On the exponent of the ideal class group of 𝑄(√-𝑑)

Abstract

Let m ( d ) m(d) be the exponent of the ideal class group of Q ( − d ) {\mathbf {Q}}(\sqrt { - d} ) , we establish the bound m ( d ) ≫ log ⁡ d log ⁡ log ⁡ d m(d) \gg \frac {{\log d}}{{\log \log d}} for almost all the discriminants d by using uniform asymptotic formulas on the number of n ≤ x n \leq x for which there exists a prime less than s for which n is a quadratic residue.