Marcelo Viana: Euler and popstar of mathematical constants – 03/21/2023 – Marcelo Viana

One of the most fascinating aspects of mathematics — I think my colleagues agree — is its ability to reveal surprising relationships between things that don’t seem to have anything to do with each other. Today’s example also testifies to Leonhard Euler’s (1707–1783) remarkable flair for questions that seem merely curious but are actually profound and generate important advances.

The problem was formulated in 1650 by the Italian priest Pietro Mengoli (1626–1686), one of the first mathematicians to work with the concept of “limit”: how much is the sum 1/1^two+1/2^two+1/3^two+1/4^two+1/5^two+1/6^two+… the inverse squares of all integers? The question requires an explanation: after all, how do you add an infinite number of numbers? Is the total not necessarily infinite, as some Greek philosophers thought?

The idea is to consider partial sums, with finite but ever-increasing amounts of terms: first 1/1^twothen 1/1^two+1/2^twothen 1/1^two+1/2^two+1/3^twothen 1/1^two+1/2^two+1/3^two+1/4^two etc. If these sums are getting closer and closer to a certain value, it is reasonable to consider this limiting value to be the sum of all terms.

Mengoli’s problem caught the attention of the best mathematicians of the day, including three members of the famous Bernoulli family, who unsuccessfully attacked it. It became known as the “Basel problem”, presumably because the Bernoullis were from that Swiss city. It was settled in 1734 by Euler, who was also from Basel, but by then already in Russia. He publicly presented the solution to the St. Petersburg Academy of Sciences on December 5, 1735. He was 28 years old and the feat brought him immediate fame.

Indeed, some of the steps of his reasoning could not be rigorously justified at the time, but were later substantiated by Karl Weierstrass (1815–1897). In any case, in 1741 Euler gave a different and quite rigorous proof. But most surprising of all was the conclusion: the exact value of the sum is π^two/6…

The number π = 3.14159265359…, everyone knows, is the length of a circle with a diameter equal to 1. It cannot be represented completely in decimal expansion: an infinite number of digits would be needed, in a sequence that seems unpredictable. What can such a number have to do with integers? Even more π^two/6, are you serious?!

Euler’s argument, which today is accessible to a good high school student, explains this relationship and is directly at the origin of the work “On the number of primes smaller than a given quantity”, from 1859, in which Bernhard Riemann (1826– 1866) revealed even deeper and more surprising relationships between π and the prime numbers.

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