Riemann Zeta Function
The Riemann zeta function is a complex analytic function of fundamental importance to number theory and mathematical analysis. Formally defined for complex numbers with a real part greater than 1, it is expressed as the infinite series . Through the process of analytic-continuation, mathematicians have extended its definition to nearly all complex numbers, revealing deep structural properties that are central to modern mathematics.
Analytic Continuation and Poles
While the original series definition converges only for , the function can be uniquely extended to the entire complex plane except for a simple pole at . This extension allows the zeta function to be studied across the complex plane, where it exhibits specific symmetries described by the functional equation. This equation relates the value of the function at to its value at , linking the behavior of the function in the right half-plane to the left half-plane.
Applications in Quantum Physics
The concept of analytic continuation extends beyond pure mathematics into theoretical physics, particularly in the regularization of divergent series.
- Regularization of Divergent Sums: The technique allows for the assignment of finite values to series that diverge in the conventional sense.
- The -1/12 Result: A profound application involves the sum of all positive integers (). While this series diverges to infinity, analytic continuation of the Riemann zeta function at yields the value .
- Physical Significance: This result is not a direct summation but a regularization technique crucial in quantum-field-theory and string theory for calculating vacuum energy and Casimir effects.
For a detailed exploration of this derivation and its physical context, see 12 for Quantum Physics Applications.