Editing Basel problem (section)

{{Short description|Sum of inverse squares of natural numbers}}
{{Pi box}}
The '''Basel problem''' is a problem in [[mathematical analysis]] with relevance to [[number theory]], concerning an infinite sum of inverse squares. It was first posed by [[Pietro Mengoli]] in 1650 and solved by [[Leonhard Euler]] in 1734,<ref>{{citation |last= Ayoub |first= Raymond |title= Euler and the zeta function |journal= Amer. Math. Monthly |volume= 81 |year= 1974 |issue= 10 |pages= 1067–86 |url= https://www.maa.org/programs/maa-awards/writing-awards/euler-and-the-zeta-function |doi= 10.2307/2319041 |jstor= 2319041 |access-date= 2021-01-25 |archive-date= 2019-08-14 |archive-url= https://web.archive.org/web/20190814233022/https://www.maa.org/programs/maa-awards/writing-awards/euler-and-the-zeta-function |url-status= dead }}</ref> and read on 5 December 1735 in [[Russian Academy of Sciences#History|''The Saint Petersburg Academy of Sciences'']].<ref>[https://scholarlycommons.pacific.edu/euler-works/41/ E41 – De summis serierum reciprocarum]</ref> Since the problem had withstood the attacks of the leading [[mathematician]]s 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 more than a century later by [[Bernhard Riemann]] in his seminal 1859 paper "[[On the Number of Primes Less Than a Given Magnitude]]", in which he defined his [[Riemann zeta function|zeta function]] and proved its basic properties. The problem is named after the city of [[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 [[Multiplicative inverse|reciprocals]] of the [[square number|squares]] of the [[natural number]]s, i.e. the precise sum of the [[Series (mathematics)|infinite series]]:
<math display="block">\sum_{n=1}^\infty \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots. </math>

The sum of the series is approximately equal to 1.644934.<ref>{{Cite OEIS|1=A013661|mode=cs2}}</ref> The Basel problem asks for the ''exact'' sum of this series (in [[closed-form expression|closed form]]), as well as a [[mathematical proof|proof]] that this sum is correct. Euler found the exact sum to be <math display="inline">\frac {\pi^2}{6}</math> and announced this discovery in 1735. His arguments were based on manipulations that were not justified at the time, although he was later proven correct. He produced an accepted proof in 1741.

The solution to this problem can be used to estimate the probability that two large [[random number]]s are [[coprime]]. Two random integers in the range from 1 to {{Mvar|n}}, in the limit as {{Mvar|n}} goes to infinity, are relatively prime with a probability that approaches <math display="inline">\frac {6}{\pi^2}</math>, the reciprocal of the solution to the Basel problem.<ref>{{citation|contribution=Chapter 9: Sneaky segments|pages=101–106|title=Circle in a Box|series=MSRI Mathematical Circles Library|first=Sam|last=Vandervelde|publisher=Mathematical Sciences Research Institute and American Mathematical Society|year=2009}}</ref>