Speaker: COL John Dvorak  ( Army University at Fort Leavenworth )

Title: Prime-Based Corrections to Riemann Zero Counting via Truncated Euler Products

Abstract: The Riemann explicit formula expresses prime counting functions as sums over zeta zeros, a cornerstone of analytic number theory since 1859. We demonstrate the computational viability of the reverse direction: using primes to approximate $S(T) = (1/\pi)\arg\zeta(1/2+iT)$, the oscillatory correction term in the zero counting function $N(T)$, via truncated Euler products. The Euler product expansion yields $S(T) = -(1/\pi)\sum_p \sum_{k=1}^{\infty} (1/k) p^{-k/2} \sin(kT\log p)$, which diverges on the critical line but exhibits slow heuristic divergence $\sim \sqrt{\log\log P_{\max}}$ under random-phase cancellation, enabling optimal finite truncation.  Performance is highly heterogeneous: 41\% of test heights require only 1,000 primes for substantial improvement (``ultra-easy''), while 7\% exhibit catastrophic failure regardless of truncation. We introduce a three-factor logistic regression model (proximity to zeros, signal amplitude $|S(T)|$, and truncation parameter) to account for these performance disparities.  This work provides a novel large-scale empirical validation of computational prime-zero duality, complementing Gonek's theoretical conditions and LeClair's ultra-high zero computations with systematic performance quantification and practical guidelines.