Search

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.

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., 45 pages

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 $(Δ-12)f=-E_{3/2}^2$ for $SL(2,\mathbb{Z})$. The construction is via a type of Poincaré 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.

We prove that all the zeros of certain meromorphic functions are on the critical line $\text{Re}(s)=1/2$, and are simple (except possibly when $s=1/2$). We prove this by relating the zeros to the discrete spectrum of an unbounded self-adjoint operator. Specifically, we show for $h(s)$ a meromorphic function with no zeros in $\text{Re}(s)>1/2$ and no poles in $\text{Re}(s)1/2$ and $\frac{h(1-s)}{h(s)}\notin L^2(1/2+i\R)$, the only zeros of $h(s)\pm h(1-s)$ are on the critical line. One instance of such a function $h$ is $h(s)=\xi(2s)$, the completed zeta-function. We use spectral theory suggested by results of Lax-Phillips and Colin de Verdi\`{e}re. This simplifies ideas of W. M\"{u}ller, J. Lagarias, M. Suzuki, H. Ki, O. Vel\'{a}squez Casta\~{n}\'{o}n, D. Hejhal, L. de Branges and P.R. Taylor., Comment: 13 pages; Thm 2 and converse results removed