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Optimal estimates for an average of Hurwitz class numbers

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Abstract

In this paper, we give an optimal estimate of an average of Hurwitz class numbers. As an application, we give an equidistribution result of the family \(\Big \{\frac{t}{2q^{\nu /2}} \ | \ \nu \in {{\mathbb {N}}}, t \in {{\mathbb {Z}}}, |t|\leqslant 2q^{\nu /2}\Big \}\) with q prime, weighted by Hurwitz class numbers. This equidistribution produces many asymptotic relations among Hurwitz class numbers. Our proof relies on the resolvent trace formula of Hecke operators on elliptic cusp forms of weight \(k\geqslant 2.\)

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Acknowledgements

The authors would like to thank the referee for careful reading.

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Correspondence to Shingo Sugiyama.

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The first author was supported by Grant-in-Aid for Research Activity Start-up 18H05835. The second author was supported by Grant-in-Aid for Scientific Research (C) 15K04795.

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Sugiyama, S., Tsuzuki, M. Optimal estimates for an average of Hurwitz class numbers. Ramanujan J 52, 91–104 (2020). https://doi.org/10.1007/s11139-019-00146-z

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  • DOI: https://doi.org/10.1007/s11139-019-00146-z

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