# Maass forms and the mock theta function f(q)

@article{Ahlgren2019MaassFA, title={Maass forms and the mock theta function f(q)}, author={Scott Ahlgren and Alexander Dunn}, journal={Mathematische Annalen}, year={2019}, pages={1-38} }

Let $$f(q):=1+\sum _{n=1}^{\infty } \alpha (n)q^n$$f(q):=1+∑n=1∞α(n)qn be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form $$\begin{aligned} \alpha (n)= \sum _{c\le \sqrt{n}} \psi (n)+O_\epsilon \left( n^\epsilon \right) , \end{aligned}$$α(n)=∑c≤nψ(n)+Oϵnϵ,where $$\psi (n)$$ψ(n) is an expression involving generalized Kloosterman sums and the I-Bessel function. Andrews conjectured that the series converges to $$\alpha (n)$$α(n…

## 8 Citations

Uniform bounds for sums of Kloosterman sums of half integral weight

- MathematicsResearch in Number Theory
- 2018

For $$m,n>0$$m,n>0 or $$mn<0$$mn<0 we estimate the sums $$\begin{aligned} \sum _{c \le x} \frac{S(m,n,c,\chi )}{c}, \end{aligned}$$∑c≤xS(m,n,c,χ)c,where the $$S(m,n,c,\chi )$$S(m,n,c,χ) are…

Bounds for Coefficients of the $f(q)$ Mock Theta Function and Applications to Partition Ranks

- Mathematics
- 2020

Explicit subconvexity for $\mathrm{GL}_2$ and some applications

- Mathematics
- 2018

We make the subconvex exponent for $\mathrm{GL}_2$ cuspidal representation in the work of Michel \& Venkatesh explicit. The result depends on an effective dependence on the `fixed' $\mathrm{GL}_2$…

Explicit subconvexity for $\mathrm{GL}_2$

- Mathematics
- 2018

We make the subconvex exponent for $\mathrm{GL}_2$ cuspidal representation in the work of Michel \& Venkatesh explicit. The result depends on an effective dependence on the `fixed' $\mathrm{GL}_2$…

Congruences modulo powers of 5 for the rank parity function

- Mathematics
- 2021

It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang,…

On the distribution of modular square roots of primes

- Mathematics
- 2020

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be…

Hybrid subconvexity and the partition function

- Mathematics
- 2022

. We give an upper bound for the error term in the Hardy-Ramanujan-Rademacher formula for the partition function. The main input is a new hybrid subconvexity bound for the central value L ( 12 ,f × (…

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