The Riemann zeta function <math>\zeta(s)</math> is defined for any complex number s with real part > 1 as:
- <math>
\zeta(s) =
\sum_{n=1}^\infin \frac{1}{n^s}
</math>
In the region {
s : Re(
s) > 1},
this
infinite series converges and defines a
holomorphic function. (In that expression, Re means the real part of a number.)
Bernhard Riemann realized that the zeta function can be extended by
analytic continuation in a unique way to a holomorphic function <math>\zeta(s)</math> defined for
all complex numbers
s with
s ≠ 1. It is this function that is the object of the
Riemann hypothesis.
The connection between this function and prime numbers was already realized by Leonhard Euler:
- <math>
\zeta(s) = \prod_{p} \frac{1}{1-p^{-s}}
</math>
an
infinite product extending over all prime numbers
p. This is a consequence of the formula for the
geometric series and the
Fundamental Theorem of Arithmetic.
The zeros of ζ(s) are important because certain path integrals involving the function ln(1/ζ(s)) can be used to approximate the prime counting function π(x) (see prime number theorem). These path integrals are computed with the residue theorem and hence knowledge of the integrand's singularities is required.
The zeta function satisfies the following functional equation:
- <math>
\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)
</math>
valid for all s in C - {0,1}. Here, Γ denotes the Gamma function. This formula is used to construct the analytic continuation in the first place. At s = 1, the zeta function has a simple pole with residue 1.
Euler was also able to calculate ζ(2k) for even integers 2k using the formula
- <math>
\zeta(2k) = \frac{B_{2k}(2\pi)^{2k}}{2(2k)!}
</math>
where
B_{2k} are the
Bernoulli numbers. From this one sees that ζ(2) = π
^{2}/6, ζ(4) = π
^{4}/90, ζ(6) = π
^{6}/945 etc. These give well-known infinite series for
π. For odd integers the case is not so simple.
Ramanujan did some great work about this.
One can express the reciprocal of the zeta function using the Möbius function μ(n) as follows:
- <math>
\frac{1}{\zeta(s)} = \sum_{n=1}^{\infin} \frac{\mu(n)}{n^s}
</math>
for every complex number
s with real part > 1. This, together with the above expression for ζ(2), can be used to prove that the probability of two random integers being
coprime is 6/π
^{2}.
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