Let $B$ be a set of positive integers such that $\sum_{b \in B} 1 / \varphi(b) < +\infty$, where $\varphi(\cdot)$ is the Euler's totient function. For any $y > 0$ put $$\delta_y := \limsup_{x \to +\infty} \frac{\#\{p \leq x : \exists b \in B,\; b > y,\; p \equiv 1 \pmod b\}}{x / \log x} ,$$ where $p$ runs over the prime numbers and $\pi(\cdot)$ is the prime counting function. By the Dirichlet theorem on prime in arithmetic progressions, the primes $p \leq x$ such that $p \equiv 1 \pmod b$ are $$\sim \frac1{\varphi(b)} \frac{x}{\log x}$$ therefore I guess that it should be $$\delta_y \ll \sum_{\substack{b \in B \\ b > y}} \frac1{\varphi(b)} ,$$ and in particular $\delta_y \to 0$ for $y \to +\infty$.

*My questions are*: (1) Is it true that $\lim_{y \to +\infty} \delta_y = 0$ ? (2) If so, can we give an upper bound like $\delta_y \ll f(y)$, for some function $f(y)$ ? (3) If not, what other hypotheses on $B$ are required in order to have $\lim_{y \to +\infty} \delta_y = 0$?

I tried to estimate from above $\delta_y$ with the Brun–Titchmarsh theorem, but the problem is that $b$ could be of the order of $x$.

Thanks for any help.