The main result of this paper is as follows. Let n_o ∈ Z^+ be a period of the covering function σ (x)= 1. If d∈Z^+ divides some n_(l≤i≤k) and d|u_o,then the number of the n_i divided by d is at least d/(d,n.)where the notation (m,n) denotes the greatest common divicsor of m and n. In the case σ (x)≡1 it gives an improvement to the well-known Znám-Newman theorem,it is also better than the already announced result of Berger,Felzenbaum and Fraenkel.
设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.
