Class number formulae for imaginary quadratic number fields \(\mathbb{Q} (\sqrt{-n})\) with \(n\) squarefree and \(n\equiv 1\pmod 4\) or \(n\equiv 2\pmod 4\) (Q1594960)

For the prime \(p\) satisfying \(p\equiv 3\pmod 4\), \textit{B. C. Berndt} and \textit{S. Chowla} showed that the Legendre symbol \((a/p)\) summed over certain subintervals of \((0,p)\) is equal to zero [Nord. Mat. Tidskr. 22, 5-8 (1974; Zbl 0281.10015)]. This result immediately leads to interesting class number formulae of \(\mathbb{Q}(\sqrt{-p})\) and is easily generalized to composite moduli \(n\equiv 3\pmod 4\). In this paper, for any positive, square-free integer \(n\), by using the Jacobi symbol \((-4n/a)\) in a subinterval of \((0,2n)\) the authors modify this method of Berndt and Chowla and obtain class number formulae relating values of the Jacobi symbol \((-4n/a)\) to the class number of \(\mathbb{Q}(\sqrt{-n})\) in either of the cases \(n\equiv 1\) or \(2\pmod 4\).