Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications

Abstract

We introduce a novel, continuous geometric paradigm to reframe the distribution of prime numbers through the variational optimization of a smooth parameter surface. By relaxing the rigid integer constraints of traditional series truncations, we construct a remainder-free parametric operator surface $\mathrm{li}(n;\tau )$ mapped uniformly across the continuous coordinate domain $(1,\infty )\times {\mathbb{R}}^{+}$. Global $C^3$ multi-variable regularity is established via an invariant septic blending polynomial $\varphi (x)=35x^4-84x^5+70x^6-20x^7$, completely neutralizing the artificial step-discontinuities and derivative kinks of standard piecewise fractional interpolations. By enforcing the strict prime truncation identity $\mathrm{li}(n;{\tau }_{\pi }(n))=\pi (n)$, we demonstrate that the discrete, non-smooth arithmetic spectrum of the prime-counting function can be mapped uniquely onto an unbroken, continuous truncation ${\tau }_{\pi }(n)$. By transferring Littlewood's unconditional oscillation bounds and applying the Riemann--von Mangoldt formula to the truncation functions $\tau (n)$. We establishing the critical line as the unique, absolute global variational minimum. Exploiting this optimization constraint, we show via proof by contradiction that any hypothetical off-line zero cluster (${\beta }_0\ne 1/2$) would mathematically forbid the truncation parameter from remaining positive and real-valued across the infinite scale horizon. To preserve the real-valued existence of the manifold, all non-trivial zeroes are shown to be constrained to this absolute global minimum, forcing $\beta =1/2$ identically and establishing variational evidence supporting a proof of the Riemann Hypothesis (RH). *NOTE: This manuscript Does Not claim to prove the RH. Details on this are covered in the conclusion section.

Total of 27 pages. NOTE: Provides un-conditional evidence to support the Riemann Hypothesis; however it is Not a proof of the RH. Although, it could provide a path to one

References in corpus (3)

• Estimates of Some Functions Over Primes without R.H.
• On Riemann's Paper, "On the Number of Primes Less Than a Given Magnitude"
• Extension of Algebraic Solutions Using The Lambert W Function