论文标题
连续素数之间存在很大差异
On large differences between consecutive primes
论文作者
论文摘要
我们表明，$$ \ sum _ {\ supAck {p_n \ in [x，2x] \\ p_ {n+1} - p_n \ ge x^{1/2}}}}}}}（p_ {n+1} - p_n） - p_n） - p_n） [x，2x] \\ p_ {n+1} - p_n \ ge x^{0.45}}}}}}}}（p_ {n+1} - p_n）\ ll x^{0.63+ε}，$ p_n $ p_n $是$ n $ n $ n $ th $ th $ th prime。证明将希思·布朗（Heath-Brown）最近的作品与哈曼（Harman）的筛子结合在一起，改善并扩大了他的结果。我们将结果应用于代表性函数，素数数字的二进制数字以及通过乘法函数的真实近似。
We show that $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{1/2}}} (p_{n+1} - p_n) \ll x^{0.57+ε}$$ and $$\sum_{\substack{p_n \in [x, 2x] \\ p_{n+1} - p_n \ge x^{0.45}}} (p_{n+1} - p_n) \ll x^{0.63+ε},$$ where $p_n$ is the $n$th prime number. The proof combines Heath-Brown's recent work with Harman's sieve, improving and extending his results. We give applications of the results to prime-representing functions, binary digits of primes and approximation of reals by multiplicative functions.
