Questions: Dirichlet Series and L-Functions

Question 1

The Euler product factorization of a Dirichlet series Σ a_n / n^s (when a_n is multiplicative) is significant primarily because:

• A. It allows the series to be evaluated at s = 1 without divergence
• B. It shows the series encodes arithmetic information prime by prime, connecting analytic properties to prime distribution
• C. It proves the series converges everywhere in the complex plane
• D. It reduces the infinite sum to a finite product

Answer: (B)

The Euler product ∏_p (1 + a_p/p^s + ...) expresses the Dirichlet series as a product over primes, with each prime contributing an independent factor. This factorization is what links analytic properties of the series (convergence, zeros, poles) to arithmetic properties of primes. The prototype is ζ(s) = ∏_p 1/(1-p^{-s}): the divergence of ζ(s) as s → 1⁺ forces a product over all primes to diverge, implying there are infinitely many primes. L-functions inherit this structure, but the character weights allow the product to detect primes in specific residue classes.

Question 2

The key analytic fact that Dirichlet uses to prove infinitely many primes in every arithmetic progression a (mod q) with gcd(a,q) = 1 is:

• A. The Riemann hypothesis for L-functions
• B. L(1, χ) ≠ 0 for non-principal Dirichlet characters χ
• C. L(s, χ) has a simple pole at s = 1 for all characters χ
• D. The Euler product for L(s, χ) converges for all s with Re(s) > 0

Answer: (B)

Dirichlet's proof works by combining L-functions for all characters mod q and extracting the sum over primes ≡ a (mod q). If any non-principal L-function vanished at s = 1, the logarithmic sum over primes in the progression a (mod q) could remain bounded even as s → 1⁺ — the argument for infinitely many primes would collapse. The non-vanishing L(1, χ) ≠ 0 is what forces the sum to diverge, guaranteeing infinitely many primes in the progression. This is the deepest analytic step in the proof.

Question 3

The Riemann zeta function ζ(s) = Σ 1/n^s is a special case of a Dirichlet series with a_n = 1 for all n.

• T. True
• F. False

Answer: True

A Dirichlet series has the form Σ a_n / n^s; setting a_n = 1 for all n gives exactly ζ(s) = Σ 1/n^s. The constant function a_n = 1 is multiplicative (1·1 = 1 when gcd(m,n) = 1), so ζ(s) has an Euler product: ζ(s) = ∏_p 1/(1 - p^{-s}). The zeta function is thus the simplest and most fundamental Dirichlet series, and all the properties of Dirichlet series — Euler products, analytic continuation, functional equations — were first discovered or illustrated for ζ(s) before being generalized.

Question 4

A Dirichlet L-function L(s, χ) for a non-principal character χ has a pole at s = 1, analogous to the pole of ζ(s) at s = 1.

• T. True
• F. False

Answer: False

This is a key distinction. The Riemann zeta function ζ(s) has a simple pole at s = 1 because the sum Σ 1/n diverges (the harmonic series). But for a non-principal character χ, the character values χ(n) oscillate and partially cancel each other. This cancellation is enough to make L(1, χ) a finite nonzero number — the series converges at s = 1. L(s, χ) extends analytically to an entire function for non-principal characters (no pole at s = 1). This contrast between ζ(s) and non-principal L-functions is central to Dirichlet's theorem.

Question 5

Explain why the Euler product factorization of L(s, χ) provides the key link between Dirichlet L-functions and the distribution of primes in arithmetic progressions.

Model answer: Because χ is completely multiplicative, L(s, χ) = ∏_p 1/(1 - χ(p)/p^s). Taking the logarithm converts this product into a sum over primes: log L(s, χ) = Σ_p χ(p)/p^s + (lower-order terms). To isolate primes ≡ a (mod q), Dirichlet takes a suitable linear combination of these log sums over all characters mod q. The orthogonality of characters ensures that terms for primes not in the target progression cancel, leaving only a sum over primes ≡ a (mod q). The non-vanishing of L(1, χ) then guarantees this sum diverges as s → 1⁺, forcing infinitely many such primes.

The Euler product is the mechanism that translates analytic information (behavior of a complex function near s = 1) into arithmetic information (existence of infinitely many primes in a progression). Without the product form, L-functions would be mere generating functions; the product over primes is what gives them number-theoretic meaning.