section_path: Analytic properties
section_name: Analytic properties
paragraph_text: Analytic properties
  
  Analytic number theory studies number theory through the lens of continuous functions, limits, infinite series, and the related mathematics of the infinite and infinitesimal.
  
  This area of study began with Leonhard Euler and his first major result, the solution to the Basel problem.The problem asked for the value of the infinite sum 1 + 1 4 + 1 9 + 1 16 + …, {\displaystyle 1+{\tfrac {1}{4}}+{\tfrac {1}{9}}+{\tfrac {1}{16}}+\dots,} which today can be recognized as the value ζ (2) {\displaystyle \zeta (2)} of the Riemann zeta function. This function is closely connected to the prime numbers and to one of the most significant unsolved problems in mathematics, the Riemann hypothesis. Euler showed that ζ (2) = π 2 / 6 {\displaystyle \zeta (2)=\pi ^{2}/6}. The reciprocal of this number, 6 / π 2 {\displaystyle 6/\pi ^{2}}, is the limiting probability that two random numbers selected uniformly from a large range are relatively prime (have no factors in common).
  
  The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the n {\displaystyle n} -th prime is known. Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
  
  with relatively prime integers a {\displaystyle a} and b {\displaystyle b} take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different linear polynomials with the same b {\displaystyle b} have approximately the same proportions of primes.Although conjectures have been formulated about the proportions of primes in higher-degree polynomials, they remain unproven, and it is unknown whether there exists a quadratic polynomial that (for integer arguments) is prime infinitely often.