Let q be a sufficiently large integer and X be a Dirichlet character modulo q. In this paper, we extend the product x（-1）=-1 L（1, X） with prime q, arising from the Kummer conjecture, to the products of some general Dirichlet series, and give some meaningful estimates for them.
Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c be two integers with （n, q） = （c, q） = 1. We denote by rn（51, 52, C; q） （δ 〈 δ1,δ2≤ 1） the number of all pairs of integers a, b satisfying ab ≡ c（mod q）, 1 〈 a ≤δ1q, 1 ≤ b≤δ2q, （a,q） = （b,q） = 1 and nt（a＋b）. The main purpose of this paper is to study the asymptotic properties of rn （δ1, δ2, c; q）, and give a sharp asymptotic formula for it.
Let p be an odd prime, c be an integer with （c,p） = 1, and let N be a positive integer withN ≤ p - 1. Denote by r（N, c;p） the number of integers a satisfying 1 ≤ a ≤ N and 2 a ＋ a, where a is an integer with 1 ≤a≤ p - 1, aa ≡ c （mod p）. It is well known that r（N, c;p） = 1/2N ＋ O（p1/2log2p）.The main purpose of this paper is to give an asymptotic formula for ∑p-1 c=1（τ（N,c;p）-1/2N）2.
