martes, 14 de marzo de 2023

Un producto infinito para la constante de Ludolph van Ceulen

Consideremos el producto infinito $$ \left(1- \frac{1}{3}+\frac{1}{3^{2}}-\cdots\right) \left(1+\frac{1}{5}+\frac{1}{5^{2}}+\cdots\right) \left(1-\frac{1}{7}+\frac{1}{7^{2}}-\cdots\right)\cdots$$ Dentro de un par de paréntesis hay alternancia de signos si y sólo el número primo que lidera a los paréntesis respectivos es congruente con $-1$ módulo $4$. Al considerar el producto de los primeros $n$ factores de ese producto infinito, los términos que se obtienen son de la forma $$ \pm \frac{1}{p_{1}^{e_{1}} \cdots p_{n}^{e_{n}}} $$ donde $p_{1}, \ldots, p_{n}$ son los primeros $n$ números primos impares y $e_{1}, \ldots, e_{n}$ son números enteros no negativos. El signo de un término dado es negativo si y sólo si la suma de los exponentes de los primos congruentes con $-1$ módulo $4$ que aparecen en la expresión $p_{1}^{e_{1}} \cdots p_{n}^{e_{n}}$ es un número impar. Puesto que todo número entero positivo $N>1$ se puede expresar como un producto de potencias de números primos de manera única (salvo el orden de las potencias), al desarrollar el producto infinito aparecerán como denominadores todos los números impares exactamente una vez. Tenemos así que \begin{eqnarray*}\left(1- \frac{1}{3}+\cdots\right) \left(1+\frac{1}{5}+\cdots\right)\left(1-\frac{1}{7}+\cdots\right) \cdots &=& 1- \frac{1}{3}+\\ && \frac{1}{5}-\frac{1}{7}+\cdots \end{eqnarray*} Luego, en vista de que $$ 1-\frac{1}{p}+\frac{1}{p^{2}} - \cdots = \frac{p}{p+1},$$ $$ 1+\frac{1}{p}+\frac{1}{p^{2}} + \cdots = \frac{p}{p-1}$$ y $$ 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+\frac{1}{13}-\frac{1}{15}+\cdots = \frac{\pi}{4},$$ se concluye que \begin{eqnarray*}\frac{\pi}{4} = \frac{3}{4} \cdot \frac{5}{4} \cdot \frac{7}{8} \cdot \frac{11}{12} \cdot \frac{13}{12} \cdots \end{eqnarray*} La identidad es fácil de recordar pues, en la derecha, los numeradores son todos los números primos impares listados en su orden natural mientras que el denominador del primo $p$ es igual a $p+1$ cuando $p \equiv -1 \pmod{4}$ e igual a $p-1$ cuando $p \equiv 1 \pmod{4}$.

NOTA. La primera vez que vi este producto infinito para $\pi$ fue en las páginas del libro Alberto Barajas: su oratoria, sus matemáticas y sus enseñanzas (SMM & IMATE UNAM, 2010); no obstante, recuerdo haberlo encontrado después en algún otro texto de teoría de números. Espero volver a dar con esa obra más adelante; por ahora sólo me queda reiterarles mis mejores deseos (atrasados) por el Día de Pi 2023... ¡Hasta la próxima!

miércoles, 1 de marzo de 2023

Some excerpts from V. I. Arnold's "Yesterday and long ago" on the origins of mathematics

«Nowadays we tend to underestimate the knowledge of the ancients, especially of scientists before the ancient Greeks. For more than a hundred years historians have known about the facts I'll discuss below, but about which mathematicians have never heard.

Thousands of years ago (before Moses), a remarkable mathematician who made a lot of discoveries lived in Egypt. He was a land surveyor (he measured land--from this follows the word "geometry"). He is known as Thot, the name which he got after his death (Thot is the name of the God who carried the souls of the dead in a boat across the Lethe river in ancient Egypt).

His first discovery was the natural series: Thot understood that there is no maximal integer (before him the numbers were bounded by the tax payed to Pharaoh). He learned how to carry out proofs based on the existence of actual infinity. His second discovery is not a mathematical one: it was the first phonetic alphabet. Before there were only hieroglyphics in Egypt, and he decreased the number of symbols to several dozen, having realized that, for example, the sound d could be represented by a simplification of the hieroglyph which mean "dog".

In Plato's "Phaedrus", Ammon (the main Egyptian God) discusses with Thot the creation of this alphabet. Thot says that the ability to write down information makes people cleverer, because there is no need to remember everything. Ammon objects that it is "the other way around, they will be more stupid because by relying on their notes they would lose the habit of remembering". They did not discuss computerization yet.

The Jewish and Phoenician alphabets originate from Thot's alphabet. From the Phoenician alphabet comes the Greek one; and later from that the Latin alphabet and only then our Cyrillic version.

The next discovery of Thot is geometry. To measure plots of land in order to estimate an expected crop, the tax it would yield, and how much water for irrigation was required from the Nile, Thot invented axioms, definitions, theorems, and drawings. The only thing he did not care about was the independence of his axioms and, as a result, he did not reach the modern level. For example, instead of one axiom on parallels, he introduced four or five axioms (each of them actually yield the others). But Thot did not prove this, he just used all of them. The honor to choose one (the fifth postulate) and to convert the others into theorems belongs to Euclid.

The measurements of the Earth's radius is among the remarkable geometrical achievements of that time (which belongs either to Thot himself or to his students). Camel caravans walked from Thebes to Memphis along a meridian, and it was not difficult to count their steps (and hence the total distance). They also measured the difference in altitude of the Sun at noon on the same day at these two Egyptian cities. With this information it is easy to calculate the radius of the Earth. It is remarkable that the relative error in this result had been only 1% (as compared with the modern value).

Greek scientists did not trust the Egyptian data claiming that the Egyptian women were publicly prostituting with crocodiles (as it [has also been] mentioned in the book "De la célébration du dimanche" of Proudhon, 1870, Paris). And thus two hundred years later the Greeks decided to measure the radius again. A ship sailed north from the mouth of the Nile to the island Rhodes. To calculate the distance they multiplied "the speed of the ship driven by a wind of an average force" with the time taken by this voyage. They obtained a radius which was two times larger than the correct one (it is easier to count camel's steps than to estimate whether the force of wind is average).

It is interesting that many centuries later a captain from Genoa came to a Catholic Queen asking for permission to sail to India by a western route (instead of the eastern one followed later by Vasco da Gama). The Queen appointed a committee of experts who said that "for such a long distance it is not possible to build a ship capable of carrying sufficient water for survival". Thus the Queen refused permission for this expedition.

But the captain persisted, and after many discussions with experts he got permission to die of thirst. (It is said that the reason for their incorrect conclusion was that the experts trusted the Greek estimate of the Earth's radius, while the captain believed in the Egyptian one.) That is how, just by chance, America was discovered.

Thot founded celestial mechanics and astrology in Egypt. If not he himself, then his ancient followers, knew the law of inverse squares (for a planets' attraction to the Sun) and Kepler's laws of planetary motion. This knowledge disappeared in the destruction of Alexandria's library by fire, where all scientific records of ancient Egypt were kept. Newton wrote that he only restored this ancient knowledge (cf. "Unknown Newton". St. Petersburg, Aleteia, 1999, pp. 731-757)...

Pythagoras was actually one of the first industrial spies in the world. He lived in Egypt for about twenty years, where Egyptian priests taught him their science. He had to take an oath never to reveal this knowledge, and that is why he did not publish anything.

Returning to Greece he told his students about geometry, and they brought this to Euclid who had not taken an oath. So it was that he published the geometry of Thot. Besides from geometry, Pythagoras brought from Egypt the idea of reincarnation (independent of the Indian version), vegetarianism based on it, and the basics of musical harmony for stringed instruments (including the formula for the tension of strings with various lengths, which have the same frequency--the required tension is proportional to the square of the length; the conditions for an octave, a third, a fifth, ...--actually, the Fourier series)...

There were other "spies" who, like Pythagoras, brought Egyptian secrets to Europe: Plato (logic and philosophy), Eudoxus (number theory [up to] Euclid's algorithm and the theory of irrational numbers including the theory of Dedekind's [cuts] and Grothendieck's rings).

The theory of irrational numbers started from the discovery of incommensurability of the diagonal of a square with its side (that is, the irrationality of √2) which was kept as a secret in the Pythagorean school. The point is that this fact undermined the importance of the arithmetical theory of fractions (and in this way of all mathematics): fractions were not sufficient for day-to-day requirements (such as measurement of lengths). Consequently, mathematicians were afraid that they would be accused of creating nonsense, and that they would be discarded or, at least, not be fed.

Thus it was necessary to create a new science--the theory of real numbers. This task (which is not simple, by the way) was solved by Eudoxus. It is surprising how close his approach is to the modern one (in this matter and also in the ... theory of divisibility). The discovery that facts like the uniqueness of expansion of an integer into prime factors require proof, is actually not less important than the proof itself, which is also not evident at all.»