History

Atle Selberg (1917-2007)

In 1949, Selberg and Erdős proved the Prime Number Theorem in an elementary way based on an inequality Selberg had proved shortly before, "elementary" meaning without the machinery of complex analysis.^[1] This was notable because the theorem had first been proven by Hadamard and de la Vallée Poussin in 1896 using complex analysis and there had been doubts that an elementary proof was possible. The Prime Number Theorem states that:
\[\lim_{x \to \infty} \; \frac{\pi(x)}{x/\log{x}} = 1, \hspace{8pt} \text{ where } \pi(x) = \text{ #primes } \leq x.\]
It seemed unreasonable that such advanced methods far from basic number theory were necessary to prove a theorem about prime numbers. Selberg closed the gap in 1949 and Tatuzawa and Iseki^[1'] gave a compact proof of Selberg's Inequality in 1951, a derivation I propose to follow in this article.^[2]

P. L. Chebyshev (1821-1894)

Mémoire sur les nombres premiers^[1] is Chebyshev's great work of 1850, where he used ingenious arguments to find the order of \(\pi(x),\) the number of prime numbers less than or equal to \(x,\) and proved Bertrand's postulate, that there is always at least one prime number strictly between \(n\) and \(2n-2\) for every integer \(n > 3.\)^[2] To this end, he introduced:
\begin{align}
\theta(x) &= \sum_{p \leq x} \log{p}\\[1em]
\psi(x) &= \theta(x) + \theta\left(x^{1/2}\right) + \theta\left(x^{1/3}\right) + \ldots,
\end{align}
where the first sum is taken over all primes less than or equal to the positive real number \(x.\) Note that the second sum has a finite number of terms, since for a given \(x, x^{1/n}\) is eventually less than 2 and for such \(n, \; \theta(x^{1/n}) = 0.\)

Complex multiplication in Argand (Essai, 1806).

The Fundamental Theorem of Algebra states that every non-constant polynomial with complex coefficients has at least one root in the complex plane. The quadratic formula provides two roots for every complex quadratic polynomial, considering that every complex number has two square roots. There are similar formulas for polynomials of degree three and four, but no general formula for polynomials of degree higher than that. This theorem has a long pedigree going back to Euler and before, d'Alembert attempting a proof in 1748. Gauss is often credited with the first proof in 1799, but it was incomplete.^[1] Argand gave a simple and direct proof in 1806 in an anonymous self-published pamphlet where he showed how to represent complex numbers geometrically and took up complex addition, multiplication, division, root taking, and absolute value.^[2]

Ratdolt's Euclid of 1482, page 1. First sentence: "Punctum est cuius ps nó est."

Like the Bible, Euclid's Elements is revered as a font of ancient wisdom and has indeed served as a kind of bible for those seeking knowledge for 2300 years. The Elements is the foremost scientific text of western civilization and has been studied assiduously since the day it was written around 300 BCE, not only for its mathematical content, but also as a template for how to think clearly. It was one of the very first printed scientific works^[1] and became a foundation stone of the revival of mathematics in the Renaissance made possible by printing, vernacular as well as Latin versions falling from the presses like autumn leaves starting in earnest in the sixteenth century.^[2] Isaac Newton, for example, studied the Elements carefully and wrote the Principia very much in a Euclidean spirit, augmented by limiting processes.^[2a]

Joseph Henry Maclagen Wedderburn (1882-1948)

Wedderburn's Theorem of 1905 is a beauty, that a finite division ring is a field. The result itself, but also the proof, which surely is enrolled in The Book, Paul Erdős's whimsical imaginary list of the finest proofs in all mathematics.^[1] A division ring is a ring with a 1 in which every non-zero element has a multiplicative inverse. Add multiplicative commutativity and you have a field. The quaternions over the reals are an example of an infinite division ring that is not a field, but Wedderburn says that such an example must be infinite — there is no example of a finite division ring that is not a field. What is striking about this proof is the range of techniques it employs, including group theory, linear algebra, the cyclotomic polynomials, Euclidean geometry, and basic facts about integers and complex numbers.^[2]

C. F. Gauss (1777-1855)

The prime numbers have been an object of fascination for a long time. These are the counting numbers having no divisors other than one and themselves:

\[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \ldots. \]

The key fact about the primes is that every natural number can be written as a product of primes, and the product is unique up to the order of the factors. Euclid proved that there are infinitely many prime numbers in 300 BC in Book IX, Proposition 20 of the Elements. Like all of Euclid, the proof is geometrical, with line segments representing numbers, but it's valid and recognizable. The modern proof goes like this:

Chebyshev's Mémoire sur les nombres premiers (1852)

§1.^[1] All questions which depend on the law of distribution of prime numbers in the series

\[ | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, \text {etc. } \]

present in general great difficulties. What we manage to conclude with a very high probability from the tables of prime numbers remains most often without rigorous proof. — For example, the tables of prime numbers lead us to believe that for \( a > 3, \) there is always a prime number greater than \( a \) and less than \( 2a-2 \) (which is known as the postulate of M. Bertrand *); but so far the proof of this proposition has failed for values ​​of \( a \) which exceed the limit of our tables. The difficulty becomes even greater when we give ourselves narrower limits, or when we ask to assign the limit of \( a \) above which the series

Emmy Noether Google Doodle for March 23, 2015

There are people great and small who sense the future with a clairvoyance not given to the rest of us, seers with a preternatural instinct for the arc of what is to come. Abraham Lincoln was one, Emmy Noether another in the world of mathematics. She was the daughter of a German mathematics professor renowned in his day,^[1] a lucky fluke in a society with little use for women as thinkers and academics. Emmy Noether revolutionized mathematics in the early twentieth century by identifying and explaining the underlying abstract principles behind the mathematics bequeathed to her. Like all geniuses, indeed all human beings, she existed in a social context. Asked to explain her deep insights, she would say "Es steht alles schon bei Dedekind" (Everything is already there in Dedekind)^[2], and that is not false modesty. Not false modesty, but modesty all the same, because she was second to none in triggering a new direction in mathematics.

David Hilbert, 1862-1943 (photo 1912)

Hilbert first proved a form of the basis theorem in 1890.^[1] It was so revolutionary at the time that Paul Gordan reportedly exclaimed, “This is not mathematics, it is theology!”.^[2] van der Waerden gave an updated and generalized proof in Moderne Algebra in 1931, crediting Hilbert for the basic idea and Emil Artin for the specifics.^[3] The proof here is updated still more, though still retaining van der Waerden's degree reduction strategy.

Hilbert Basis Theorem. Let \( R \) be a commutative ring with \( 1 \). If \( R \) is Noetherian, then \( R[x] \) is Noetherian as well.

Proof. Recall that a ring is Noetherian if its ideals satisfy the Ascending Chain Condition, or equivalently, if every ideal is finitely generated.

B. L. van der Waerden in 1930 (fl. 1903-1996)

A truism in mathematics, and perhaps most subjects, is that its disseminators are as important as its creators. Niccolò Tartaglia (1499-1557) comes to mind, a great and prolific encyclopedist whose works were consulted in his own day and for generations to come. Perhaps the premiere example is Frans van Schooten and his associates, who did so much to organize and present Descartes' deep but scattered and somewhat opaque geometric approach in La Geometrie. van Schooten's "appendices" amplified La Geometrie massively, the 1659-61 edition becoming a textbook for Leibnitz and Newton and the rock that calculus was built on. Of course the disseminators are mathematicians too, among the select few who expand the new path as well as trod it ("Cartesian" coordinates appear nowhere in Descartes, for example, but were introduced by his acolytes). They cast a great net, subsuming existing mathematics into the new framework as they organize and simplify, all the while spreading the word and making it available to a broad audience of practitioners, many having already worked obscurely to clear patches of brambles and more than ready for the big breakthrough.