Two sign corrections in the explicit four-point correlation formula of Conrey and Snaith

Conrey and Snaith derived, from the ratios conjecture, explicit formulas for the n-point correlation functions of the zeros of the Riemann zeta function, with their lower-order arithmetic terms. We show that two signs in their explicit four-point formula are inconsistent with the general definitions of the same paper. In the arithmetic factor A*, eq. (206) and its copy (211) of arXiv:0803.2795v1 (eqs. (5.40) and (5.45) of the published version), the bracket 1 + Y₁ + Y₂ should read 1 − Y₁ − Y₂; and the first term of eq. (202) (published (5.36)) should carry a plus sign. The first correction follows from the definitions by a residue computation, the second from the cumulant structure of the prime sums. With the printed bracket the Euler product defining A* diverges, and in any truncation to finitely many primes the four-point function acquires a double pole where two of its arguments coincide. With both corrections the formula reproduces, to rounding, the four-point function of a model with three primes, which can be computed exactly. Code for all checks is included in this record (sign_corrections_cs08_source_code.zip).