Riemann zeta function ζ(s)

Definition

The Riemann zeta function ζ(s) is a complex function defined for complex numbers s with real part greater than 1 as the infinite series $$ ext{ζ}(s) = rac{1}{1^s} + rac{1}{2^s} + rac{1}{3^s} + rac{1}{4^s} + ext{...}$$ It plays a crucial role in number theory, particularly in understanding the distribution of prime numbers and is linked to properties like Dirichlet convolution through its relationship with arithmetic functions.

5 Must Know Facts For Your Next Test

1. The Riemann zeta function can be analytically continued to other values of s, except for a simple pole at s = 1.
2. The critical line, where the real part of s is 1/2, is significant in relation to the Riemann Hypothesis, which posits that all non-trivial zeros lie on this line.
3. The function has a deep relationship with prime numbers, as shown by the Euler product formula that expresses ζ(s) as a product over all prime numbers.
4. The values of the zeta function at negative integers are related to the Bernoulli numbers, providing connections between number theory and special values.
5. The functional equation of the Riemann zeta function relates values at s and 1 - s, showing symmetry and playing a key role in its analysis.

Review Questions

• How does the Riemann zeta function relate to Dirichlet convolution and its properties?
  • The Riemann zeta function is fundamentally linked to Dirichlet convolution through its representation in terms of arithmetic functions. Specifically, if you take two multiplicative functions, their Dirichlet convolution can be expressed in terms of the zeta function. This property underlines how important ζ(s) is when studying sums involving prime numbers or any arithmetic functions that are closely tied to multiplication.
• Discuss how the Euler product formula enhances our understanding of the connection between the Riemann zeta function and prime numbers.
  • The Euler product formula reveals that the Riemann zeta function can be expressed as an infinite product over all prime numbers. This connection shows that ζ(s) captures information about the distribution of primes and provides insights into their density. By linking an infinite sum to an infinite product, it highlights how primes play a critical role in number theory, emphasizing their fundamental nature in forming integers.
• Evaluate the implications of the Riemann Hypothesis in relation to the zeros of the Riemann zeta function and its effects on number theory.
  • The Riemann Hypothesis asserts that all non-trivial zeros of the Riemann zeta function lie on the critical line where the real part is 1/2. If proven true, it would have profound implications for number theory, particularly in understanding the distribution of prime numbers. Such a result would sharpen estimates related to primes and could lead to breakthroughs in various problems concerning their density and distribution, ultimately transforming our grasp of primes and their behavior.

Related terms

Dirichlet convolution: A binary operation on arithmetic functions that combines two functions into a new function, typically denoted by * and defined for all positive integers.

Euler product formula: A representation of the Riemann zeta function as an infinite product over prime numbers, highlighting its deep connection with the distribution of primes.

Analytic continuation: A technique in complex analysis that allows the extension of the domain of a given analytic function beyond its original radius of convergence.