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Jay Jorgenson

Publications and source records attributed to Jay Jorgenson.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

Let $(X,\chi,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $\chi$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,\chi,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $\chi$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $\Delta_{2k}$ as well that of the scalar Laplacian $\Delta_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\det\Delta_{2k_n}$ for a sequence $(X_{n}, \chi_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\det\Delta_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_{\alpha}$ which depends only on $\alpha=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.

math.SP↗
arXiv2026

Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring

We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations.

math.SP↗
arXiv2025

An explicit construction of heat kernels and Green's functions in measure spaces

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $\sigma$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.

math.CA↗
arXiv2025

Determinants of twisted Laplacians and the twisted Selberg zeta function

Let $X$ be an orbisurface, meaning a compact hyperbolic Riemann surface possibly with a finite number of elliptic points, and let $X_1$ denote its unit tangent bundle. We consider the twisted Selberg zeta function $Z(s;\rho)$ associated to a representation $\rho: \pi_1(X_1) \to \text{GL}(V_\rho)$. We prove a relation between the twisted Selberg zeta function $Z(s;\rho)$ and the regularized determinant of the twisted Laplacian associated to $\rho$. These results can be viewed as a generalization of a result due to Sarnak who considered the trivial character. Yet our proof is different, as it is based on evaluation of the Laplace-Mellin type integral transformations. Going further, we explicitly compute the multiplicative constant, which we call the torsion factor, and express its dependence on parameters which determine the representation. We study the asymptotic behavior of the constant for a sequence of non-unitary representations introduced by Yamaguchi and prove that the asymptotic behavior of this constant as the dimension of the representation tends to infinity is the same as the behavior of the higher-dimensional Reidemeister torsion on $X_1$ (up to an absolute constant).

math.SP↗
arXiv2025

On a generating function of Niebur-Poincar\'e series

Let $\Gamma\subset PSL_2(\mathbb{R})$ be a Fuchsian group of the first kind which has a cusp $i\infty$ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincar\'e series $\{F_{m,s}(\tau)\}_{m\ge 1}$ associated to $i\infty$. We prove a relation between the continuation of this generating function to $s=1$ with the resolvent kernel associated to the hyperbolic Laplacian and the non-holomorphic Eisenstein series associated to $i\infty$, also at $s=1$. Secondly, we show that, for any $s\in \mathbb{N}$, the generating function equals Poincar\'e type series involving polylogarithms. We also consider a generating function formed with derivatives in $s$ of the Niebur--Poincar\'e series and prove that the continuation of the generating function at $s=1$ can be expressed in terms of $\Gamma$-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series.

math.NT↗
arXiv2024

The discrete analogue of the Gaussian

This paper illustrates the utility of the heat kernel on $\mathbb{Z}$ as the discrete analogue of the Gaussian density function. It is the two-variable function $K_{\mathbb{Z}}(t,x)=e^{-2t}I_{x}(2t)$ involving a Bessel function and variables $x\in\mathbb{Z}$ and real $t\geq 0$. Like its classic counterpart it appears in many mathematical and physical contexts and has a wealth of applications. Some of these will be reviewed here, concerning Bessel integrals, trigonometric sums, hypergeometric functions and asymptotics of discrete models appearing in statistical and quantum physics. Moreover, we prove a new local limit theorem for sums of integer-valued random variables, obtain novel special values of the spectral zeta function of Bethe lattices, and provide a discussion on how $e^{-2t}I_{x}(2t)$ could be useful in differential privacy.

math-ph↗
arXiv2024

Constructing heat kernels on infinite graphs

Let $G$ be an infinite, edge- and vertex-weighted graph with certain reasonable restrictions. We construct the heat kernel of the associated Laplacian using an adaptation of the parametrix approach due to Minakshisundaram-Pleijel in the setting of Riemannian geometry. This is partly motivated by the wish to relate the heat kernels of a graph and a subgraph, or of a domain and a discretization of it. As an application, assuming that the graph is locally finite, we express the heat kernel $H_G(x,y;t)$ as a Taylor series with the lead term being $a(x,y)t^r$, where $r$ is the combinatorial distance between $x$ and $y$ and $a(x,y)$ depends (explicitly) upon edge and vertex weights. In the case $G$ is the regular $(q+1)$-tree with $q\geq 1$, our construction reproves different explicit formulas due to Chung-Yau and to Chinta-Jorgenson-Karlsson. Assuming uniform boundedness of the combinatorial vertex degree, we show that a dilated Gaussian depending on any distance metric on $G$, which is uniformly bounded from below can be taken as a parametrix in our construction. Our work extends in part the recent articles [LNY21, CJKS23] in that the graphs are infinite and weighted.

math.AP↗
arXiv2024

On the functional equation of twisted Ruelle zeta function and Fried's conjecture

Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $\chi$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,\chi)$ be the associated Ruelle zeta function, and $\varphi(s,\chi)$ the determinant of the scattering matrix. We prove the functional equation that $R(s,\chi)\varphi(s,\chi) = R(-s,\chi)\varphi(s,\chi)H(s,\chi)$ where $H(s,\chi)$ is a meromorphic function of order one explicitly determined using the topological data of $M$ and of $\chi$, and the trigonometric function $\sin(s)$. From this, we determine the order of the divisor of $R(s,\chi)$ at $s=0$ and compute the lead coefficient in its Laurent expansion at $s=0$. When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of $R(0,\chi)$.

math.NT↗
arXiv2023

The parametrix construction of the heat kernel on a graph

In this paper we develop the parametrix approach for constructing the heat kernel on a graph $G$. In particular, we highlight two specific cases. First, we consider the case when $G$ is embedded in a Eulidean domain or manifold $\Omega$, and we use a heat kernel associated to $\Omega$ to obtain a formula for the heat kernel on $G$. Second, we consider when $G$ is a subgraph of a larger graph $\widetilde{G}$, and we obtain a formula for the heat kernel on $G$ from the heat kernel on $\widetilde{G}$ restricted to $G$.

math.AP↗
arXiv2023

The resolvent kernel on the discrete circle and twisted cosecant sums

Let $X_m$ denote the discrete circle with $m$ vertices. For $x,y\in X_{m}$ and complex $s$, let $G_{X_m,\chi_{\beta}}(x,y;s)$ be the resolvent kernel associated to the combinatorial Laplacian which acts on the space of functions on $X_{m}$ that are twisted by a character $\chi_{\beta}$. We will compute $G_{X_m,\chi_{\beta}}(x,y;s)$ in two different ways. First, using the spectral expansion of the Laplacian, we show that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a generating function for certain trigonometric sums involving powers of the cosecant function; by choosing $\beta$ or $s$ appropriately, the sums in question involve powers of the secant function. Second, by viewing $X_{m}$ as a quotient space of $\mathbb{Z}$, we prove that $G_{X_m,\chi_{\beta}}(x,y;s)$ is a rational function which is given in terms of Chebyshev polynomials. From the existence and uniqueness of $G_{X_m,\chi_{\beta}}(x,y;s)$, these two evaluations are equal. From the resulting identity, we obtain a means by which one can obtain explicit evaluations of cosecant and secant sums. The identities we prove depend on a number of parameters, and when we specialize the values of these parameters we obtain several previously known formulas. Going further, we derive a recursion formula for special values of the $L$-functions associated to the cycle graph $X_{m}$, thus answering a question from arXiv:2212.13687v1.

math.CO↗
arXiv2022

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

In this article we develop a general method by which one can explicitly evaluate certain sums of $n$-th powers of products of $d\geq 1$ elementary trigonometric functions evaluated at $\mathbf{m}=(m_1,\ldots,m_d)$-th roots of unity. Our approach is to first identify the individual terms in the expression under consideration as eigenvalues of a discrete Laplace operator associated to a graph whose vertices form a $d$-dimensional discrete torus $G_{\mathbf{m}}$ which depends on $\mathbf{m}$. The sums in question are then related to the $n$-th step of a Markov chain on $G_{\mathbf{m}}$. The Markov chain admits the interpretation as a particular random walk, also viewed as a discrete time and discrete space heat diffusion, so then the sum in question is related to special values of the associated heat kernel. Our evaluation follows by deriving a combinatorial expression for the heat kernel, which is obtained by periodizing the heat kernel on the infinite lattice $\mathbb{Z}^{d}$ which covers $G_{\mathbf{m}}$.

math.CO↗
arXiv2022

Discrete diffusion-type equation on regular graphs and its applications

We derive an explicit formula for the fundamental solution $K_{T_{q+1}}(x,x_{0};t)$ to the discrete-time diffusion equation on the $(q+1)$-regular tree $T_{q+1}$ in terms of the discrete $I$-Bessel function. We then use the formula to derive an explicit expression for the fundamental solution $K_{X}(x,x_{0};t)$ to the discrete-time diffusion equation on any $(q+1)$-regular graph $X$. Going further, we develop three applications. The first one is to derive a general trace formula that relates the spectral data on $X$ to its topological data. Though we emphasize the results in the case when $X$ is finite, our method also applies when $X$ has a countably infinite number of vertices. As a second application, we obtain a closed-form expression for the return time probability distribution of the uniform random walk on any $(q+1)$-regular graph. The expression is obtained by relating $K_{X}(x,x_{0};t)$ to the uniform random walk on a $(q+1)$-regular graph. We then show that if $\{X_{h}\}$ is a sequence of $(q+1)$-regular graphs whose number of vertices goes to infinity and which satisfies a certain natural geometric condition, then the limit of the return time probability distributions from $\{X_{h}\}$ is equal to the return time probability distribution on the tree $T_{q+1}$. As a third application, we derive formulas which express the number of distinct closed irreducible walks without tails on a finite graph $X$ in terms of moments of the spectrum of its adjacency matrix.

math.PR↗
arXiv2021

An integer factorization algorithm which uses diffusion as a computational engine

In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be considered as being comparable to a quantum step for an algorithm which runs on a quantum computer. With this, we prove that our factorization algorithm uses at most $O((\log N)^{2})$ deterministic steps and at most $O((\log N)^{2})$ diffusion steps with an implied constant which is effective. By comparison, Shor's algorithm is known to use at most $O((\log N)^{2}\log (\log N) \log (\log \log N))$ quantum steps on a quantum computer. As an example of our algorithm, we simulate the diffusion computer algorithm on a desktop computer and obtain factorizations of $N=33$ and $N=1363$.

quant-ph↗
arXiv2021

Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formulas

Let $X$ be a smooth, compact, projective Kähler variety and $D$ be a divisor of a holomorphic form $F$, and assume that $D$ is smooth up to codimension two. Let $ω$ be a Kähler form on $X$ and $K_{X}$ the corresponding heat kernel which is associated to the Laplacian that acts on the space of smooth functions on $X$. Using various integral transforms of $K_{X}$, we will construct a meromorphic function in a complex variable $s$ whose special value at $s=0$ is the log-norm of $F$ with respect to $μ$. In the case when $X$ is the quotient of a symmetric space, then the function we construct is a generalization of the so-called elliptic Eisenstein series which has been defined and studied for finite volume Riemann surfaces.

math.NT↗
arXiv2021

Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space

In Cogdell et al., \it LMS Lecture Notes Series \bf 459, \rm 393--427 (2020), \rm the authors proved an analogue of Kronecker's limit formula associated to any divisor $\mathcal D$ which is smooth in codimension one on any smooth Kähler manifold $X$. In the present article, we apply the aforementioned Kronecker limit formula in the case when $X$ is complex projective space $\CC\PP^n$ for $n \geq 2$ and $\mathcal D$ is a hyperplane, meaning the divisor of a linear form $P_D({z})$ for ${z} = (\mathcal{Z}_{j}) \in \CC\PP^n$. Our main result is an explicit evaluation of the Mahler measure of $P_{D}$ as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the $L^{2}$-norm of the vector of coefficients of $P_{D}$.

math.NT↗
arXiv2021

Kronecker limit functions and an extension of the Rohrlich-Jensen formula

In 1984 Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert$ of a $\text{\rm PSL}(2,\ZZ)$ modular form $f$ in terms of the Dedekind Delta function evaluated at the divisor of $f$. Recently, Bringmann-Kane re-interpreted the Rohrlich-Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert$ and extended the result to compute a regularized inner product of $\log \Vert f \Vert$ with what amounts to powers of the Hauptmoduli of $\text{\rm PSL}(2,\ZZ)$. In the present article, we revisit the Rohrlich-Jensen formula and prove that it can be viewed as a regularized inner product of special values of two Poincaré series, one of which is the Niebur-Poincaré series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group $Γ$ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when $Γ$ is the full modular group $\text{\rm PSL}(2,\ZZ)$, thus reproving the theorems from \cite{BK19}; and second when $Γ$ is an Atkin-Lehner group $Γ_{0}(N)^+$, where explicit computations are given for certain genus zero, one and two levels.

math.NT↗
arXiv2020

Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encode the spectrum of $Δ$ in such a way that they can be used to define the regularized determinant of $Δ-z(1-z)I$. The resulting formula for the regularized determinant of $Δ-z(1-z)I$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, which could not be seen in previous works, due to a different definition of the regularized determinant.

math.NT↗
arXiv2019

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.

math.NT↗