Analytic Continuation of the Riemann Zeta Function | Number Theory

Riemann discovered a functional equation relating ζ(s) to ζ(1-s): ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s). This formula reflects values across the line Re(s) = 1/2. Using this, Riemann could compute ζ(-1) by reflecting ζ(2) = π²/6 through the formula, yielding ζ(-1) = -1/12. This explains why some physicists claim 1 + 2 + 3 + 4 + ... = -1/12 in certain contexts.

The Riemann zeta function satisfies a functional equation that relates ζ(s) to ζ(1-s). For s with real part between 0 and 1, ζ(s) = 2^s π^(s-1) sin(πs/2) Γ(1-s) ζ(1-s), where Γ is the gamma function. This equation allows the zeta function to be defined for all complex numbers except s = 1, where it has a simple pole.

The video presents the functional equation that extends the Riemann zeta function to all complex numbers except s = 1. The equation is ζ(s) = 2^s × π^(s-1) × sin(πs/2) × Γ(1-s) × ζ(1-s), where Γ is the gamma function (a generalization of the factorial). This equation allows calculating ζ(s) for values where the original series doesn't converge. The gamma function Γ(n) = (n-1)! for positive integers n. This functional equation is the key to understanding how the zeta function can be extended beyond its original domain of convergence.

The Riemann zeta function satisfies a functional equation relating ζ(1-s) to ζ(s): ζ(1-s) = 2^(1-s)π^(-s)cos(πs/2)Γ(s)ζ(s). This equation allows computation of ζ(s) for values of s with real part less than 1 by relating them to values with real part greater than 1. The gamma function Γ(s) appears in this relationship, which generalizes the factorial function to complex numbers.

The Riemann zeta function satisfies the functional equation ζ(s) = Γ(s)π^(-s)sin(πs/2)ζ(1-s). This relates values of ζ at s to values at 1-s, connecting the region Re(s) > 1 (where the series converges) with Re(s) < 0. The gamma function's poles at non-positive integers explain why we get information about even positive integers (ζ(2k)) but not odd positive integers (ζ(2k+1)), which remain mysterious.

This section explains the zeros of the Riemann zeta function. Trivial zeros occur at negative even integers (-2, -4, -6, ...). Non-trivial zeros are complex numbers where ζ(s) = 0 and are located in the critical strip (0 < Re(s) < 1). The section introduces the critical line at Re(s) = 1/2, which is the central focus of the Riemann Hypothesis. The functional equation relates ζ(s) to ζ(1-s), allowing the function to be defined for all complex numbers except s=1.

The Riemann Zeta function has two categories of zeros: trivial zeros at negative even integers (-2, -4, -6, ...) found easily via the functional equation, and non-trivial zeros within the critical strip where 0 < Re(s) < 1. Remarkably, all known non-trivial zeros have real part exactly 1/2, lying on the critical line. Computational verification has confirmed this pattern for over 10^13 zeros, yet no proof exists that all non-trivial zeros must lie on this line. The zeta function connects to prime numbers through Euler's product formula: ζ(s) = product over primes p of (1 - 1/p^s)^(-1). Riemann developed an explicit formula expressing the prime-counting function π(x) as Riemann's function minus a sum over non-trivial zeros. This profound connection bridges number theory and complex analysis, with the Riemann Hypothesis remaining one of mathematics' most important unsolved problems.

Riemann factorized the zeta function as ζ(s) = s/(s-1) × 2^s π^(s-1) Γ(1-s) × ∏ρ (1 - s/ρ), where ρ represents non-trivial zeros. This shows zeros completely determine the function. The non-trivial zeros lie in the critical strip 0 < Re(s) < 1 and are symmetric about Re(s)=1/2. Their distribution controls the error in prime number approximations, making their location fundamental to understanding prime distribution.

The Riemann Zeta function ζ(s) is defined for complex numbers s with real part greater than 1 as an infinite series: ζ(s) = Σ(n=1 to ∞) of 1/n^s. The function can be analytically continued to all complex numbers except s = 1, where it has a simple pole. It has two types of zeros: trivial zeros at negative even integers (-2, -4, -6, ...) and non-trivial zeros in the critical strip where 0 < Re(s) < 1. The Euler Product Formula connects the Zeta function to all prime numbers: ζ(s) = Π(p prime) of 1/(1 - p^(-s)).

The Riemann zeta function ζ(s) = Σn^(-s) connects prime numbers to complex analysis. Euler showed it equals the product over primes (1 - p^(-s))^(-1). Trivial zeros occur at negative even integers; non-trivial zeros lie in the critical strip 0 < Re(s) < 1. The Riemann Hypothesis conjectures all non-trivial zeros lie on Re(s)=1/2. Dirichlet L-functions generalize this with characters χ(n), leading to the Generalized Riemann Hypothesis.

The Riemann Hypothesis, one of mathematics' most famous unsolved problems worth $1 million, proposes that all non-trivial zeros of the Riemann Zeta function lie on the critical line Re(s) = 1/2 in the complex plane; this hypothesis connects the distribution of prime numbers to the behavior of the zeta function, which assigns values to complex numbers through analytic continuation beyond its original domain of convergence.

The Riemann Hypothesis states that all non-trivial zeros of the Riemann Zeta Function lie on the critical line Re(s) = 1/2 in the complex plane. This hypothesis, proposed by Bernhard Riemann in 1859, is one of the seven Millennium Prize Problems with a $1 million prize. The hypothesis is significant because it would provide the best possible error term in the Prime Number Theorem, meaning the prime numbers are distributed as regularly as possible with no single wave dominating the distribution.

Riemann discovered that all non-trivial zeros appear to lie on the critical line where the real part equals 1/2. This is the Riemann Hypothesis: 'All non-trivial zeros of the zeta function have real part 1/2.' Despite being stated in one line, it remains unproven after 165 years. Riemann showed that prime distribution can be reconstructed by combining oscillations from each zero point. If all zeros lie on the critical line, oscillations remain harmonious and prime distribution is stable. If even one zero deviates, the prime rhythm would break down.

The Riemann Hypothesis, which conjectures that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2, is connected to a mysterious new geometry called F1 (geometry over the field with one element), which may provide a path to proving this famous unsolved problem; the hypothesis links prime numbers to the zeros of the zeta function, and studying L-functions and their properties—including positivity criteria, explicit formulas, and analogies with function field zeta functions—represents key approaches to understanding and potentially proving this conjecture.

The prime number theorem is equivalent to the statement that the Riemann zeta function has no zeros on the line Re(s) = 1. The video explains this connection and shows how the behavior of the zeta function determines the distribution of prime numbers. This connection is one of the most important in number theory.

The Prime Number Theorem is equivalent to the assertion that ζ(s) has no zeros on the line Re(s) = 1. The explicit formula shows ψ(x) = x - ∑_{ρ} x^ρ/ρ + O(x log x / T). If ζ(s) had a zero on Re(s) = 1, then the sum ∑_{ρ} x^ρ/ρ would contain terms growing like x^1, contradicting ψ(x) ≈ x. Conversely, if ψ(x) = x + o(x), then the zero-free region implies no zeros on Re(s) = 1. The logarithmic expansion log ζ(s) = ∑_{p} ∑_{k=1}^∞ p^{-ks}/k + regular function for Re(s) > 1/2 connects prime distribution to analytic properties. Assuming ζ(1 + 2it₀) = 0 leads to contradictory asymptotic behaviors, proving no zeros on the critical line and completing the Prime Number Theorem proof.

This segment completes the proof by establishing that g(s) is analytic on Re(s) ≥ 1 and that ζ(s) has no zeros on Re(s) = 1. At s = 1, both ζ(s) and sζ(s) have simple poles with residue 1, causing singularities to cancel in g(s) = -s/ζ'(s) - 1/(s-1), leaving an analytic function. For other points on Re(s) = 1, assuming a root of order a leads to contradictions when examining limits of εκ(1+ε±iα) as ε → 0+, because the resulting expressions must be non-negative while coefficients force them to be negative for a ≥ 1. This eliminates all possible zeros, completing the proof that g(s) is analytic on Re(s) ≥ 1.

The Riemann zeta function ζ(s) does not vanish on the line Re(s) = 1 (except at s = 1 where it has a pole). This is proven using Landau's theorem, which states that for σ > 1 and t ∈ ℝ, the product ζ(σ)³ζ(σ + it)⁴ζ(σ + 2it) > 1. The proof proceeds by contradiction: assuming ζ(1 + it) = 0 leads to a contradiction when taking the limit as σ → 1⁺, since the left side remains bounded while the right side tends to infinity. This result is crucial for the proof of the Prime Number Theorem.

This segment develops the complete connection between the Prime Number Theorem and the zeros of the Riemann Zeta function. The video combines ζ'(s)/ζ(s) = -∑_p (log p)/(p^s - 1) with s × Newman's integral = ∑_p (log p)/(p^s - 1) to derive the fundamental equation: ζ'(s)/ζ(s) + 1/(s-1) = -s × ∫₁^∞ (θ(x) - x)/x^(1+s) dx. This holds for Re(s) > 1, but the Prime Number Theorem requires extending it to s = 1. The term -1/(s-1) has a pole at s = 1, but ζ(s) also has a pole there, and the combination is holomorphic at s = 1. The expression can only diverge where ζ(s) = 0, so the Prime Number Theorem requires showing ζ(s) ≠ 0 for Re(s) = 1. This is the first non-trivial zero-free region for the Zeta function, and the critical line Re(s) = 1 is the boundary where the Prime Number Theorem can be proven.