广义Lehmer数的渐近行为（英文）

Asymptotic behavior of the number of generalized Lehmer numbers

整数a称为模p的Lehmer数是指1≤a≤p-1且a+a^(-1)为奇数,其中a^(-1)表示a模p的逆.令M_p为模p的Lehmer数的个数.1994年,张证明了■.设整数c≥2,整数d∈[0,c-1].对每个素数p≡1(mod c),如果a+a^(-1)≡d(mod c),则称整数a为关于模p的(c,d)-Lehmer数.令M_(c,d,p)表示模p的(c,d)-Lehmer数的个数.本文得到■,推广了张的结果.

A Lehmer number modulo an odd prime p is an integer a with 1≤a≤p-1 and a+a-1 is odd.Let M p denote the number of Lehmer number modulo p,Zhang has shown that M p=p-1/2+O(p^1/2 log^3p)in 1994.Let c≥2 be an positive integer,d∈[0,c-1]be an integer.For every p≡1(mod c),a is called to be a(c,d)-Lehmer number modulo p,if a+a-1≡d(mod c).Let M c,d,p denote the number of(c,d)-Lehmer number modulo p.In this paper,it is proved that M c,d,p=p-1 c+O(p^1/2 log^2p).If c=2,d=1,the Zhang’s formula is recovered.