TY - JOUR

T1 - Maass forms and the mock theta function f(q)

AU - Ahlgren, Scott

AU - Dunn, Alexander

N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/8/6

Y1 - 2019/8/6

N2 - Let 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 α(n)=∑c≤nψ(n)+Oϵ(nϵ),where ψ(n) is an expression involving generalized Kloosterman sums and the I-Bessel function. Andrews conjectured that the series converges to α(n) when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews’ formula, and we also prove the second of these conjectures. Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on an average estimate for the Fourier coefficients of such forms which gives a power savings in the spectral parameter. As a further application of this result, we derive a formula which expresses α(n) with small error as a sum of exponential terms over imaginary quadratic points (this is similar in spirit to a recent result of Masri). We also obtain a bound for the size of the error term incurred by truncating Rademacher’s analytic formula for the ordinary partition function which improves a result of the first author and Andersen when 24 n- 23 is squarefree.

AB - Let 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 α(n)=∑c≤nψ(n)+Oϵ(nϵ),where ψ(n) is an expression involving generalized Kloosterman sums and the I-Bessel function. Andrews conjectured that the series converges to α(n) when extended to infinity, and that it does not converge absolutely. Bringmann and Ono proved the first of these conjectures. Here we obtain a power savings bound for the error in Andrews’ formula, and we also prove the second of these conjectures. Our methods depend on the spectral theory of Maass forms of half-integral weight, and in particular on an average estimate for the Fourier coefficients of such forms which gives a power savings in the spectral parameter. As a further application of this result, we derive a formula which expresses α(n) with small error as a sum of exponential terms over imaginary quadratic points (this is similar in spirit to a recent result of Masri). We also obtain a bound for the size of the error term incurred by truncating Rademacher’s analytic formula for the ordinary partition function which improves a result of the first author and Andersen when 24 n- 23 is squarefree.

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U2 - 10.1007/s00208-019-01829-0

DO - 10.1007/s00208-019-01829-0

M3 - Article

AN - SCOPUS:85064808273

SN - 0025-5831

VL - 374

SP - 1681

EP - 1718

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -