# Effective Congruences for Mock Theta Functions

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## Abstract

**:**

## 1. Introduction and Statement of the Results

**Theorem**

**1.**

**Remarks**

**1.**

- Theorem 1 is a special case of Theorem 5 in Section 3, a more general result that applies to a weight of 1/2 harmonic Maass forms, whose holomorphic parts have algebraic coefficients and whose non-holomorphic parts are period integrals of a weight of $3/2$ unary theta series. The next section will set up all the notation and preliminary results to state and prove the general theorem, as well as how Theorem 1 follows from it.
- The other computable parameters will be described toward the end of Section 3. Briefly, they involve computing the level of a certain half-integral weight modular form from the work of Treneer [2] as well as the order of vanishing at the cusps; the constants from the results of Lichtenstein [1]; and, if we do not assume the Generalized Riemann Hypothesis, the constant of Lagarias, Montgomery and Odlyzko [19].

## 2. Nuts and Bolts

#### 2.1. Harmonic Maass Forms

- For every $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \Gamma $, we have:$$f\left(\frac{az+b}{cz+d}\right)={\left(\frac{c}{d}\right)}^{2k}{\epsilon}_{d}^{-2k}\phantom{\rule{0.166667em}{0ex}}\chi (d)\phantom{\rule{0.166667em}{0ex}}{(cz+d)}^{k}\phantom{\rule{0.166667em}{0ex}}f(z)$$$${\epsilon}_{d}:=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{0.277778em}{0ex}}d\equiv 1\phantom{\rule{3.33333pt}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\hfill \\ i\hfill & \mathrm{if}\phantom{\rule{0.277778em}{0ex}}d\equiv 3\phantom{\rule{3.33333pt}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}4\hfill \end{array}\right.$$
- We have ${\Delta}_{k}f=0$.
- There is a polynomial, ${P}_{f}={\sum}_{n\ge 0}{c}^{+}(n){q}^{n}\in {q}^{-1}]$, such that: $f(z)-{P}_{f}(z)=O({e}^{-\u03f5y})$ as $y\to +\infty $ for some $\u03f5>0$. Analogous conditions are required at all cusps.

#### 2.2. Elements of the Proof

**Remarks**

**2.**

**Lemma**

**2.**

**Theorem**

**3.**

**Theorem**

**4.**

**Remarks**

**3.**

**Remarks**

**4.**

## 3. Statement of the General Theorem and Its Proof

**Theorem**

**5.**

- (i)
- Let Q be an odd prime with $\left(\frac{{\delta}_{i}}{Q}\right)=1$ for $1\le i\le h$. Then, we have:$$\widehat{f}=\sum \widehat{a}(n){q}^{n}:=\sum _{\left(\frac{-n}{Q}\right)=-1}{a}^{+}(n){q}^{n}\in {M}_{\frac{1}{2}}^{!}({\Gamma}_{0}(4N{Q}^{3}),\chi )$$
- (ii)
- Define $\alpha =\alpha (\widehat{f},\ell )$ and $\beta =\beta (\widehat{f},\ell )$, as in Equation (2.2). Then, there exists a cusp form:$$g\in {S}_{\frac{1}{2}+{\ell}^{\beta}\left(\frac{{\ell}^{2}-1}{2}\right)}({\Gamma}_{0}(4N{Q}^{3}{\ell}^{2}),\chi {\psi}_{\ell}^{\alpha})$$$$g\equiv \sum _{\ell \nmid n}\widehat{a}({\ell}^{\alpha}n){q}^{n}\phantom{\rule{10.0pt}{0ex}}(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}{\ell}^{j})$$$$\widehat{a}({p}^{3}{\ell}^{\alpha}n)\equiv 0\phantom{\rule{10.0pt}{0ex}}(\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}{\ell}^{j})$$
- (iii)
- Define: $S:={S}_{{\ell}^{\beta}({\ell}^{2}-1)}({\Gamma}_{0}(2N{Q}^{3}{\ell}^{2})),$ and let $B=B(S,{\ell}^{j})$ and $L=L(S,\ell )$, as given in Theorem 4 above. Then, the smallest prime, p, for which Equation (3.1) holds satisfies:$$p\le 2{({L}^{B-1}{B}^{B})}^{{A}_{1}}$$$$p\le 280{B}^{2}{(logB+logL)}^{2}$$

- To prove ($ii$), apply Theorem 3 to the weakly holomorphic modular form, $\widehat{f}$.
- Statement ($iii$) is immediate from Theorem 4. ☐

**Remarks**

**5.**

**Remarks**

**6.**

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Andersen, N.; Friedlander, H.; Fuller, J.; Goodson, H.
Effective Congruences for Mock Theta Functions. *Mathematics* **2013**, *1*, 100-110.
https://doi.org/10.3390/math1030100

**AMA Style**

Andersen N, Friedlander H, Fuller J, Goodson H.
Effective Congruences for Mock Theta Functions. *Mathematics*. 2013; 1(3):100-110.
https://doi.org/10.3390/math1030100

**Chicago/Turabian Style**

Andersen, Nickolas, Holley Friedlander, Jeremy Fuller, and Heidi Goodson.
2013. "Effective Congruences for Mock Theta Functions" *Mathematics* 1, no. 3: 100-110.
https://doi.org/10.3390/math1030100