The Basel Problem

Initially posed in 1650 by Italian mathematician Pietro Mengoli, the Basel problem was solved by Euler in 1734, instantly elevating his status in the mathematical community. The problem seeks to find the sum of the reciprocals of the squares of the natural numbers:

Many prominent mathematicians, including John Wallis and Gottfried Leibniz, attempted to derive a formula for this sum but were unsuccessful. Euler, at the young age of 28, provided a solution that astonished the mathematical world with its simplicity and beauty. While his initial proof lacked rigor, it was celebrated for its originality.

Euler's innovative approach involved expressing the sinc(?) function as a product of its zeros.

To grasp this concept, consider a quartic polynomial represented as a product of its zeros:

Expanding this expression yields:

Euler's technique involved applying a similar expansion to transcendental functions.

Transcendental Functions

These functions do not satisfy polynomial equations like the one above. Notable examples include the exponential function, trigonometric functions, and the logarithmic function.

The sinc(?) function has the following roots:

Euler then expressed sinc(?) in the same manner as the quartic polynomial:

Utilizing the fundamental identity

and recognizing that each root has a corresponding negative, he could express:

The next step involved multiplying out the terms but focusing solely on the quadratic component:

Taylor Series

Taylor series provide a means to express functions as infinite sums of terms, where each term is derived from the function's derivatives at a specific point.

The seven Taylor series depicted above have the following algebraic expressions:

The Taylor expansion for the sinc(?) function is:

This equation can be thought of as a "pseudo-polynomial" with infinite degrees and roots, as indicated in previous equations.

Comparing Both Results

By comparing the quadratic term expression and the Taylor series expansion, we arrive at our desired conclusion:

Additionally, Euler's derivation yields the renowned Wallis product. By substituting x = 1/2 into the previous equation and inverting it, we obtain:

A Rigorous Proof

In this concluding section, we will explore a rigorous proof of Euler’s result, credited to Daniel P. Giesy. Consider the function:

This function is typically expressed in a different form. One defines the number E(n) and calculates it by integrating the expression following the second equality of the previous equation:

It is evident that the sum on the right is zero for even k. Thus, substituting k with (2k-1) allows us to consider only terms where the subindex of E is odd:

To complete the proof, one must show that this expression equals zero. While this demonstration is quite intricate and not particularly enlightening, it will be omitted. The resulting expression is:

Through some straightforward algebraic manipulations, we arrive back at the previous conclusion:

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