Euler product and zeta function

What was discovered

An identity expressing the sum of 1/n^s as a product over all the primes, uniting analysis and arithmetic; from it he deduced that the sum of the reciprocals of the primes diverges. Euler also conjectured the functional equation of zeta.

Euler manipulated divergent series heuristically; Riemann extended zeta to the complex plane in 1859.

What it led to

• the Riemann hypothesis
• the prime number theorem
• L-functions and the Langlands programme

Built on

• Infinitude of primesc. 300 BCE
• Basel problem1735
• infinitude of primes

Made possible

• Riemann hypothesis1859

Links