Questions: Introduction to the Riemann Zeta Function

Question 1

The Euler product formula expresses ζ(s) as a product over all primes. What deep fact about integers does this identity encode?

• A. It encodes the fact that primes are infinite in number
• B. It encodes the fundamental theorem of arithmetic — every positive integer factors uniquely into primes
• C. It encodes the distribution of primes in arithmetic progressions
• D. It encodes the fact that the harmonic series diverges

Answer: (B)

The Euler product ζ(s) = ∏ₚ 1/(1−p⁻ˢ) works precisely because every positive integer has a unique prime factorization. Each factor in the product is a geometric series for one prime, and multiplying all of them together recovers the sum over all n⁻ˢ with each integer counted exactly once — one for each unique factorization. The identity would fail if unique factorization failed. It is the analytic restatement of the fundamental theorem of arithmetic.

Question 2

The Riemann Hypothesis states that all nontrivial zeros of ζ(s) lie on the line Re(s) = 1/2. Why does the location of these zeros matter for number theory?

• A. It determines whether the series Σ 1/n^s converges for Re(s) > 1
• B. It determines how accurately π(x) ≈ x/ln(x) approximates the prime counting function — zeros off the critical line would produce larger oscillations in prime distribution
• C. It determines whether ζ(s) can be analytically continued beyond Re(s) > 1
• D. It settles whether there are infinitely many twin primes

Answer: (B)

The prime counting function π(x) can be written via an explicit formula involving sums over the nontrivial zeros of ζ(s). Each zero contributes an oscillatory correction term. If all zeros lie on Re(s) = 1/2, the oscillations are as small as possible and the Prime Number Theorem approximation holds with the best possible error bound. Zeros farther from the critical line would cause larger 'lumpiness' in the prime distribution. This is why the RH, despite being a statement about a complex function, is fundamentally a statement about how regularly primes are distributed.

Question 3

The Riemann zeta function ζ(s) is defined by the series Σ 1/n^s for most complex numbers s ≠ 1.

• T. True
• F. False

Answer: False

The series Σ 1/n^s converges only for Re(s) > 1. For all other values of s (except s = 1, where ζ has a pole), the function is defined through analytic continuation — a process of extending a function beyond its original domain of convergence. The analytically continued ζ(s) agrees with the series where the series converges, but the series itself is not defined for Re(s) ≤ 1. This distinction is crucial: ζ(−2) = 0 is a well-defined result from the continued function, not from the series.

Question 4

The Euler product for ζ(s) is a direct analytic consequence of the unique factorization of integers into primes.

• T. True
• F. False

Answer: True

This is exactly right. The product ∏ₚ 1/(1−p⁻ˢ) expands each factor into a geometric series and, by unique factorization, the product of all these series produces exactly one term n⁻ˢ for each positive integer n. If unique factorization did not hold (as it fails in some number rings), the Euler product identity would not equal the Dirichlet series. The Euler product is therefore not just a formula but an encoding of the multiplicative structure of the integers.

Question 5

What does it mean to analytically continue the Riemann zeta function, and why is this step necessary?

Model answer: Analytic continuation means extending ζ(s) — originally defined by a convergent series only for Re(s) > 1 — to a meromorphic function on all of ℂ. The extension is unique (by the identity theorem for analytic functions) and gives the same values where the series converges. It is necessary because the most important properties of ζ — its nontrivial zeros, the functional equation, and the connection to prime distribution — only become visible in the full complex plane. Asking about zeros for Re(s) between 0 and 1 is meaningless without continuation, yet this is precisely where the Riemann Hypothesis lives.

Analytic continuation is like discovering that a recipe for positive inputs secretly works for all inputs — with reinterpretation. The series Σ 1/n^s cannot be evaluated at s = −1, but the analytically continued function gives ζ(−1) = −1/12, a result that appears in string theory and regularization. The step is necessary because the nontrivial zeros, the functional equation relating ζ(s) to ζ(1−s), and the connection between zero locations and prime distribution all require ζ to be defined across the entire critical strip 0 < Re(s) < 1.