Global Growth of the Riemann Xi Function
Abstract
The classical pole-removed xi function satisfies a uniform exponential bound on the complex plane.
Theorem 1.1 (A uniform log-linear growth bound).
Proof. Machine-checked in Lean as D5/S3/Analytic/Resolvent/XiGlobalGrowth.xi_reading_norm_le_exp_log_linear (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. Jeffrey C. Lagarias (2007). Li Coefficients for Automorphic L-Functions. DOI: 10.5802/aif.2311.
Commentary.
There is one positive real constant C such that the displayed bound holds for every complex s. Here xi is the classical Riemann xi function, with xi(0) = xi(1) = 1/2. Its exponent is C times (1 + norm(s)) times (1 + log(1 + norm(s))).
Lagarias’s order-one theorem is compatible with this estimate. The proof uses the symmetric theta-tail Mellin integral for the pole-removed completion. Continuity controls the compact initial interval, while exponential theta decay controls the remaining half-line. A logarithmic estimate bounds both complex powers uniformly in s by an integrable exponential majorant. Integration and the polynomial xi factor then give the asserted constant.
The estimate is unconditional and includes both endpoints. Since 1 + log(r) is at most 3 times r^(1/2) for r at least one, it also gives the exp(C r^(3/2)) bound used by the normalized resolvent. No identity between Li coefficients and an infinite zero sum is asserted.