关于1,1/2,…,1/n的一类初等对称函数的2-adic赋值  

On the 2-adic valuation of a class of elementary symmetric functions of 1,1/2,…,1/n

作　　者：邱敏 林宗兵 谭千蓉 QIU Min;LIN Zong-Bing;TAN Qian-Rong(School of Science,Xihua University,Chengdu 610039,China;School of Mathematics and Computer Science,Panzhihua University,Panzhihua 617000,China)

机构地区：[1]西华大学理学院,成都610039 [2]攀枝花学院数学与计算机学院,攀枝花617000

出　　处：《四川大学学报（自然科学版）》2023年第1期13-18,共6页Journal of Sichuan University(Natural Science Edition)

基　　金：西华大学校人才引进项目(RZ2000001366)。

摘　　要：设n和k为正整数且n≥k.本文考虑关于1,1/2,…,1/n的第k次初等对称函数H(n,k)∑(i 1,...,i_(k))∈Z^(k)1≤i_(1)<…<i_(k)≤n 1/i_(1)…i_(k)的2-adic赋值.设p为素数.2015年,Lengyel证明v_(p)(H(n,k))>-k log_(p)n+O_(k)(1),其中v_(p)(H(n,k))表示H(n,k)的p-adic赋值,O_(k)(1)表示一个依赖于k的常数.2017年,Leonetti和Sanna猜想:对所有足够大的正整数n,总存在一个正的常数c=c(p,k),使得v_(p)(H(n,k))<-clogn,并对不超过x的正整数n证明了当n的p-adic表示是以k-1的p-adic表示为起始值时,除了至多3x^(0.835)个例外之外此猜想是正确的.本文给出了H(n,2)的2-adic赋值的确切值或下界,部分验证了上述猜想.For any integers n≥k≥1,the k-th elementary symmetric function of 1,1/2,…,1/n,denoted by H(n,k),is defined by H(n,k)∑(i_(1),…,i_(k))∈Z^(k) 1≤i_(1)<…<i_(k)≤n 1/i_(1)…i_(k).Let p be a prime.In 2015,Lengyel proved that v_(p)(H(n,k))>-k log_(p)n+O_(k)(1),where v_(p)(H(n,k))represents the p-adic valuation of H(n,k)and O_(k)(1)denotes a constant depending on k.In 2017,Leonetti and Sanna conjectured that there exists a positive constant c=c(p,k)such that v_(p)(H(n,k))<-c log n for all large n and proved this conjecture affirmativly for all n≤x whose base p representations start with the base p representation of k-1 but at most 3x^(0.835) exceptions.In this paper,we give the exact value or a lower bound of the 2-adic valuation of H(n,2),thus gives a partial support to above conjecture.

关 键 词：初等对称函数 2-adic赋值 p-adic分析 

分 类 号：O156.1[理学—数学]