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We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg-Kloosterman sums for discrete subgroups of $\PSL_2(\mathbb C)$ using Poincar\'e series., Comment: 14 pages

We prove exact identities for convolution sums of divisor functions of the form $\sum_{n_1 \in \mathbb{Z} \smallsetminus \{0,n\}}\varphi(n_1,n-n_1)\sigma_{2m_1}(n_1)\sigma_{2m_2}(n-n_1)$ where $\varphi(n_1,n_2)$ is a Laurent polynomial with logarithms for which the sum is absolutely convergent. Such identities are motivated by computations in string theory and prove and generalize a conjecture of Chester, Green, Pufu, Wang, and Wen from \cite{CGPWW}. Originally, it was suspected that such sums, suitably extended to $n_1\in\{0,n\}$ should vanish, but in this paper we find that in general they give Fourier coefficients of holomorphic cusp forms., Comment: 12 pages

In \cite{CGPWW2021}, it was conjectured that a particular shifted sum of even divisor sums vanishes, and in \cite{SDK}, a formal argument was 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 \cite{GMV2015} and have applications to subconvexity bounds of $L$-functions. In this article, we generalize the argument from~\cite{SDK} and rigorously evaluate shifted convolution of the divisor functions of the form $\displaystyle \sum_{\stackrel{n_1+n_2=n}{n_1, n_2 \in \mathbb{Z} \setminus \{0\}}} \sigma_{k}(n_1) \sigma_{\ell}(n_2) |n_1|^R $ and $\displaystyle \sum_{\stackrel{n_1+n_2=n}{n_1, n_2 \in \mathbb{Z} \setminus \{0\} }} \sigma_{k}(n_1) \sigma_{\ell}(n_2) |n_1|^Q\log|n_1| $ where $\sigma_\nu(n) = \sum_{d \divides n} d^\nu$. In doing so, we derive exact identities for these sums and conjecture that particular sums similar to but different from the one found in \cite{CGPWW2021} will also vanish.

We complete the program, initiated in a 2015 paper of Green, Miller, and Vanhove, of directly constructing the automorphic solution to the string theory $D^6 R^4$ differential equation $(\Delta-12)f=-E_{3/2}^2$ for $SL(2,\mathbb{Z})$. The construction is via a type of Poincar\'e series, and requires explicitly evaluating a particular double integral. We also show how to derive the predicted vanishing of one type of term appearing in $f$'s Fourier expansion, confirming a conjecture made by Chester, Green, Pufu, Wang, and Wen motivated by Yang-Mills theory.

In this article, we find the full Fourier expansion for the generalized non-holomorphic Eisenstein series for certain values of parameters. 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., Comment: 45 pages