Hölder regularity of arithmetic Fourier series arising from modular forms

2013/11/04 by Izabela Petrykiewicz, Petrykiewicz, Izabela
Mathematics · #11J70 #26A15 #65T60 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Number Theory (math.NT) #Primary 42A16 #Secondary 11F03 #advanced mathematical theories #math.CA #math.NT #msc:11F03 #msc:11J70 #msc:26A15 #msc:42A16 #msc:65T60

paper · pdf · doi:10.48550/arxiv.1311.0655

19 pages, added references, improved results

openalex publication_date 2013/11/04 · arxiv created 2014/05/22 · arxiv updated 2014/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a modular form which is not a cusp form M_k(z)=∑_n=0^∞r_ne^2πinz of weight k ≥ 4, we define the series M_k,s(x)=∑_n=1^∞(r_n)/(n^s)sin(2πnx), which converges for all x∈ℝ when s>k. In this paper, we compute the Hölder regularity exponent of M_k,s at irrational points. In our analysis we apply wavelets methods proposed by Jaffard in 1996 in the study of the Riemann series. We find that the Hölder regularity exponent at a point x is related to the fine diophantine properties of x, in a very precise way.

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