.” This is now known to be true for the first roots.
The plot above shows the real and imaginary parts of (i.e., values of along the critical line) as is varied from 0 to 35.
Now consider this John Cook’s take…
where p is a positive integer. Here looking at what happens when p becomes a negative integer and we let n go to infinity.
If p < -1, then the limit as n goes to infinity of Sp(n) is ζ(-p). That is, for s > 1, the Riemann-Zeta function ζ(s) is defined by
We don’t have to limit ourselves to real numbers s > 1; the definition holds for complex numbers s with real part greater than 1. That’ll be important below.
When s is a positive even number, there’s a formula for ζ(s) in terms of the Bernoulli numbers:
The best-known special case of this formula is that
1 + 1/4 + 1/9 + 1/16 + … = π2 / 6.
It’s a famous open problem to find a closed-form expression for ζ(3) or any other odd argument.
The formula relating the zeta function and Bernoulli tells us a couple things about the Bernoulli numbers. First, for n ≥ 1 the Bernoulli numbers with index 2n alternate sign. Second, by looking at the sum defining ζ(2n) we can see that it is approximately 1 for large n. This tells us that for large n, |B2n| is approximately (2n)! / 22n-1 π2n.
We said above that the sum defining the Riemann zeta function is valid for complex numbers s with real part greater than 1. There is a unique analytic extension of the zeta function to the rest of the complex plane, except at s = 1. The zeta function is defined, for example, at negative integers, but the sum defining zeta in the half-plane Re(s) > 1 is not valid.
One must have seen the equation
1 + 2 + 3 + … = -1/12.
This is an abuse of notation. The sum on the left clearly diverges to infinity. But if the sum defining ζ(s) for Re(s) > 1 were valid for s = -1 (which it is not) then the left side would equal ζ(-1). The analytic continuation of ζ is valid at -1, and in fact ζ(-1) = -1/12. So the equation above is true if you interpret the left side, not as an ordinary sum, but as a way of writing ζ(-1). The same approach could be used to make sense of similar equations such as
12 + 22 + 32 + … = 0
13 + 23 + 33 + … = 1/120.