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arxiv: 2606.28657 · v1 · pith:LVAK7BG2new · submitted 2026-06-27 · 🧮 math.NT

Riesz Means of Quadratic Class Numbers

Olivia Beckwith , Tushar Karmakar This is my paper

Pith reviewed 2026-06-30 09:15 UTC · model grok-4.3

classification 🧮 math.NT
keywords Riesz meansquadratic class numbersHurwitz class numberssesquiharmonic Maass formsmock modular formsL-functionsweight 1/2 forms
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The pith

An asymptotic formula holds for the weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to prove an asymptotic formula for a weighted Riesz mean that combines Hurwitz class numbers with the class numbers of real quadratic fields. The proof proceeds by defining L-functions attached to weight one-half sesquiharmonic Maass forms of moderate growth and establishing a general identity for the Riesz means of the associated generalized mock modular forms. This identity extends an earlier result that covered only ordinary mock modular forms. The general identity is then specialized to a sesquiharmonic Maass form constructed by Duke, Imamoglu and Toth. A sympathetic reader would care because class numbers record the arithmetic size of quadratic fields and their averaged sums often encode information about L-functions and distribution questions in number theory.

Core claim

By constructing L-functions for weight 1/2 sesquiharmonic Maass forms of moderate growth and proving a Riesz-mean formula for the corresponding generalized mock modular forms, an asymptotic formula is obtained for the weighted sum of Hurwitz class numbers H(n) and the class numbers of real quadratic fields, applied to the Duke-Imamoglu-Toth form.

What carries the argument

L-functions for weight 1/2 sesquiharmonic Maass forms of moderate growth, which produce the Riesz-mean formula for generalized mock modular forms.

If this is right

  • The weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers admits an explicit main term plus a smaller error term.
  • The Riesz-mean identity extends from ordinary mock modular forms to generalized mock modular forms arising from sesquiharmonic Maass forms.
  • Specialization to the Duke-Imamoglu-Toth form produces a concrete asymptotic involving the indicated class numbers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same L-function construction could be tested on other moderate-growth sesquiharmonic forms to obtain Riesz means for different arithmetic sequences.
  • If the moderate-growth hypothesis can be verified more broadly, the method may connect Riesz means of class numbers to other mean-value problems in the theory of Maass forms.

Load-bearing premise

The Riesz-mean formula for generalized mock modular forms applies without further analytic continuation when the input is the specific sesquiharmonic Maass form of Duke, Imamoglu and Toth.

What would settle it

Numerical computation of the weighted Riesz mean up to a large cutoff N; the computed value deviates from the predicted main term by more than the claimed error term.

read the original abstract

We prove an asymptotic formula for a weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers. To do this, we introduce L-functions for weight $\frac {1}{2} $ sesquiharmonic Maass forms of moderate growth and prove a formula for the Riesz means of the corresponding generalized mock modular forms, generalizing a recent result of the first author with Diamantis, Gupta, Rolen, and Thalagoda for mock modular forms. We then apply this formula to a sesquiharmonic Maass form that was first introduced by Duke, Imamo\={g}lu, and T\'{o}th.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proves an asymptotic formula for a weighted Riesz mean of Hurwitz class numbers and real quadratic class numbers. It introduces L-functions for weight 1/2 sesquiharmonic Maass forms of moderate growth, establishes a Riesz-mean formula for the associated generalized mock modular forms (generalizing prior work on mock modular forms), and applies the result to the Duke-Imamoğlu-Tóth sesquiharmonic Maass form.

Significance. If the claims hold, the work extends spectral methods and mock-modular-form techniques to sesquiharmonic Maass forms, producing new asymptotics for class numbers. The generalization of the Riesz-mean identity constitutes a technical advance in the analytic theory of these forms.

major comments (1)
  1. [Application to the Duke-Imamoğlu-Tóth form] Application to the Duke-Imamoğlu-Tóth form: the Riesz-mean identity and the L-function construction both require that the specific Duke-Imamoğlu-Tóth sesquiharmonic Maass form satisfy the moderate-growth hypothesis used to define the L-functions. The manuscript provides no explicit growth estimate, bound, or reference confirming that this form meets the hypothesis without additional analytic continuation.
minor comments (1)
  1. [Abstract and §1] The abstract and introduction should clarify the precise weight and growth conditions under which the new L-functions are defined, to avoid ambiguity with existing constructions for harmonic Maass forms.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting this point about the application to the Duke-Imamoğlu-Tóth form. We address the comment below.

read point-by-point responses

  1. Referee: Application to the Duke-Imamoğlu-Tóth form: the Riesz-mean identity and the L-function construction both require that the specific Duke-Imamoğlu-Tóth sesquiharmonic Maass form satisfy the moderate-growth hypothesis used to define the L-functions. The manuscript provides no explicit growth estimate, bound, or reference confirming that this form meets the hypothesis without additional analytic continuation.

    Authors: We agree that an explicit confirmation would strengthen the presentation. The Duke-Imamoğlu-Tóth sesquiharmonic Maass form is constructed in their 2013 paper via the spectral theory of Maass forms on the modular surface and satisfies the moderate-growth condition by virtue of its definition as a weight-1/2 form of moderate growth (see Duke-Imamoğlu-Tóth, "A sesquiharmonic Maass form and the class number generating function", Theorem 1 and the growth estimates in Section 3). Nevertheless, the manuscript does not cite this fact explicitly. We will add a sentence in the application section (near the statement of the main theorem) referencing the relevant growth result from Duke-Imamoğlu-Tóth and confirming that the form meets the moderate-growth hypothesis required for the L-function construction and Riesz-mean identity. This revision will be made without altering any proofs. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation introduces new objects and generalizes external result

full rationale

The paper defines new L-functions attached to weight 1/2 sesquiharmonic Maass forms of moderate growth, proves a Riesz-mean identity for the associated generalized mock modular forms (explicitly generalizing a cited theorem of Beckwith-Diamantis-Gupta-Rolen-Thalagoda), and applies the identity to the Duke-Imamoglu-Toth form. No equation in the supplied abstract or description reduces the target asymptotic to a parameter fitted inside the paper or to a quantity defined in terms of the conclusion itself. The moderate-growth hypothesis is an input assumption rather than a derived output, and the self-citation is to a prior independent theorem rather than a load-bearing uniqueness claim. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on standard properties of Maass forms and mock modular forms from prior literature, plus the new construction of L-functions whose analytic properties are asserted but not detailed in the abstract.

axioms (1)
  • domain assumption Weight 1/2 sesquiharmonic Maass forms of moderate growth admit associated L-functions with the expected functional equation and growth properties needed for the Riesz-mean formula.
    Invoked when the authors introduce the L-functions and apply the general formula to the Duke-Imamoglu-Toth form.

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