ζ(s) = ∑ 1/n^s

Definition

The Riemann zeta function, denoted as ζ(s), is a complex function defined by the series $$ζ(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ for complex numbers s with real part greater than 1. This function is crucial in number theory, particularly in understanding the distribution of prime numbers and their relationship to the zeros of the function.

5 Must Know Facts For Your Next Test

1. The Riemann zeta function converges for values of s with a real part greater than 1, but it can be analytically continued to other values except for s = 1 where it has a simple pole.
2. The non-trivial zeros of the zeta function, located in the critical strip where 0 < Re(s) < 1, are intimately connected to the distribution of prime numbers.
3. The famous Riemann Hypothesis posits that all non-trivial zeros of ζ(s) lie on the critical line where Re(s) = 1/2.
4. The zeta function can be expressed using an Euler product formula for s greater than 1, which shows its deep connection with prime numbers: $$ζ(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}.$$
5. Values of ζ(s) at positive even integers relate to Bernoulli numbers and can be calculated using formulas involving pi, particularly for example $$ζ(2) = \frac{\pi^2}{6}$$.

Review Questions

• How does the definition of the Riemann zeta function relate to prime numbers and their distribution?
  • The Riemann zeta function encapsulates important properties of prime numbers through its series representation. The Euler product formula, which connects ζ(s) to primes, reveals how primes influence the behavior of the zeta function. As such, studying the zeros of ζ(s) gives insights into the distribution of primes via results like the Prime Number Theorem.
• What role does analytic continuation play in understanding the Riemann zeta function beyond its initial definition?
  • Analytic continuation is essential for extending the domain of the Riemann zeta function beyond its region of convergence (Re(s) > 1). This technique allows us to explore values of s where the original series diverges. By doing so, we can study properties such as its poles and zeros throughout the entire complex plane, providing a deeper understanding of its implications in number theory.
• Evaluate the significance of the critical line in relation to the Riemann Hypothesis and non-trivial zeros of ζ(s).
  • The critical line, defined by Re(s) = 1/2, holds great significance due to the conjecture that all non-trivial zeros of ζ(s) lie on this line. The Riemann Hypothesis, one of mathematics' most famous unsolved problems, posits this assertion. Understanding whether these zeros are aligned on the critical line has profound implications on our comprehension of prime number distribution and can potentially lead to breakthroughs in analytic number theory.

Related terms

Prime Number Theorem: A fundamental theorem that describes the asymptotic distribution of prime numbers, showing that the number of primes less than a given number n approximates to $$\frac{n}{\log(n)}$$.

Analytic Continuation: A technique used to extend the domain of a given analytic function beyond its original region of convergence, allowing functions like the Riemann zeta function to be defined for values of s where the series does not converge.

Critical Line: The line in the complex plane where the real part of s is equal to 1/2, which is significant because it is where all non-trivial zeros of the Riemann zeta function are conjectured to lie, according to the Riemann Hypothesis.