8.1 Proof of non-vanishing of zeta function on Re(s) = 1

The Riemann zeta function is a cornerstone of analytic number theory. Its non-vanishing on the line Re(s) = 1 is crucial for understanding prime number distribution and proving the Prime Number Theorem.

This section dives into the proof of this non-vanishing property. We'll use contradiction, logarithmic derivatives, and Dirichlet series to show why the zeta function can't be zero when the real part of s equals 1.

Riemann Zeta Function Properties

Definition and Convergence

• Riemann zeta function defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for complex s with Re(s) > 1
• Converges absolutely for Re(s) > 1 due to comparison with p-series
• Extends analytically to entire complex plane except for simple pole at s = 1
• Functional equation relates values of zeta(s) to zeta(1-s)

Euler Product Representation

• Euler product formula expresses zeta function as product over primes: $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}$ for Re(s) > 1
• Demonstrates deep connection between zeta function and prime numbers
• Provides key insight into distribution of primes
• Allows for alternative proofs of certain zeta function properties

Analytic Continuation and Critical Strip

• Analytic continuation extends zeta function beyond its initial domain of convergence
• Achieved through various methods (Riemann's integral formula, Hankel contour)
• Critical strip refers to vertical strip in complex plane where 0 ≤ Re(s) ≤ 1
• Contains all non-trivial zeros of zeta function
• Riemann Hypothesis conjectures all non-trivial zeros lie on critical line Re(s) = 1/2

Non-vanishing on Re(s) = 1

Importance of Re(s) = 1 Line

• Real part of s equal to 1 forms boundary of absolute convergence for zeta function
• Non-vanishing on this line crucial for Prime Number Theorem
• Connects to behavior of primes through Euler product representation
• Studying zeta function on this line provides insights into distribution of primes

Proof Strategy and Key Concepts

• Non-vanishing proven by contradiction, assuming zeta(s) = 0 for some s with Re(s) = 1
• Logarithmic derivative of zeta function plays central role in proof
• Defined as $-\frac{\zeta'(s)}{\zeta(s)}$ where ζ'(s) is derivative of zeta function
• Relates to sum of prime powers through Euler product formula

Dirichlet Series and Convergence

• Dirichlet series form $\sum_{n=1}^{\infty} \frac{a_n}{n^s}$ generalizes zeta function
• Convergence of Dirichlet series depends on behavior of coefficients a_n
• For zeta function, all a_n = 1, leading to convergence for Re(s) > 1
• Proof uses properties of Dirichlet series to derive contradiction from assumed zero