Arithmetic limiting laws over shifted primes: dimension zero, Fourier decay, and absolute continuity
Vipin Singh Sehrawat
For primes $p$, let $c(p)=\frac{\varphi(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$. Let $μ_G$ be its classical shifted-totient limiting law over primes and $μ_f$ its image under $x\mapsto-\log x$. Both laws have Hausdorff dimension zero, packing dimension one on every positive-mass Borel set, and almost-everywhere lower and upper local dimensions $(0,1)$. The four totient and divisor-sum laws over integers and shifted primes, and their logarithmic images, have lower and upper Shannon information dimensions in $(0.5503615,0.570489495)$. For convolution powers $μ_f^{*k}$, both information dimensions tend to one as $k\to\infty$, while each finite power has Hausdorff dimension zero. $μ_G$ gives full mass to absolutely normal Liouville numbers. Both Fourier transforms are $O((\log|τ|)^{-1/3}(\log\log|τ|)^{1/3})$. The empirical laws of $c(p)$ and $\varphi(p-1)/(p-1)$ for $p\le x$ have sharp Kolmogorov order $1/\log x$ in fixed reduced progressions; over unrestricted primes, they and their logarithmic images converge in $1$-Wasserstein distance at rate $O_B((\log x)^{-B})$ for every fixed $B>0$, possibly with ineffective constants. For $μ_f$, we obtain concentration expansions to every fixed inverse-logarithmic order, quantitative maximizer localization near $\log3$, and Fourier resonance bounds. For $U_T$ uniform on $[0,T]$, $(\log T)\widehat{μ_f}(U_T)$ converges in distribution to a circularly symmetric random Euler product, with convergence of modulus moments for every fixed real $s>-1$. Strongly additive functions with prime values $(\log\ell)^{-A}$ have absolutely continuous integer and shifted-prime laws for $A>0$. We give certified number-theoretic transform prime searches. A lattice appendix determines second and third distinct nonzero squared norms of type-$A$ tensor products and gives cyclotomic smoothing bounds. These results are unconditional.
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