Visualizing the Riemann zeta function and analytic continuation | #Sciencefather #mathawards #Research

 The Riemann zeta function, a cornerstone of analytic number theory, extends beyond its classical definition as an infinite series to a more profound domain through analytic continuation. This function, originally defined for complex numbers with real part greater than 1, plays a crucial role in understanding the distribution of prime numbers and the celebrated Riemann Hypothesis.

Graphical Representation

• 3D Surface Plots: Representing the modulus of ζ(s) over the complex plane, revealing the critical line (Re(s) = 1/2) and the non-trivial zeros.
• Phase Portraits: Highlighting contours where the function changes phase, exposing intricate patterns of zeros.
• Argand Diagrams: Displaying how ζ(s) evolves dynamically with s, illustrating analytic continuation and singularities.

Analytic Continuation and Extension

The Riemann zeta function is initially defined as:

$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad \text{for Re}(s) > 1.$$

However, through analytic continuation, it extends to most of the complex plane, excluding $s = 1$, where it has a simple pole. This extension enables deep exploration of the critical strip (0 < Re(s) < 1), where the zeros of ζ(s) hold profound implications for number theory.

Implications in Mathematics

• The Riemann Hypothesis, one of the most famous unsolved problems, conjectures that all non-trivial zeros of ζ(s) lie on the critical line.
• Connections to quantum chaos, prime number distribution, and random matrix theory further elevate the significance of ζ(s) beyond pure mathematics.

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