Rank Amplification for Shifted Equal Values of Euler's Totient Function

Abstract

Let $S_h^\varphi(x)$ denote the number of integers $n\le x$ for which $\varphi(n)=\varphi(n+h)$. For the unit shift, we prove $S_1^\varphi(x)\ll x\exp{-(1/2-o(1))\sqrt{\log x,\log_2 x}}$. More generally, put $A=\log_3 x+\log_4 x-\log 2$, $G=\sqrt{\log x,A}$, and $V=\log x/G$. For every fixed integer $J\ge 1$, uniformly for $1\le h\le \exp{G/\sqrt{J}}$, we obtain $S_h^\varphi(x)=D_{h,>Y_J}^\varphi(x)+O_J(x\exp{-\sqrt{J},G+o_J(V)})$, where $Y_J=\exp{\sqrt{J},G}$. Here $D_{h,>Y_J}^\varphi(x)$ is the above-cutoff part of the classical Graham--Holt--Pomerance same-support family; it is empty for odd $h$. A moving choice $J\asymp \log_2 x/\log_3 x$ gives the unit-shift estimate and an analogous decomposition for a uniform range of shifts. The proof combines the smooth-totient theorem of Banks--Friedlander--Pomerance--Shparlinski with labelled supplier systems, a shifted divisor convolution, and an injective encoding of large supplier products into weighted friable tuples. The results of this paper have been formally verified in Lean.