论文标题
默森纳数字的互惠总和
The reciprocal sum of divisors of Mersenne numbers
论文作者
论文摘要
我们调查了有关梅尔森数字$ 2^n-1 $的互惠总和或主要除数总和的各种问题。在Elliott-Halberstam的猜想和广义的Riemann假设上，我们确定$ \ max_ {n \ le x} \ sum_ {p \ mid 2^n-1} 1/p $ to $ o（1）$和$ \ max_和$ \ max_ { $ 1+o（1）$，为$ x \ to \ infty $。这是有条件地完善了ERDS和ERDőS-KISS-POMERANE的早期估计。有条件地（仅）GRH，我们还确定$ \ sum 1/d $在$ 1+o（1）$以内，其中$ d $在所有$ 2^n-1 $的数字上运行，对于某些$ n \ le x $。这有条件地证实了pomerance的猜想，并回答了Murty-Rosen-Silverman的问题。最后，我们证明$ \ sum_ {p \中间2^n-1} 1/p $和$ \ sum_ {d \中间2^n-1} 1/d $在概率数字理论的意义上允许连续分布函数。
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $n\le x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $\sum_{p\mid 2^n-1} 1/p$ and $\sum_{d\mid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
