Averages and moments associated to class numbers of imaginary quadratic fields
DR Heath-Brown, LB Pierce - Compositio Mathematica, 2017 - cambridge.org
DR Heath-Brown, LB Pierce
Compositio Mathematica, 2017•cambridge.orgFor any odd prime $\ell $, let $ h_ {\ell}(-d) $ denote the $\ell $-part of the class number of
the imaginary quadratic field $\mathbb {Q}(\sqrt {-d}) $. Nontrivial pointwise upper bounds
are known only for $\ell= 3$; nontrivial upper bounds for averages of $ h_ {\ell}(-d) $ have
previously been known only for $\ell= 3, 5$. In this paper we prove nontrivial upper bounds
for the average of $ h_ {\ell}(-d) $ for all primes $\ell\geqslant 7$, as well as nontrivial upper
bounds for certain higher moments for all primes $\ell\geqslant 3$.
the imaginary quadratic field $\mathbb {Q}(\sqrt {-d}) $. Nontrivial pointwise upper bounds
are known only for $\ell= 3$; nontrivial upper bounds for averages of $ h_ {\ell}(-d) $ have
previously been known only for $\ell= 3, 5$. In this paper we prove nontrivial upper bounds
for the average of $ h_ {\ell}(-d) $ for all primes $\ell\geqslant 7$, as well as nontrivial upper
bounds for certain higher moments for all primes $\ell\geqslant 3$.
Abstract
For any odd prime , let denote the -part of the class number of the imaginary quadratic field . Nontrivial pointwise upper bounds are known only for ; nontrivial upper bounds for averages of have previously been known only for . In this paper we prove nontrivial upper bounds for the average of for all primes , as well as nontrivial upper bounds for certain higher moments for all primes .
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