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University of Iowa

1. Czarnecki, Kyle Jeffrey. Resonance sums for Rankin-Selberg products.

Degree: PhD, Mathematics, 2016, University of Iowa

Consider either (i) f = f1 f2 for two Maass cusp forms for SLm(ℤ) and SLm′(ℤ), respectively, with 2 ≤ m ≤ m′, or (ii) f= f1 f2 f3 for three weight 2k holomorphic cusp forms for SL2(ℤ). Let λf(n) be the normalized coefficients of the associated L-function L(s, f), which is either (i) the Rankin-Selberg L-function L(s, f1 ×f2), or (ii) the Rankin triple product L-function L(s, f1 ×f2 ×f3). First, we derive a Voronoi-type summation formula for λf (n) involving the Meijer G-function. As an application we obtain the asymptotics for the smoothly weighted average of λf (n) against e(αnβ), i.e. the asymptotics for the associated resonance sums. Let ℓ be the degree of L(s, f). When β = 1/ℓ and α is close or equal to ±ℓq 1/ℓ for a positive integer q, the average has a main term of size |λf (q)|X 1/2ℓ+1/2 . Otherwise, when α is fixed and 0 < β < 1/ℓ it is shown that this average decays rapidly. Similar results have been established for individual SLm(ℤ) automorphic cusp forms and are due to the oscillatory nature of the coefficients λf (n). Advisors/Committee Members: Ye, Yangbo (supervisor).

Subjects/Keywords: publicabstract; exponential sums; Fourier-Whittaker; Meijer G-function; Rankin-Selber; resonance sums; Mathematics

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APA (6th Edition):

Czarnecki, K. J. (2016). Resonance sums for Rankin-Selberg products. (Doctoral Dissertation). University of Iowa. Retrieved from

Chicago Manual of Style (16th Edition):

Czarnecki, Kyle Jeffrey. “Resonance sums for Rankin-Selberg products.” 2016. Doctoral Dissertation, University of Iowa. Accessed August 04, 2020.

MLA Handbook (7th Edition):

Czarnecki, Kyle Jeffrey. “Resonance sums for Rankin-Selberg products.” 2016. Web. 04 Aug 2020.


Czarnecki KJ. Resonance sums for Rankin-Selberg products. [Internet] [Doctoral dissertation]. University of Iowa; 2016. [cited 2020 Aug 04]. Available from:

Council of Science Editors:

Czarnecki KJ. Resonance sums for Rankin-Selberg products. [Doctoral Dissertation]. University of Iowa; 2016. Available from: