section_path: History/Irrationality and transcendence
section_name: Irrationality and transcendence
paragraph_text: Irrationality and transcendence
  
  Not all mathematical advances relating to π were aimed at increasing the accuracy of approximations. When Euler solved the Basel problem in 1735, finding the exact value of the sum of the reciprocal squares, he established a connection between π and the prime numbers that later contributed to the development and study of the Riemann zeta function:
  
  π 2 6 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ {\displaystyle {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }
  
  Swiss scientist Johann Heinrich Lambert in 1768 proved that π is irrational, meaning it is not equal to the quotient of any two integers. Lambert's proof exploited a continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π is also irrational. In 1882, German mathematician Ferdinand von Lindemann proved that π is transcendental, confirming a conjecture made by both Legendre and Euler. Hardy and Wright states that "the proofs were afterwards modified and simplified by Hilbert, Hurwitz, and other writers".