اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe reciprocal sum of divisors of Mersenne numbers
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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 $max_{nle x} sum_{p mid 2^n-1} 1/p$ to within $o(1)$ and $max_{nle x} sum_{dmid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $xtoinfty$. This refines, conditionally, earlier estimates of ErdH{o}s and ErdH{o}s-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 $nle x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $sum_{pmid 2^n-1} 1/p$ and $sum_{dmid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
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