Weierstrass and Hadamard products

1. Weierstrass products 2. Poisson-Jensen formula 3. Hadamard products Apart from factorization of polynomials, after Euler's sin πz πz = ∞ n=1 1 − z 2 n 2 there is Euler's product for Γ(z), which he used as the definition of the Gamma function: ∞ 0 e −t t z dt t = Γ(z) = 1 z e γz ∞ n=1 1 + z n e −z/n where the Euler-Mascheroni constant γ is essentially defined by this relation. The integral (Euler's) converges for Re (z) > 0, while the product (Weierstrass') converges for all complex z except non-positive integers. Because the exponential factors are linear, and can cancel, 1 Γ(z) · Γ(−z) = −z 2 ∞ n=1 1 − z 2 n 2 = − z π · sin πz Linear exponential factors are exploited in Riemann's explicit formula [Riemann 1859], derived from equality of the Euler product and Hadamard product [Hadamard 1893] for the zeta function ζ(s) = n 1 n s for Re (s) > 1: p prime 1 1 − p −s = ζ(s) = e a+bs s − 1 · ρ 1 − s ρ e s/ρ · ∞ n=1 1 + s 2n e −s/2n where the product expansion of Γ(s 2) is visible, corresponding to trivial zeros of ζ(s) at negative even integers, and ρ ranges over all other, non-trivial zeros, known to be in the critical strip 0 < Re (s) < 1. The hard part of the proof (below) of Hadamard's theorem is essentially that of [Ahlfors 1953/1966], with various rearrangements. A somewhat different argument is in [Lang 1993]. Some standard folkloric proofs of supporting facts about harmonic functions are recalled.