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On correlations between class numbers of imaginary quadratic fields

Volume 185 / 2018

V. Vinay Kumaraswamy Acta Arithmetica 185 (2018), 211-231 MSC: Primary 11E25; Secondary 11R29, 11P55. DOI: 10.4064/aa170319-13-12 Published online: 6 July 2018


Let $h(-n)$ be the class number of the imaginary quadratic field with fundamental discriminant $-n$. We establish an asymptotic formula for correlations involving $h(-n)$ and $h(-n-l)$, over fundamental discriminants that avoid the congruence class $1\pmod{8}$. Our result is uniform in the shift $l$, and the proof uses an identity of Gauss relating $h(-n)$ to representations of integers as sums of three squares. We also prove analogous results on correlations involving $r_Q(n)$, the number of representations of an integer $n$ by an integral positive definite quadratic form $Q$.


  • V. Vinay KumaraswamySchool of Mathematics
    University of Bristol
    Bristol, BS8 1TW, UK

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