For each $$m\ge 1$$ m ≥ 1 , there exists $$H=H(m)>0$$ H = H ( m ) > 0 such that the set $$\begin{aligned} \bigg \{\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { is prime and }[p+1,p+H]\text { contains at least }m\text { primes}\bigg \} \end{aligned}$$ { ϕ ( p + 1 ) ϕ ( p - 1 ) : p is prime and [ p + 1 , p + H ] contains at least m primes } is dense in $$[0,+\infty )$$ [ 0 , + ∞ ) , where $$\phi $$ ϕ denotes the Euler totient function. This gives a unconditional weak form of a recent result of Garcia, Luca, Shi and Udell, which was proved under the assumption of a conjecture of Dickson.

中文翻译：

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总商数和质数之间的小差距

对于每个 $$m\ge 1$$ m ≥ 1 ，存在 $$H=H(m)>0$$ H = H ( m ) > 0 使得集合 $$\begin{aligned} \bigg \ {\frac{\phi (p+1)}{\phi (p-1)}:\,p\text { 是素数且 }[p+1,p+H]\text { 至少包含 }m\ text { primes}\bigg \} \end{aligned}$$ { ϕ ( p + 1 ) ϕ ( p - 1 ) : p 是素数，[ p + 1 , p + H ] 包含至少 m 个素数 } 是密集的在 $$[0,+\infty )$$ [ 0 , + ∞ ) 中，其中 $$\phi $$ ϕ 表示欧拉整体函数。这给出了 Garcia、Luca、Shi 和 Udell 的最近结果的无条件弱形式，该结果在 Dickson 猜想的假设下得到证明。