Quantitative Sato--Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH

# Quantitative Sato--Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH **Version:** v3.0 **Specific Version DOI:** 10.5281/zenodo.22112048 **Author:** Byoungwoo Lee **Series relation:** Major-version successor to *Effective Entropy--Spectral Sato--Tate v2.2c: Quantitative Convergence under Symmetric--Power Analytic Inputs* (previous specific-version DOI: 10.5281/zenodo.17959752). ## Description Let $E/\mathbf{Q}$ be a fixed non-CM elliptic curve. This version rebuilds the earlier effective entropy--spectral Sato--Tate framework around a direct arithmetic localization theorem for the literal moving prime interval $(X,X+h]$. The central result is a finite-depth hard-window localization principle. For angular cutoff $M$, edge width $\Delta$, and window length $h$, assuming RH for $\zeta(s)$ and GRH for $L(s,\operatorname{Sym}^m E)$ only for $1\le m\le M$, the paper proves $$\sup_{I\subseteq[0,\pi]}\left|\mu_{X,h}^{\sharp}(I)-\mu_{\mathrm{ST}}(I)\right|\ll_E\frac{1}{M}+M\mathcal R_M(X,h,\Delta),$$ where $$\mathcal R_M(X,h,\Delta)=\frac{\sqrt X}{h}\,\mathcal L_{M,\Delta}\left(1+\log\frac{h}{\Delta}\right)+\frac{\Delta}{h}+\frac{h}{X\log X}.$$ Here $\mathcal L_{M,\Delta}$ is logarithmic in the conductor, $M$, and $X/\Delta$. In the near-square-root regime $$h=\sqrt X(\log X)^A,\qquad A>1,$$ choosing $$\Delta=\sqrt X\log X,\qquadM\asymp \frac{(\log X)^{(A-1)/2}}{\sqrt{\log\log X}}$$ gives $$\sup_{I\subseteq[0,\pi]}\left|\mu_{X,h}^{\sharp}(I)-\mu_{\mathrm{ST}}(I)\right|\ll_{E,A}\frac{\sqrt{\log\log X}}{(\log X)^{(A-1)/2}}.$$ The proof uses a two-scale plateau in the prime variable. Its Mellin transform has transition heights $X/h$ and $X/\Delta$, leading to low-, middle-, and high-zero ranges in the symmetric-power explicit formula. The hard interval is recovered by Brun--Titchmarsh control of the edge layers. Secondary consequences include a truncated character-energy estimate and a heat-regularized relative-entropy estimate. The raw empirical measure is atomic, so no unsmoothed KL divergence is asserted. After the radial $\mathrm{SU}(2)$ heat semigroup is applied, Parseval gives an exact $\chi^2$ identity and the heat kernel itself controls the high modes. ## Major changes from v2.2c Version v3.0 is a major mathematical rebuild rather than a minor revision. - Replaces the global empirical measure $p\le X$ by the literal moving hard prime interval $(X,X+h]$.- Replaces abstract low/high-mode and Rankin--Selberg packages by a direct finite-depth symmetric-power explicit formula.- Makes the analytic depth explicit: angular resolution through $M$ uses symmetric-power GRH only through degree $M$.- Introduces independent window and edge parameters $h$ and $\Delta$, with Mellin transition scales $X/h$ and $X/\Delta$.- Proves the near-square-root specialization $h=\sqrt X(\log X)^A$ for every fixed $A>1$.- Treats entropy only after explicit heat regularization; no artificial arithmetic evolution and no arithmetic high-frequency tail hypothesis are used.- Supplies a rigorous contour closure, absolute convergence of the zero sum, bad-prime coefficient control, and explicit trivial-zero bookkeeping. ## Relation to companion work The companion preprint *Mixed Frobenius Character Energies for Joint Chebotarev--Sato--Tate Data: Normally Monomial Realizations and Heat-Smoothed Entropy Decay*, Version v0.6r2, DOI 10.5281/zenodo.21743631, develops product-character energy and fixed-heat entropy for joint Chebotarev--Sato--Tate data. The present v3.0 paper is independent of that paper's automorphic-induction machinery; the overlap is the character/Parseval heat-regularization viewpoint. ## Claim boundary This work does **not** claim a new class of explicit-formula method or the first theorem on short-interval Sato--Tate. The claimed contribution is the finite-depth hard-window localization theorem, its near-square-root specialization under symmetric-power GRH, and the secondary character-energy and heat-entropy consequences. ## Suggested citation note For reproducibility, cite the specific v3.0 Version DOI **10.5281/zenodo.22112048**. The Zenodo Concept DOI may instead be used when citing the evolving series as a whole.