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Holomorphic projection for sesquiharmonic Maass forms

Michael Allen, Olivia Beckwith, Vaishavi Sharma

Abstract

We study the holomorphic projection of mixed mock modular forms involving sesquiharmonic Maass forms. As a special case, we numerically express the holomorphic projection of a function involving real quadratic class numbers multiplied by a certain theta function in terms of eta quotients. We also analyze certain shifted convolution $L$-series involving mock modular forms and bound certain shifted convolution sums.

Holomorphic projection for sesquiharmonic Maass forms

Abstract

We study the holomorphic projection of mixed mock modular forms involving sesquiharmonic Maass forms. As a special case, we numerically express the holomorphic projection of a function involving real quadratic class numbers multiplied by a certain theta function in terms of eta quotients. We also analyze certain shifted convolution -series involving mock modular forms and bound certain shifted convolution sums.

Paper Structure

This paper contains 13 sections, 9 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Modular forms
  4. Holomorphic unary theta functions
  5. Holomorphic Projection
  6. Harmonic Maass forms
  7. Sesquiharmonic Maass forms
  8. Shifted convolution sums
  9. Holomorphic projection calculation
  10. Lemmas
  11. Fourier calculation of projection of mixed sesquiharmonic forms
  12. Proof of Theorem \ref{['thm:AAS-projection']}
  13. Example

Key Result

Theorem 1.1

Let $\chi$ be an odd Dirichlet character modulo $m$. The function $\sum_{h=1}^{\infty} r_{\chi}(h) q^h$ belongs to the space $S_2( \Gamma_0(4m^2))$.

Figures (2)

  • Figure 1: Values of $\sum_{m \ll X} H(m^2-14) m\chi_4(m)/X^{5/4}$, where each $x$-value in the plot corresponds to $X = (2x+1)^2$. Note that this suggests a stronger bound than the $4/3+\epsilon$ given above.
  • Figure 2: The numerical column gives approximations to the nearest $10,000^{th}$ for $r_{\chi_4}(k)$ using the holomorphic projection computed in \ref{['thm:proj-computation']}, the expected column gives $r_{\chi_4}(k)$ computed as a linear combination of Fourier coefficients of modular forms as in the right-hand side of \ref{['eq:approximated-projection']}.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 2.1: Proposition 6.2 GrossZagier
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • proof
  • Theorem 2.6: Theorem 2 AhlgrenAndersenSamart
  • proof
  • proof
  • ...and 12 more