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Kim Klinger-Logan

Publications and source records attributed to Kim Klinger-Logan.

10 recordsLinked to original sources
arXiv2025

Convolution identities for complex-indexed divisor functions and modular graph functions

We find exact identities for sums of the form \begin{equation*}\label{eq:convsumabs} \sum_{\stackrel{n_1+n_2 = n}{n_1 \in \mathbb{Z} \setminus \{ 0, n \} }} Q(n_1,n_2) \sigma_{-r_1}(n_1) \sigma_{-r_2}(n_2), \end{equation*} where $n\in\mathbb{N}$, $r_1,r_2\in\mathbb{C}$, $Q$ is a combination of hypergeometric functions, and $\sigma_{a}(x)$ denotes the divisor function. Specifically, we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their $L$-values. This result expands upon previous work with Radchenko in which such identities were found for divisor functions with even integer index \cite{FKLR} and encompasses results of Jacobi \cite{motohashi1994binary} and Diamantis and O'Sullivan in \cite{diamantis2010kernels, o2023identities} for divisor functions with odd integer index. The proof of our result expresses these sums in terms of Estermann zeta functions and uses trace formulae. In addition, we use a regularization of divergent convolution sums to provide a mathematical explanation for $L$-values (non-critical in the sense of Deligne) appearing in modular graph functions \cite{DKS2021_2}.

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arXiv2025

A Dedekind-Rademacher cocycle for Bianchi groups

We construct a generalization of the Dedekind-Rademacher cocycle to congruence subgroups of $\mathrm{SL}_2(\mathbb C)$, and derive some of its basic properties. In particular, we show that it parametrizes a family of $L$-values and prove the integrality of these values.

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arXiv2023

Convolution identities for divisor sums and modular forms

We prove exact identities for convolution sums of divisor functions of the form $\sum_{n_1 \in \mathbb{Z} \smallsetminus \{0,n\}}φ(n_1,n-n_1)σ_{2m_1}(n_1)σ_{2m_2}(n-n_1)$ where $φ(n_1,n_2)$ 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.

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arXiv2023

Shifted convolution sums motivated by string theory

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\}}} σ_{k}(n_1) σ_{\ell}(n_2) |n_1|^R $ and $\displaystyle \sum_{\stackrel{n_1+n_2=n}{n_1, n_2 \in \mathbb{Z} \setminus \{0\} }} σ_{k}(n_1) σ_{\ell}(n_2) |n_1|^Q\log|n_1| $ where $σ_ν(n) = \sum_{d \divides n} d^ν$. 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.

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arXiv2022

The $D^6 R^4$ interaction as a Poincar\'e series, and a related shifted convolution sum

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 $SL(2,\Z)$. The construction is via a type of Poincar\'e series, and requires explicitly evaluating a particular double integral. We also show how to use double Dirichlet series to formally 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 (and later proved rigorously by Fedosova, Klinger-Logan, and Radchenko using the Gross-Zagier Holomorphic Projection Lemma.).

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arXiv2022

Whittaker Fourier type solutions to differential equations arising from string 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.

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arXiv2021

A spectral interpretation of zeros of certain functions

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 $\text{Re}(s)<1/2$, real-valued on $\R$, $\frac{h(1-s)}{h(s)}\ll |s|^{1-ε}$ 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 $h(s)=ξ(2s)$, the completed zeta-function. We use spectral theory suggested by results of Lax-Phillips and Colin de Verdière. This simplifies ideas of W. Müller, J. Lagarias, M. Suzuki, H. Ki, O. Velásquez Castañón, D. Hejhal, L. de Branges and P.R. Taylor.

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arXiv2021

Linear Operators, the Hurwitz Zeta Function and Dirichlet $L$-Functions

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity \cite{HilbertProb}. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a {\it non}-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies \cite{RHequiv}. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other $L$-functions does converge.

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arXiv2018

Differential equations in automorphic forms

Physicists such as Green, Vanhove, et al show that differential equations involving automorphic forms govern the behavior of gravitons. One particular point of interest is solutions to $(Δ-λ)u=E_α E_β$ on an arithmetic quotient of the exceptional group $E_8$. We establish that the existence of a solution to $(Δ-λ)u=E_αE_β$ on the simpler space $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ for certain values of $α$ and $β$ depends on nontrivial zeros of the Riemann zeta function $ζ(s)$. Further, when such a solution exists, we use spectral theory to solve $(Δ-λ)u=E_αE_β$ on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$ and provide proof of the meromorphic continuation of the solution. The construction of such a solution uses Arthur truncation, the Maass-Selberg formula, and automorphic Sobolev spaces.

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