Riemann zeta function

Definition

The Riemann zeta function is a complex function defined as $$\\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$ for complex numbers $s$ with real part greater than 1. This function plays a central role in number theory, particularly in the distribution of prime numbers and has deep connections to other mathematical areas through its generalizations and properties.

5 Must Know Facts For Your Next Test

1. The Riemann zeta function can be analytically continued to other values of $s$, except at $s=1$ where it has a simple pole.
2. The non-trivial zeros of the Riemann zeta function, which occur in the critical strip where the real part of $s$ is between 0 and 1, are crucial in understanding the distribution of prime numbers.
3. The famous Riemann Hypothesis conjectures that all non-trivial zeros of the Riemann zeta function lie on the critical line where the real part of $s$ is 1/2.
4. The functional equation relating the values of the Riemann zeta function at $s$ and $1-s$ shows deep symmetry and links its values across different regions of the complex plane.
5. Generalizations of the Riemann zeta function include Dirichlet L-functions and Dedekind zeta functions, which have applications in various branches of mathematics such as algebraic number theory.

Review Questions

• How does the analytic continuation of the Riemann zeta function expand its utility in number theory?
  • Analytic continuation allows the Riemann zeta function to be defined beyond its initial series representation, which only converges for real parts greater than 1. By extending this function to other complex values, it becomes essential for studying properties like the distribution of prime numbers through its non-trivial zeros. This extension makes it possible to explore deeper relationships between prime numbers and other mathematical structures.
• Discuss the significance of the non-trivial zeros of the Riemann zeta function in relation to the prime number theorem.
  • The non-trivial zeros of the Riemann zeta function play a pivotal role in understanding how prime numbers are distributed among integers. The prime number theorem connects this distribution with the logarithmic integral and suggests that primes become less frequent as numbers grow larger. The locations of these zeros influence the error term in estimates of prime counts, revealing a profound link between number theory and complex analysis.
• Evaluate how generalizations like L-functions build on concepts established by the Riemann zeta function and their implications in modern mathematics.
  • Generalizations such as L-functions extend concepts from the Riemann zeta function into broader contexts, connecting to various areas like automorphic forms and Galois representations. These extensions preserve key properties like functional equations and play a vital role in modern number theory, particularly in conjectures about primes and modular forms. Their implications are far-reaching, impacting not only pure mathematics but also applications in cryptography and computational number theory.

Related terms

Analytic continuation: A technique used to extend the domain of a given analytic function beyond its initial domain, allowing for the Riemann zeta function to be defined on the entire complex plane except for a simple pole at $s=1$.

Prime number theorem: A fundamental theorem that describes the asymptotic distribution of prime numbers, which is deeply linked to the non-trivial zeros of the Riemann zeta function.

L-functions: A broad class of functions that generalize the Riemann zeta function and are used in various areas of number theory, with connections to properties like automorphic forms and Galois representations.