The Riemann zeta function, denoted as \zeta(s), is introduced for a complex variable s = \sigma + i t through a Dirichlet series. The series converges absolutely for all s within the open half-plane where the real part of s exceeds one:
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad \Re(s) > 1
A relationship with number theory is established by the Euler product representation, which reformulates the function as an infinite product extending over all prime numbers p. This identity creates a bridge between the methods of complex analysis and properties of integers:
\zeta(s) = \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1}
The scope of the zeta function is extended to the entire complex plane, \mathbb{C}, by the process of analytic continuation. This extended function is meromorphic, possessing only a single simple pole at s=1 with a residue of 1.
The function exhibits zeros at the negative even integers, s \in \{-2, -4, -6, \dots\}, which are designated as the trivial zeros. Any other zeros must lie within the critical strip 0 < \Re(s) < 1.
The Riemann Hypothesis, an still unproven conjecture in modern mathematics, states that all these non-trivial zeros are located on the critical line specified by \Re(s) = 1/2. The precise locations of these zeros are connected to the distribution of prime numbers.
MURPHY, Terrence P., 2020. A Study of Bernhard Riemann’s 1859 Paper. Paramount Ridge Press. ISBN 978-0-9961671-3-0.
EDWARDS, Harold M., 2001. Riemann’s Zeta Function. Dover Ed edition. Mineola, NY: Dover Publications. ISBN 978-0-486-41740-0.
TITCHMARSH, E. C., 1987. The Theory Of The Riemann Zeta-Function. Subsequent edition. Oxford Oxfordshire : New York: Oxford University Press, U.S.A. ISBN 978-0-19-853369-6.