We systematically derive congruences for the sums ∑_(j=1)~([kp/M」) 1/j~2 modulo p and for the sums ∑_(j=1)~([kp/M]) 1/j modulo p~2, for all integers M ≥ 1 that divide 24 and integers k with 1 ≤ k ≤ M and gcd(M,k) = 1. While many of these congruences are well-known,otheεs are new in the forms given. The congruences involve Fermat quotients, Euler numbers, Bernoulli polynomials, and some particular classes of generalized Bernoulli numbers belonging to quadratic characters.
展开▼
