If Riemann hypothesis is true, functions {Riemann zeta function} can find number of primes less than N. Riemann zeta function is Dirichlet series. For complex numbers, zeta(z) = 1^-z + 2^-z + 3^-z + ..., which converges if z real part > 1, zeta(z) = 0, and z = -2, -4, -6, ... If imaginary numbers are input to zeta function, output can equal 0. Riemann zeta function equals infinity if z = 0 or 1.
Riemann hypothesis
Riemann zeta function converges if z real part = -0.5 {Riemann hypothesis} {Riemann problem}. This has no proof yet.
primes
If Riemann hypothesis is true, equation-zero locations give prime-number locations.
properties
For numbers x and N, zeta(x) = 1 + 1/2^x + 1/3^x + ... + 1/N^x. If x = 1, zeta is harmonic series. For x = 2, zeta converges to (pi)^2/6 [Euler, 1748], so sum of rational fractions gives transcendental number.
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Date Modified: 2022.0224