On the regularity of fractional integrals of modular forms

Pastor, Carlos

doi:10.1090/tran/7418

On the regularity of fractional integrals of modular forms
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Trans. Amer. Math. Soc. 372 (2019), 829-857 Request permission

Abstract:

In this paper we study some local and global regularity properties of Fourier series obtained as fractional integrals of modular forms. In particular we characterize the differentiability at rational points, determine their Hölder exponent everywhere (using several definitions), and compute the associated spectrum of singularities. We also show that these functions satisfy an approximate functional equation and use it to discuss the graphs of “Riemann’s example” and of fractional integrals of cusp forms for $\Gamma _0(N)$. We include some computer plots.

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