Heading: Riemann Zeta-Function
Definition
The Riemann zeta-function \zeta(s) is defined by
for a complex variable s=\sigma+it with \sigma>1 . It arises in the study of the distribution of prime numbers because of the product formula
which is also valid for \sigma>1. This product is called the Euler product for \zeta(s).
Riemann, in his famous 1859 work, Uber die Anzahl der Primzahlen unter eine gegebener Grösse, showed that \zeta(s) is a meromorphic function in the whole complex plane, with it's only singularity a simple pole at s=1 with residue 1. He proved the functional equation
with both sides analytic for all s except 0 and 1.
Zeros
It follows that \zeta(s)=0 for s=-2,-4,-6,.... The negative even integers are called the trivial zeros of \zeta(s). It also follows from the functional equation and Euler product that the non-trivial zeros of \zeta are symmetrically located with respect to the real line t=0 and with respect to the critical line \sigma=\frac{1}{2}, and that all of these non-trivial zeros must be located in the critical strip 0\le \sigma\le 1.
Riemann also concluded that the number N(T) of these zeros in the critical strip with positive ordinates smaller than T satisfies
This formula was rigorously proven by von Mangoldt in 1895.
After numerical investigations into the location of the zeros of \zeta(s) Riemann conjectured that all of the non-trivial zeros of \zeta(s) are on the critical line \sigma=\frac{1}{2} . This conjecture is the famous Riemann Hypothesis which is still unproven.
It is known by work of van de Lune and te Riele that the first 1,500,000,000 of the zeros of \zeta(s) are simple and on the critical line. Also, it has been shown by Conrey, using the technique invented by Levinson, that at least 40% of the zeros of \zeta(s) are simple zeros on the critical line.
A precise connection between zeros of \zeta(s) and primes is given by the explicit formula, enunciated by Riemann and proven by von Mangoldt. In this formula, \psi(x) is a weighted counting function for prime powers. The weight function is given by \Lambda(n) which is defined to be \log p if n is a positive integral power of the prime p, and is defined to be 0 otherwise. Then,
for x not equal to a prime power. The sum over zeros is not absolutely convergent and should be interpreted as a limit as T\rightarrow \infty of the sum for |\gamma|<T.
The Prime Number Theorem is the assertion that
and is easily shown to be equivalent to the assertion that \psi(x)\sim x as x\rightarrow\infty. This assertion is equivalent to the assertion that \zeta(1+it)\ne 0, i.e. that all of the non-trivial zeros of \zeta(s) are strictly inside the critical strip. This fact was proven in 1896 independently by Hadamard and de la Vallée Poussin.
Lindelöf Hypothesis
A consequence of the Riemann Hypothesis is the assertion that
as t\rightarrow\infty for any \epsilon>0. This assertion is known as the Lindelöf Hypothesis. Nigel Watt showed that
improving on ideas of Bombieri and Iwaniec who had the exponent 9/56. Huxley has given a further improvement lowering the exponent to 89/570.
Moments
The Lindelöf Hypothesis would be a consequence of the bound for the 2kth moment
The above is known to hold for k=1 and 2.
More precisely, it is conjectured that there are constants c_k such that
It is further believed that c_k=a_kf_k where f_k is something simple and where a_k can be defined implicitly by the asymptotic formula
Here d_k(n) denotes the k-fold divisor function with generating function given by \zeta(s)^k=\sum d_k(n)/n^s. Alternatively, a_k can be defined more explicitly by
where e(\theta)=\exp(2\pi i \theta). It is known that f_1=1 (Hardy and Littlewood), f_s=2 (Ingham), and it is conjectured that f_3=42 (Conrey and Ghosh) and f_4=24024 (Conrey and Gonek).