Let $N_{k} = 2 \cdot 3 \dots p_{k}$ and ${a_{1} < a_{2} < \dots < a_{ϕ (N_{k})}}$ be the integers $< N_{k}$ which are relatively prime to $N_{k}$ . Then, for any $c \geq 0$ , the limit $\frac{# { a _{i} - a _{i - 1} \leq c \frac{N _{k}}{ϕ ( N _{k} )} : 2 \leq i \leq ϕ ( N _{k} )}}{ϕ ( N _{k} )}$ exists and is a continuous function of $c$ .

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number theory · solved · 0 attempts

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