Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Waibel, Fabian

doi:10.1007/s11139-017-9934-1

Cited by 10 publications

(10 citation statements)

References 19 publications

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“…Then, θpQ, zq ´θpspn Q, zq P U K Ď S 3{2 pN, χq. For gpzq " ř n bpnqepnzq P U K , we have by [27,Theorem 1] for n " tv 2 w 2 with squarefree t and pw, N q " 1 that…”

mentioning

confidence: 99%

Uniform bounds for norms of theta series and arithmetic applications

Waibel¹

2020

Preprint

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“…Then, θpQ, zq ´θpspn Q, zq P U K Ď S 3{2 pN, χq. For gpzq " ř n bpnqepnzq P U K , we have by [27,Theorem 1] for n " tv 2 w 2 with squarefree t and pw, N q " 1 that…”

mentioning

confidence: 99%

Uniform bounds for norms of theta series and arithmetic applications

Waibel¹

2020

Preprint

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“…For the theta-multiplier, Iwaniec [Iwa87, Theorem 3] obtained a bound for the sums K pN q µ pn; xq averaged over the level. This was extended to the case of twists by a quadratic character by Waibel [Wai17,§3] (this is also implicit in the work of Duke [Duk88]). Combining these results, we have Proposition 5.2.…”

Section: Three Estimates For the Coefficients Of Maass Formsmentioning

confidence: 98%

Maass forms and the mock theta function f(q)

Ahlgren

Dunn

2019

Math. Ann.

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Let f pqq :" 1`ř 8 n"1 αpnqq n be the well-known third order mock theta of Ramanujan. In 1964, George Andrews proved an asymptotic formula of the form αpnq "where ψpnq is an expression involving generalized Kloosterman sums and the I-Bessel function. Andrews conjectured that the series converges to αpnq 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 αpnq 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 24n´23 is squarefree.

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“…see e.g. [73,Lemma 4] in the case n = m, the general case being analogous. In order to simplify the notation we assume that D 1 , D 2 are fundamental discriminants.…”

Section: A Relative Trace Formula For Pairs Of Heegner Periodsmentioning

confidence: 99%

A symplectic restriction problem

Blomer¹,

Corbett

2021

Math. Ann.

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We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

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Fourier coefficients of half-integral weight cusp forms and Waring’s problem

Abstract: Abstract. Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N . As an application, we consider a Waring-type problem with sums of mixed powers.

Cited by 10 publications

References 19 publications

Uniform bounds for norms of theta series and arithmetic applications

Uniform bounds for norms of theta series and arithmetic applications

Maass forms and the mock theta function f(q)

A symplectic restriction problem

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