The Riemann zeta function
The rational numbers a/q form a fractal on the real line. If you cut them off at q {less than or equal to} Q (in the lowest terms form), they are fairly uniform, with obvious gaps around 1 and 1/2 and so on. Is the distribution as uniform as it can be, given that there are these gaps? This is one of the ramifications of the Riemann Hypothesis. The Riemann zeta function is a function of a complex variable, {zeta}(s), which acts as a generating function for the integers, the perfect squares, the prime numbers, and also for the combinatorial inclusion-exclusion M"obius function of sieve theory.
It has three properties:
D. a Dirichlet series,
E. an Euler product over primes, which means that log {zeta}(s) is also a Dirichlet series,
F. a functional equation of reflection type.
The Riemann Hypothesis says that all the zeros in the complex plane are on the reflection axis, except for zeros at -2, -4,... on the other symmetry axis. Weaker information than this would still have interesting consequences for prime numbers. Unlike most well-known functions, {zeta}(s) does not satisfy an algebraic differential equation, but it appears as the Titchmarsh function or scattering matrix in the Fourier theory of two by two matrices.
Reference
- M.N. Huxley, Exponetial sums and the Riemann zeta functions V Proc. London Math. Soc., (3) 90 (2005), 1-41
- M. C. Lettington, Fleck’s congruence, associated magic squares and a zeta identity, Funct. Approx. Comment Math. (2) 45 (2011), 165-205.
- M. C. Lettington, A trio of Bernoulli relations, their implications for the Ramanujan polynomials and the zeta constants, arXiv:1203.1753, (2012), 29pp.
- E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann Zeta Function, Oxford 1987