The shifted-prime Erdős-Wintner law for primitive-root determinant densities: extremal order, dimension zero, and Fourier decay

Abstract

For a prime $p$, let $c(p)=\frac{\phi (p-1)}{p-1}{\prod }_{j\ge 1}(1-p^{-j})$, the limiting density of matrices over ${\mathbb{F}}_p$ with primitive-root determinant. Its limiting law over the primes is the classical continuous shifted-totient law on $[0,1/2]$. We prove Hausdorff dimension zero and vanishing lower and upper dyadic $L^q$ dimensions for $q>1$. Its image ${\mu }_f$ under $x\mapsto -\log x$ is Rajchman. As $T\to \infty$, for $U_T$ uniform on $[0,T]$, $\log |\widehat{{\mu }_f}(U_T)|/\log \log T\to -1$ in probability. For every $A>0$, $|\widehat{{\mu }_f}(\tau )|\le (\log \log T{)}^4/\log T$ outside a subset of $[0,T]$ of relative measure $O_A((\log T{)}^{-A})$. As $h\downarrow 0$, ${\sup }_a{\mu }_f([a,a+h])={\mathfrak{S}}_2{e}^{-\gamma }/\log (1/h)+O({\log }^{-2}(1/h))$, where ${\mathfrak{S}}_2$ is the twin-prime singular series; maximizing left endpoints lie within $h$ of $\log 3$ for small $h$. We prove ${\min }_{p\le x}c(p)\sim e^{-\gamma }/\log \log x$ and ${\mathrm{limsup}}_{p\to \infty }(c(p)\log \log p{)}^{-1}=e^{\gamma }$. The limiting law of $\log (\phi (p+1)/\phi (p-1))$ has support $\mathbb{R}$ and Hausdorff dimension zero. For the classical law of $\sigma (p-1)/(p-1)$ on $[3/2,\infty )$, we prove dimension zero, a sharp left-endpoint asymptotic, and a Rajchman logarithmic image. Its odd-prime component has an entire Mellin transform of order one. Partial-factorization bounds yield certified asymptotic searches for fully splitting negacyclic number-theoretic transform primes with prescribed reciprocal-density bounds at fixed power-of-two length. We determine the second distinct squared norm of $A_{n_1}\otimes \cdots \otimes A_{n_k}$ for $k,n_i\ge 2$, yielding exact cyclotomic codifferent shell gaps and a uniform smoothing asymptotic at $\epsilon =2^{-c\phi (m)}$ for $c>2{\log }_2(1+\sqrt{6})$. These results are unconditional. An explicit unproved exponent-pair hypothesis yields $|\widehat{{\mu }_f}(\tau )|=O(1/\log \log |\tau |)$.

Substantial revision: sharp minimum-density asymptotic; two-sided typical Fourier decay and stronger exceptional-set bounds; sharper $\sigma$-endpoint remainder; revised prior-work attribution and applications. Removed the standalone $\sigma$-extremal section