Table of contents

1 The Riemann zeta function 2 History of this proof 3 What you need to know 4 The proof 5 References 6 External Links

The Riemann zeta function

The Riemann zeta function ζ(s) is one of the most important functions in mathematics, because of its relationship to the distribution of the prime numbers. The function is defined for any complex number s with real part > 1 by the following formula:

History of this proof

The origin of the proof is unclear. It appeared in the journal Eureka in 1982, attributed to John Scholes, but Scholes claims he learned the proof from Peter Swinnerton-Dyer, and in any case he maintains the proof was "common knowledge at Cambridge in the late 1960's".

What you need to know

To understand the proof, you will need to understand the following facts:

• De Moivre's formula states that for any real number x and any integer n,

• The binomial theorem, which states that for any real numbers x and y and any nonnegative integer n,

• The function cot² x is 1-1 on the interval (0, π/2).
  • Proof: Suppose cot² x = cot² y for some x, y in the interval (0, π/2). Using the definition of cotangent cot x = (cos x)/(sin x) and the trigonometric identity cos² x = 1 − sin² x, we see that (sin² x)(1 − sin² y) = (sin² y)(1 − sin² x). Subtracting (sin² x)(sin² y) from each side, we have sin² x = sin² y. Since the sine function is always nonnegative on the interval (0, π/2), this means sin x = sin y, but it is geometrically evident (by looking at the unit circle, e.g.) that the sine function is 1-1 on the interval (0, π/2), so that x = y.

• If p(t) is a polynomial of degree m, then p has no more than m distinct roots.
  • Proof: This is a consequence of the fundamental theorem of algebra.

• If p(t) = a_mt^m + a_{m − 1}t^{m − 1} + ... + a₁t + a₀, then the sum of the roots of p (counting multiplicities) is −a_{m − 1}/a_m.
  • Proof: If a_m = 1, then p(t) = product of all (t − s), where s ranges over all roots of p. Expanding this product, we see the coefficient of t^{m − 1} is minus the sum of all the roots. If a_m ≠ 1, then we can divide each term by it, obtaining a new polynomial with the same roots, whose leading coefficient is now 1; applying the preceding argument shows that the sum of the roots of the original p(t) = sum of the roots of this new polynomial = −a_{m − 1}/a_m.

• The trigonometric identity csc² x = 1 + cot² x.
  • Proof: This follows from the fundamental identity 1 = sin² x + cos² x after dividing through by sin² x.

• For any real number x with 0 < x < π/2, we have the inequalities cot² x < 1/x² < csc² x.
  • Proof: First note that 0 < sin x < x < tan x. This can be seen by considering the following picture: [there is a way of "showing" 0 < sin x < x < tan x by drawing a picture on the unit circle...if someone can create a picture showing this and put it here, it would be really great!] Now, take the reciprocal of everything and square. Remember that the inequality switches direction.

• Let a, b, and c be any real numbers, with a and c not both zero; then the limit of the function (am + b)/(am + c) as m approaches infinity is 1.
  • Proof: Divide each term by m, and get (a + b/m)/(a + c/m). If we divide a fixed number by a larger and larger number, the quotient approaches zero; thus, both the numerator and the denominator above tend to a, and so their quotient tends to 1.

• The squeeze theorem, which states that if a function is "squeezed" between two other functions, and each of those two functions approach a common limit, then the "squeezed" function also approaches that same limit.
  • Proof: See the article for a thorough discussion and proof.

References

• Proofs From the Book, Martin Aigner, Gunter Ziegler, Springer, ISBN 3540678654
• Riemann's Zeta Function, Harold M. Edwards, Dover, ISBN 0486417409