2 The Riemann zeta function

The Riemann zeta function ζ(s) is one of the most important functions in mathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures, because of its relationship to the distribution of the prime numberIn mathematics, a prime number or prime for short, is a natural number whose only distinct positive divisors are 1 and itself; otherwise it is called a composite number . Hence a prime number has exactly two divisors. The number 1 is neither prime nor coms. The function is defined for any complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher s with real part > 1 by the following formula:

Taking s = 2, we see that ζ(2) is equal to the sum of the reciprocals of the squares of the positive integers:

How do we know it converges at all? We can demonstrate this with the following inequality:

This gives us the upper bound ζ(2) < 2, but the exact value ζ(2) = π²/6 was unknown for some time, until Leonhard Euler computed it in 1735. It can be shown that ζ(s) has a nice expression in terms of the Bernoulli numbers whenever s is a positive even integer.

3 A rigorous proof

The following argument proves the identity ζ(2) = π²/6, where ζ(s) is the Riemann zeta function. It is by far the simplest proof yet available; while most proofs utilise results from advanced mathematics, such as Fourier analysis, complex analysis, and multivariable calculus, the following does not even require single-variable calculus (although a single limit is taken at the end).

3.1 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 1960s".

3.2 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,

Proof: This can be proved from Euler's formula; see the article for more details.

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

where we have the binomial coefficients

(See factorial)

Proof: This requires mathematical induction and some properties of the binomial coefficients.

• 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:

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.

3.3 The proof

The main idea behind the proof is to bound the partial sums

between two expressions, each of which will tend to π²/6 as m approaches infinity. The two expressions are derived from identities involving the cotangent and cosecant functions. These identities are in turn derived from De Moivre's formula, and we now turn to establishing these identities.

Let x be a real number with 0 < x < π/2, and let n be a positive integer. Then from De Moivre's formula and the definition of the cotangent function, we have

From the binomial theorem, we have

Combining the two equations and equating imaginary parts gives the identity

We take this identity and set n = 2m + 1, where m is a positive integer, and x = rπ/(2m + 1), where r = 1, 2, ..., m. Then nx = rπ, so that sin(nx) = 0, and so

This equation holds for each of the values x = rπ/(2m + 1), where r = 1, 2, ..., m. These values of x are distinct numbers strictly between 0 and π/2. Since the function cot²(x) is 1-1 on the interval (0, π/2), the numbers cot²(x) = cot²(rπ/(2m + 1)) are therefore distinct for r = 1, 2, ..., m. But by the above equation, each of these m distinct numbers is a root of the mth degree polynomial

This means that the numbers x = cot²(rπ/(2m + 1)), for r = 1, 2, ..., m are precisely the roots of the polynomial p(t). But we can calculate the sum of the roots directly by examining the coefficients, and the comparison shows that

Substituting the identity csc² x = cot² x + 1, we have

Now consider the inequality cot² x < 1/x² < csc² x. If we add up all these inequalities for each of the numbers x = rπ/(2m + 1), and if we use the two identities above, we get

Multiplying through by (π/(2m + 1))², this becomes

As m approaches infinity, the left and right hand expressions each approach π²/6, so by the squeeze theorem,

and this completes the proof. Q.E.D.

4 References

• Number Theory: An Approach Through History, Andre Weil, Springer, BooksEnthsiast.com
• Euler: The Master of Us All, William Dunham, MAA, BooksEnthsiast.com
• Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, John Derbyshire, Joseph Henry Press, BooksEnthsiast.com
• Proofs From the Book, Martin Aigner, Gunter Ziegler, Springer, BooksEnthsiast.com
• Riemann's Zeta Function, Harold M. Edwards, Dover, BooksEnthsiast.com

5 External links

• A page dedicated to ζ(2)
• Euler's solution of the Basel problem -- the longer story (PDF)
• How Euler did it (PDF)
• The infinite series of Euler and the Bernoulli's spice up a calculus class (PDF)
• Fourteen proofs of the evaluation of ζ(2), compiled by Robin Chapman
  • PDF version
  • PS version
  • DVI version

Number theory

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