Arithmetic and Asymptotic Properties of Restricted Totient Sums

Abstract

This article extends our previous study on the summatory behavior of Euler's totient function $\varphi(n)$. We investigate two complementary restricted sums, $\Upsilon(x,p)=\sum_{\substack{k\le x\\\gcd(k,p)=1}}\varphi(k)$ and $\Delta(x,p)=\sum_{\substack{k\le x\\p\mid k}}\varphi(k)$, which satisfy the decomposition $\Psi(x)=\sum_{k\le x}\varphi(k)=\Upsilon(x,p)+\Delta(x,p)$. We establish recurrence formulas, congruence relations, and generating function identities for $\Delta(x,p)$. In particular, we prove that $\Delta(x,p)\equiv 0\pmod{p-1}$ for every prime $p$, and we derive the asymptotic expansion $\Delta(x,p)=\dfrac{3}{\pi^{2}(p+1)}\,x^{2}+O(x\log x)$. Furthermore, we study average orders, connections with $\omega(n)$, and relations with divisor structures. These results refine the analytic understanding of totients in arithmetic progressions and complement the classical asymptotic theory of $\Psi(x)$.