Arithmetic and Asymptotic Properties of Restricted Totient Sums

This article extends our previous study on the summatory behavior of Euler’s totient function $\phi (n)$. We investigate two complementary restricted sums, $\mathrm{{\rm Y}}(x,p)={\sum }_{\begin{array}{c}k\le x\\ \gcd (k,p)=1\end{array}}\phi (k)$ and $\mathrm{\Delta }(x,p)={\sum }_{\begin{array}{c}k\le x\\ p\mid k\end{array}}\phi (k)$, which satisfy the decomposition $\mathrm{\Psi }(x)={\sum }_{k\le x}\phi (k)=\mathrm{{\rm Y}}(x,p)+\mathrm{\Delta }(x,p)$. We establish recurrence formulas, congruence relations, and generating function identities for $\mathrm{\Delta }(x,p)$. In particular, we prove that $\mathrm{\Delta }(x,p)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}p-1)$ for every prime $p$, and we derive the asymptotic expansion $\mathrm{\Delta }(x,p)={\displaystyle \frac{3}{{\pi }^2(p+1)}}{x}^2+O(x\log x)$. 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 $\mathrm{\Psi }(x)$.