ζ(s)

Definition

The Riemann zeta function, denoted as ζ(s), is a complex function that plays a crucial role in number theory, particularly in the study of the distribution of prime numbers. It is defined for complex numbers s with a real part greater than 1 as the infinite series $$ ext{ζ}(s) = rac{1}{1^s} + rac{1}{2^s} + rac{1}{3^s} + ext{...}$$ and can be analytically continued to other values of s, except for s = 1 where it has a simple pole. The function's deep connections to prime numbers are highlighted by its Euler product formula, which expresses ζ(s) as an infinite product over prime numbers.

5 Must Know Facts For Your Next Test

1. The Riemann zeta function is initially defined by the series $$ ext{ζ}(s) = rac{1}{n^s}$$ for $$n = 1, 2, 3, ...$$ when the real part of s is greater than 1.
2. The analytic continuation allows ζ(s) to be defined for all complex numbers s except for s = 1, where it has a simple pole.
3. One of the most famous properties of ζ(s) is its relationship with prime numbers, encapsulated in the Euler product formula: $$ ext{ζ}(s) = rac{1}{ ext{p}^s}$$ for all primes p.
4. The Riemann Hypothesis conjectures that all non-trivial zeros of ζ(s) lie on the critical line where the real part of s equals 1/2.
5. The zeta function also has special values at even positive integers, leading to deep results in both number theory and mathematical analysis.

Review Questions

• How does the definition of ζ(s) relate to its role in number theory?
  • The Riemann zeta function ζ(s) is defined through an infinite series that converges for complex numbers s with a real part greater than 1. This definition connects it to the study of prime numbers via its Euler product formula, which shows how ζ(s) can be expressed as an infinite product over primes. This relationship allows mathematicians to use properties of ζ(s) to derive results about primes and their distribution.
• Discuss how analytic continuation impacts the study and application of the Riemann zeta function.
  • Analytic continuation extends the domain of ζ(s) beyond its initial definition, allowing it to be evaluated at points where the original series does not converge. This extension means that researchers can explore properties and implications of ζ(s) across a broader range of complex values. The ability to analyze ζ(s) at these points has led to significant insights in number theory, particularly concerning the distribution of prime numbers and the Riemann Hypothesis.
• Evaluate the implications of the Riemann Hypothesis in relation to ζ(s) and its zeros.
  • The Riemann Hypothesis posits that all non-trivial zeros of the Riemann zeta function ζ(s) lie on the critical line where the real part of s equals 1/2. If proven true, this would have profound implications for number theory, especially regarding the distribution of prime numbers. The hypothesis connects deeply with various fields in mathematics, suggesting that understanding these zeros could unlock new pathways in analytic number theory and even impact areas like cryptography and complex analysis.

Related terms

Analytic Continuation: The process of extending the domain of a given analytic function beyond its original region of convergence.

Euler Product Formula: A representation of the Riemann zeta function as an infinite product over all prime numbers, highlighting the connection between ζ(s) and primes.

Critical Line: The line in the complex plane defined by the real part of s being equal to 1/2, which is significant in the context of the Riemann Hypothesis.