A Modified Legendre Product for Primes in Consecutive Square Intervals

Let I_n = [n², (n+1)²] and d = 2n+1. We study the modified Legendre product

E(d) = d · ∏{p ≤ d/2} (1 - 1/p) · ∏{d/2 < p ≤ d} (1 - d/p²),

obtained by replacing the usual local density 1/p by d/p² in the upper prime range. This modification is not an upper-bound sieve: primes in (d/2, d] may have two multiples in I_n, and the new factor removes more density than the standard Legendre factor. Its role is instead that of a finite-range heuristic interpolant.

We show that E(d) has the same asymptotic Mertens–Buchstab bias as the standard Legendre product,

π(I_n) / E(d) → e^γ/2,

while changing the lower-order approach to that limit. A two-term asymptotic expansion predicts the computed mean ratio to within 5×10^{-4} from n = 10^4 to 10^9. An exhaustive census for 3 ≤ n ≤ 250,000 finds exactly 7,543 violations of π(I_n) ≤ E(d), the last at n = 77,433; we conjecture that no later violation occurs. The remaining asymptotic discrepancy is the classical difference between the Mertens density e^{-γ}/log z and the Buchstab density ω(2)/log z, with ω(2) = 1/2. Thus the modified product gives a striking finite-range fit, but no new sieve-theoretic mechanism for primes between consecutive squares.