Congruences relating class numbers of quadratic orders and Zagier's sums

Abstract We prove a congruence modulo 16 relating the class numbers h ( − 4 p ) , h ( 16 p ) of quadratic orders and Zagier's sum m ( 4 p ) associated to 4 p , when p ≡ 1 (mod 4) is a prime. This gives an analogy to Chua-Gunby-Park-Yuan's congruence established when p ≡ 3 (mod 4), and generalizes a recent work by Cheng and Guo. In particular, when p ≡ 1 (mod 4) is a prime, it is shown that the class number h ( − 4 p ) is divisible by 16 if and only if the Zagier sum m ( 4 p ) is divisible by 16.