Basel problem

The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up years later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family, who unsuccessfully attacked the problem.

The Basel problem asks for the precise summation of the reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:

:$$sum_{n=1}^infin frac{1}{n^2} =lim_{n o +infty}left(frac{1}{1^2} + frac{1}{2^2} + cdots + frac{1}{n^2} ight).

The series is approximately equal to 1.644934 OEIS|id=A013661. The Basel problem asks for the "exact" sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be $$pi^2/6 and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, and it was not until 1741 that he was able to produce a truly rigorous proof.

Euler attacks the problem

Euler's original "derivation" of the value $$pi^2/6 is clever and original. He essentially extended observations about finite polynomials and assumed that these same properties hold true for infinite series. Of course, Euler's original reasoning requires justification, but even without justification, by simply obtaining the correct value, he was able to verify it numerically against partial sums of the series. The agreement he observed gave him sufficient confidence to announce his result to the mathematical community.

To follow Euler's argument, recall the Taylor series expansion of the sine function

:$$sin(x) = x - frac{x^3}{3!} + frac{x^5}{5!} - frac{x^7}{7!} + cdots.

Dividing through by "x", we have

:$$frac{sin(x)}{x} = 1 - frac{x^2}{3!} + frac{x^4}{5!} - frac{x^6}{7!} + cdots.

Now, the roots (zeros) of sin("x")/"x" occur precisely at $$x = ncdotpi where $$n = pm1, pm2, pm3, dots,.Let us assume we can express this infinite series as a product of linear factors given by its roots, just as we do for finite polynomials:

:$$egin{align}frac{sin(x)}{x} & {} =left(1 - frac{x}{pi} ight)left(1 + frac{x}{pi} ight)left(1 - frac{x}{2pi} ight)left(1 + frac{x}{2pi} ight)left(1 - frac{x}{3pi} ight)left(1 + frac{x}{3pi} ight) cdots \& {} = left(1 - frac{x^2}{pi^2} ight)left(1 - frac{x^2}{4pi^2} ight)left(1 - frac{x^2}{9pi^2} ight) cdots.end{align}

If we formally multiply out this product and collect all the "x"² terms, we see that the "x"² coefficient of sin("x")/"x" is

:$$-left(frac{1}{pi^2} + frac{1}{4pi^2} + frac{1}{9pi^2} + cdots ight) =-frac{1}{pi^2}sum_{n=1}^{infty}frac{1}{n^2}.

But from the original infinite series expansion of sin("x")/"x", the coefficient of "x"² is −1/(3!) = −1/6. These two coefficients must be equal; thus,

:$$-frac{1}{6} =-frac{1}{pi^2}sum_{n=1}^{infty}frac{1}{n^2}.

Multiplying through both sides of this equation by −π² gives the sum of the reciprocals of the positive square integers.

:$$sum_{n=1}^{infty}frac{1}{n^2} = frac{pi^2}{6}.

Q.E.D.

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:

:$$zeta(s) =sum_{n=1}^infin frac{1}{n^s}.

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

:$$zeta(2) =sum_{n=1}^infin frac{1}{n^2} =frac{1}{1^2} + frac{1}{2^2} + frac{1}{3^2} + frac{1}{4^2} + cdots = frac{pi^2}{6} approx 1.644934.

Convergence can be proven with the following inequality:

:$$sum_{n=1}^N frac{1}{n^2} < 1 + sum_{n=2}^N frac{1}{n(n-1)}= 1 + sum_{n=2}^N left( frac{1}{n-1} - frac{1}{n} ight)= 1 + 1 - frac{1}{N} ; stackrel{N o infty}{longrightarrow} ; 2.

This gives us the upper bound ζ(2) < 2, and because the infinite sum has only positive terms, it must converge. It can be shown that ζ("s") has a nice expression in terms of the Bernoulli numbers whenever "s" is a positive even integer. With $$s=2n:

:$$zeta(2n)=frac{(2pi)^{2n}(-1)^{n+1}B_{2n{2cdot(2n)!}

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 most elementary proof yet available; while most proofs use 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).

History of this proof

The proof goes back to Augustin Louis Cauchy (Cours d'Analyse, 1821, Note VIII). In 1954 this proof appeared in the book of Akiva and Isaak Yaglom "Nonelementary Problems in an Elementary Exposition". Later, in 1982, it appeared in the journal "Eureka", 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".

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",

:$$(cos x + isin x)^n = cos (nx) + i sin (nx). ,!

:Proof: This can be proved from Euler's formula; see the article for more details.
* The binomial theorem, which states that for any complex numbers "x" and "y" and any nonnegative integer "n",

:$$(x+y)^n=sum_{k=0}^n{n choose k}x^{n-k}y^k

:where the binomial coefficients can be expressed using the factorial by

:$${n choose k}=frac{n!}{k!(n-k)!}.

:Proof: See the proof in the article about the binomial theorem, it uses mathematical induction and some properties of the binomial coefficients.
* The function cot² "x" is one-to-one 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"). Adding (sin² "x")(sin² "y") to 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 one-to-one on the interval (0, π/2), so that "x" = "y".
* If "p"("t") = "a"_"m""t"^"m" + "a"_{"m" − 1}"t"^{"m" − 1} + ... + "a"₁"t" + "a"₀, where "a"_"m" ≠ 0, is a polynomial of degree "m" with distinct real roots "t"₁, "t"₂, ... , "t"_m, then "p"("t") = "a"_"m"("t" − "t"₁)("t" − "t"₂) ... ("t" − "t"_"m") and "t"₁ + "t"₂ + ... + "t"_"m" = −"a"_"m"−1/"a"_"m".
** Proof: The product representation of the polynomial is a consequence of the factor theorem, which in turn follows from polynomial long division. The formula for the sum of the roots is a special case of Viète's formulas and follows by expanding the product representation of the polynomial and comparing the coefficient of "t"^"m-1".
* 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/"x"² < csc² "x".
** Proof: First note that 0 < sin "x" < "x" < tan "x". This can be seen by considering the following picture:

:To see that 0 < sin "x" < "x", observe that in the picture, sin θ is the length of the line AC, and θ is the length of the circular arc AD.

:To see that "x" < tan "x", observe that the area of the triangle OAE is tan(θ)/2, the area of the sector OAD is θ/2, and that the sector is contained within the triangle.

: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" ≠ 0; then the limit of the sequence ("am" + "b")/("am" + "c") tends to 1 as "m" approaches infinity.
** Proof: Divide each term by "m" to 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.

The proof

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

:$$sum_{k=1}^m frac{1}{k^2} = frac{1}{1^2} + frac{1}{2^2} + cdots + frac{1}{m^2}

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

:$$frac{cos (nx) + i sin (nx)}{(sin x)^n} = frac{(cos x + isin x)^n}{(sin x)^n} = left(frac{cos x + i sin x}{sin x} ight)^n = (cot x + i)^n.

From the binomial theorem, we have

:$$(cot x + i)^n = {n choose 0} cot^n x + {n choose 1} (cot^{n-1} x)i + cdots + {n choose {n-1 (cot x)i^{n-1} + {n choose n} i^n

:$$= left [ {n choose 0} cot^n x - {n choose 2} cot^{n-2} x pm cdots ight] ; + ; ileft [ {n choose 1} cot^{n-1} x - {n choose 3} cot^{n-3} x pm cdots ight] .

Combining the two equations and equating imaginary parts gives the identity

:$$frac{sin (nx)}{(sin x)^n} = left [ {n choose 1} cot^{n-1} x - {n choose 3} cot^{n-3} x pm cdots ight] .

We take this identity, fix a positive integer "m", set "n" = 2"m" + 1 and consider "x"_"r" = "r" π/(2"m" + 1) for "r" = 1, 2, ..., "m". Then "nx"_"r" is a multiple of π and therefore a zero of the sine function, and so

:$$0 = 2m+1} choose 1} cot^{2m} x_r - 2m+1} choose 3} cot^{2m-2} x_r pm cdots + (-1)^m2m+1} choose {2m+1

for every "r" = 1, 2, ..., "m". The values "x"₁, ..., "x"_"m" are distinct numbers in the interval (0, π/2). Since the function cot² "x" is one-to-one on this interval, the numbers "t"_"r" = cot² "x"_"r" are distinct for "r" = 1, 2, ..., "m". By the above equation, these "m" numbers are the roots of the "m"th degree polynomial

:$$p(t) := 2m+1} choose 1}t^m - 2m+1} choose 3}t^{m-1} pm cdots + (-1)^m2m+1} choose {2m+1.

By Viète's formulas we can calculate the sum of the roots directly by examining the first two coefficients of the polynomial, and this comparison shows that

:$$cot ^2 x_1 + cot ^2 x_2 + cdots + cot ^2 x_m= frac{inom{2m+1}3} {inom{2m+1}1}= frac{2m(2m-1)}6.

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

:$$csc ^2 x_1 + csc ^2 x_2 + cdots + csc ^2 x_m=frac{2m(2m-1)}6 + m = frac{2m(2m+2)}6.

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

:$$frac{2m(2m-1)}6 < left( frac{2m+1}{pi} ight) ^2 + left( frac{2m+1}{2 pi} ight) ^2 + cdots + left( frac{2m+1}{m pi} ight) ^2 < frac{2m(2m+2)}6.

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

:$$frac{pi ^2}{6}left(frac{2m}{2m+1} ight)left(frac{2m-1}{2m+1} ight) < frac{1}{1^2} + frac{1}{2^2} + cdots + frac{1}{m^2} < frac{pi ^2}{6}left(frac{2m}{2m+1} ight)left(frac{2m+2}{2m+1} ight).

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

:$$zeta(2) =sum_{k=1}^infin frac{1}{k^2} =lim_{m o infty}left(frac{1}{1^2} + frac{1}{2^2} + cdots + frac{1}{m^2} ight) = frac{pi ^2}{6}

and this completes the proof.

A slicker proof from Fourier series

Let $$f(x)=x over the interval $$x in (-pi,pi). The Fourier series for this function (worked out in that article) is

:$$f(x) = 2sum_{n=1}^{infty}frac{(-1)^{n+1{n} sin(nx).

Then, using Parseval's identity, we have that

:$${pi^2 over 6} = {1 over 2pi} int_{-pi}^{pi} f^2(x) , dx = sum_{n=1}^{infty} {1 over n^2},

as required.

References

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External links

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* [http://www.maths.ex.ac.uk/~rjc/etc/zeta2.pdf Fourteen proofs of the evaluation of ζ(2)] , compiled by Robin Chapman