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In this article, we find the full Fourier expansion for solutions of a certain differential equation. When such an f is fully automorphic these functions are referred to as generalized non-holomorphic Eisenstein series. We give a connection of the boundary condition on such Fourier series with convolution formulas on the divisor functions. Additionally, we discuss a possible relation with the differential Galois theory.
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Shifted convolution sums motivated by string theory
It was conjectured by physicists that a particular shifted sum of even divisor sums vanishes, and a formal argument was later given for this vanishing. Shifted convolution sums of this form appear when computing the Fourier expansion of coefficients for the low energy scattering amplitudes in type IIB string theory and have applications to subconvexity bounds of L-functions. In this article, we generalize the argument from and rigorously evaluate shifted convolution of the divisor functions. In doing so, we derive exact identities for these sums and conjecture particular identities similar to but different from the one originally found.
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- PAR ID:
- 10536430
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Journal of Number Theory
- Volume:
- 260
- Issue:
- C
- ISSN:
- 0022-314X
- Page Range / eLocation ID:
- 151 to 172
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jnt.2024.01.012
