12.1 Distribution of zeros of the zeta function

The distribution of zeros of the Riemann zeta function is crucial for understanding prime numbers. This section dives into the critical strip, where all non-trivial zeros lie, and explores the symmetry and patterns of these zeros.

We'll look at the Riemann-von Mangoldt formula, which estimates the number of zeros up to a certain height. We'll also examine the density of zeros and how the Hadamard product relates the zeta function to its zeros.

Zeros of the Zeta Function

Critical Strip and Trivial Zeros

• Critical strip defined as the vertical strip in the complex plane where 0 < Re(s) < 1
• Critical strip contains all non-trivial zeros of the Riemann zeta function
• Trivial zeros occur at negative even integers (-2, -4, -6, ...)
• Trivial zeros result from the functional equation of the zeta function
• Trivial zeros do not provide significant information about prime number distribution

Non-trivial Zeros and Symmetry

• Non-trivial zeros lie within the critical strip
• All known non-trivial zeros have real part exactly 1/2
• Riemann Hypothesis conjectures that all non-trivial zeros have real part 1/2
• Symmetry of zeros observed across the critical line Re(s) = 1/2
• Zeros occur in complex conjugate pairs (if s is a zero, so is 1 - s)
• Symmetry property aids in the study of zero distribution and function behavior

Density and Distribution of Zeros

Riemann-von Mangoldt Formula

• Riemann-von Mangoldt formula estimates the number of non-trivial zeros up to height T
• Formula states: N(T) = (T/2π) log(T/2π) - T/2π + O(log T)
• N(T) represents the number of zeros β + iγ with 0 < γ ≤ T
• Formula provides crucial insight into the distribution of zeros along the critical line
• Accuracy of the formula improves as T increases

Density of Zeros and Hadamard Product

• Density of zeros increases logarithmically as we move up the critical strip
• Average spacing between consecutive zeros decreases as the imaginary part increases
• Hadamard product expresses the zeta function in terms of its zeros
• Hadamard product formula: $ζ(s) = e^{B + Cs} s(s-1) \prod_ρ (1 - s/ρ) e^{s/ρ}$
• Product runs over all non-trivial zeros ρ of the zeta function
• B and C are constants related to the Euler-Mascheroni constant
• Hadamard product provides a powerful tool for studying the zeta function's behavior

Properties of the Zeta Function

Functional Equation and Analytical Continuation

• Functional equation relates values of ζ(s) to values of ζ(1-s)
• Functional equation: $ζ(s) = 2^s π^{s-1} sin(πs/2) Γ(1-s) ζ(1-s)$
• Equation allows for the analytical continuation of the zeta function to the entire complex plane
• Reveals the symmetry of the zeta function about the critical line
• Plays a crucial role in understanding the distribution of zeros
• Provides a powerful tool for studying the zeta function's behavior in different regions of the complex plane