论文标题
在Hölder地图和主要差距上
On Hölder maps and prime gaps
论文作者
论文摘要
令$ p_n $表示$ n $ th prime，并考虑$ 1/n \ mapsto 1/p_n $的功能，将正整数的倒数映射到Primes的倒数上。我们表明，此功能的Hölder连续性等同于连续素数之间差距的Cramér类型估算的参数化家族。在这里，参数化来自Hölder指数。特别是，我们表明Cramér的猜想等于地图$ 1/n \ mapsto 1/p_n $是Lipschitz。另一方面，我们表明，反地图$ 1/p_n \ mapsto 1/n $是所有订单的Hölder，但不是Lipshitz，这与Cramér的猜想无关。
Let $p_n$ denote the $n$th prime, and consider the function $1/n \mapsto 1/p_n$ which maps the reciprocals of the positive integers bijectively to the reciprocals of the primes. We show that Hölder continuity of this function is equivalent to a parameterised family of Cramér type estimates on the gaps between successive primes. Here the parameterisation comes from the Hölder exponent. In particular, we show that Cramér's conjecture is equivalent to the map $1/n \mapsto 1/p_n$ being Lipschitz. On the other hand, we show that the inverse map $1/p_n \mapsto 1/n$ is Hölder of all orders but not Lipshitz and this is independent of Cramér's conjecture.
