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,[1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences.[2] Since the problem had withstood the attacks of the leading mathematicians 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 zeta function and proved its basic properties. The problem is named after 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 reciprocals of the squares of the natural numbers, i.e. the precise sum of the infinite series:
The sum of the series is approximately equal to 1.644934.[3] The Basel problem asks for the exact sum of this series (in closed form), as well as a proof that this sum is correct. Euler found the exact sum to be 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 numbers are coprime. Two random integers in the range from 1 to , in the limit as goes to infinity, are relatively prime with a probability that approaches , the reciprocal of the solution to the Basel problem.[4]
^Ayoub, Raymond (1974), "Euler and the zeta function", Amer. Math. Monthly, 81 (10): 1067–86, doi:10.2307/2319041, JSTOR 2319041
^E41 – De summis serierum reciprocarum
^Sloane, N. J. A. (ed.), "Sequence A013661", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
^Vandervelde, Sam (2009), "Chapter 9: Sneaky segments", Circle in a Box, MSRI Mathematical Circles Library, Mathematical Sciences Research Institute and American Mathematical Society, pp. 101–106
The Baselproblem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed...
simple solution for this infinite series was a famous problem in mathematics called the Baselproblem. Leonhard Euler solved it in 1735 when he showed it...
unsolved problems in number theory and analysis, including the Baselproblem, which had remained unsolved for 150 years. The Baselproblem consists of...
Leonhard Euler and his first major result, the solution to the Baselproblem. The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 +...
computed by Euler. The first of them, ζ(2), provides a solution to the Baselproblem. In 1979 Roger Apéry proved the irrationality of ζ(3). The values at...
the squares of the first three equals the square of the fourth. The Baselproblem, solved by Euler in terms of π {\displaystyle \pi } , asked for an exact...
proofs of the Baselproblem Proof that e is irrational (also showing the irrationality of certain related numbers) Hilbert's third problem Sylvester–Gallai...
is not considered a solution to the ancient problem of squaring the circle. This is because the problem is to construct the area using a compass and...
1730s for real values of s, in conjunction with his solution to the Baselproblem. He also proved that it equals the Euler product ζ ( s ) = ∏ p prime...
Basel III is the third Basel Accord, a framework that sets international standards for bank capital adequacy, stress testing, and liquidity requirements...
theorem for Fourier series. This example leads to a solution of the Baselproblem. A proof that a Fourier series is a valid representation of any periodic...
Chronology A History of Pi In culture Indiana pi bill Pi Day Related topics Squaring the circle Baselproblem Six nines in π Other topics related to π v t e...
example of an Euler product, and the evaluation of ζ(2) as π2/6 is the Baselproblem, solved by Leonhard Euler in 1735. There is no way to choose a positive...
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given...
this series to the expansion of the infinite product form to solve the Baselproblem. The product of 1-D sinc functions readily provides a multivariate sinc...
}{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}}} which is a proof of the Baselproblem. The same argument works for all f ( x ) = x − 2 n {\displaystyle f(x)=x^{-2n}}...
Basel II is the second of the Basel Accords, which are recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision...
{(-1)^{n}z^{2n+1}}{2n+1}}.} His use of power series enabled him to solve the famous Baselproblem in 1735: lim n → ∞ ( 1 1 2 + 1 2 2 + 1 3 2 + ⋯ + 1 n 2 ) = π 2 6 . {\displaystyle...
diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the Baselproblem and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}}...
Proofs of Fermat's theorem on sums of two squares Riemann zeta function Baselproblem on ζ(2) Hurwitz zeta function Bernoulli number Agoh–Giuga conjecture...
The Basel Committee on Banking Supervision (BCBS) is a committee of banking supervisory authorities that was established by the central bank governors...
Global Information
