Complete Monotonicity, Laplace Transforms, and Refined Turán‐Type Inequalities for Shifted Ratios of the Riemann Zeta Function

We first establish a general positivity principle for functions whose logarithms possess Dirichlet‐series expansions with nonnegative coefficients. If log  Z ( x ) = ∑ _{n ≥2} c _n n ^{− x} , c _n ≥ 0, then, for r , s > 0, the shifted cross‐ratio Q _{r , s} ( x ) = ( Z ( x ) Z ( x + r + s ))/( Z ( x + r ) Z ( x + s )) has a positive logarithmic Dirichlet expansion, a positive discrete Laplace representation, is strictly completely monotonic and admits a Dirichlet series with nonnegative coefficients. The same framework yields monotonicity in the shifts, separate concavity of log Q _{r , s} , and positive mixed shift derivative. Applying the principle to the Riemann zeta function gives log R _{r , s} ( x ) = ∑ _p ∑ _{k ≥1} ((1 − p ^{− k r} )(1 − p ^{− k s} )/ k ) p ^{− k x} . We identify the first three asymptotic terms of both log R _{r , s} and R _{r , s} − 1, including the 4 ^{− x} contribution. For the associated Turán difference Δ _{r , s} ( x ) = ζ ( x ) ζ ( x + r + s ) − ζ ( x + r ) ζ ( x + s ), we derive the explicit coefficients d _N ( r , s ) = (1/2)∑ _{d ∣ N} ( d ^{− r} − ( N / d ) ^{− r} )( d ^{− s} − ( N / d ) ^{− s} ) > 0. This gives, in one step, a positive Dirichlet series, a positive discrete Laplace representation, strict complete monotonicity, finite‐truncation lower bounds, and an arbitrary‐order Dirichlet asymptotic hierarchy for Δ _{r , s} .