We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an iterated diffusion process on a finite graph. Concretely, a diffusion step is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an ℓ^1–normalized state vector, followed by an optional readout of selected coordinates. Let N≥ 3 be an odd integer which is neither prime nor a prime power, and let b∈(ℤ/Nℤ)^∗ have odd multiplicative order r= ord_N(b). We construct, without knowing r in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup ⟨ b⟩ and whose edges correspond to the powers b^± 2^t for t≤⌊log_2 N⌋+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 conceptual and model-based. We replace unitary ℓ^2 evolution by Markovian ℓ^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.

Let (X,χ,k) be a triple consisting of a smooth, compact hyperbolic Riemann surface X of genus g, and an m dimensional unitary multiplier system χ 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,χ,k). The error term we obtain is explicit with effectively computable constants which depend solely on the genus of X, the dimension of χ, the length of shortest geodesic on X and the smallest non-zero eigenvalues of the weighted Laplacian Δ_2k as well that of the scalar Laplacian Δ_0. Our second result studies the asymptotic behavior of the spectral determinant _2k_n for a sequence (X_n, χ_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_2k_n/vol(X_n) converges to a constant C_α which depends only on α=lim_n→∞ 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.

Let G be an infinite, edge-weighted and vertex-weighted graph, with certain reasonable restrictions which will be described in the paper. 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 kernel of a graph and any one of its subgraphs, or the wish to have an explicit expression of the heat kernel related to Gaussian-type estimates for any graph metric bounded from below. 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. We discuss several applications of parametrix construction. For example, 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 >= 1, our construction reproves different explicit formulas in papers by Chung-Yau, Cowling-Meda-Setti, and Chinta-Jorgenson-Karlsson.

Let Γ⊂ PSL_2(ℝ) be a Fuchsian group of the first kind which has a cusp i∞ of width one. In this paper, we first consider a generating function formed with the Niebur–Poincaré series {F_m,s(τ)}_m≥ 1 associated to i∞. 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∞, also at s=1. Secondly, we show that, for any s∈ℕ, the generating function equals Poincaré type series involving polylogarithms. We also consider a generating function formed with derivatives in s of the Niebur–Poincaré series and prove that the continuation of the generating function at s=1 can be expressed in terms of Γ-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series.

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;ρ) associated to a representation ρ: π_1(X_1) →GL(V_ρ). We prove a relation between the twisted Selberg zeta function Z(s;ρ) and the regularized determinant of the twisted Laplacian associated to ρ. 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).

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 𝔛 with Borel σ-algebra 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.

This paper illustrates the utility of the heat kernel on Z as the discrete analogue of the Gaussian density function. The heat kernel on Z is the two-variable function KZ(t,x)=e-2tIx(2t) where Ix(2t) is a Bessel function, with variables x is an element of Z and t >= 0. Like its classic counterpart, KZ(t,x) appears in many mathematical and physical contexts and has a wealth of applications. Some of these applications will be reviewed here, and they concern 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-2tIx(2t) could be useful in differential privacy.

Let M be a finite volume hyperbolic Riemann surface with arbitrary signature, and let χ be an arbitrary m-dimensional multiplier system of weight k. Let R(s,χ) be the associated Ruelle zeta function, and φ(s,χ) the determinant of the scattering matrix. We prove the functional equation that R(s,χ)φ(s,χ) = R(-s,χ)φ(s,χ)H(s,χ) where H(s,χ) is a meromorphic function of order one explicitly determined using the topological data of M and of χ, and the trigonometric function sin(s). From this, we determine the order of the divisor of R(s,χ) 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,χ).

In this paper we present a general unifying principle for computing finite trigonometric sums of types that arise in physics and number theory. We obtain formulas that are more general than previous expressions and deduce linear recursions, which are computationally more efficient than the degree two recursions proved by Zagier in [Za96]. As an application, we provide an answer to a question posed by Xie-Zhao-Zhao in [XZZ22 concerning special values of Dirichlet L-functions. The proofs use the combinatorial Laplacian on cyclic graphs and their twisted coverings. The techniques therefore connect the trigonometric sums to spectral invariants of graphs and open up for future investigations.

Let G be a finite, weighted graph, and let T be a time-scale with a fixed point t 0 such that sup T = infinity. . In this paper, we construct the heat kernel on G in time-scale T in terms of a certain convolution series involving the heat operator acting on a parametrix, which is a fairly general function depending on the vertex set of G and the time variable t is an element of T. We develop some applications by choosing different parametrices and various time-scales. The results we obtain here extend, in part, aspects of the recent articles in that the time-scale considered in this paper is arbitrary.

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 $G$.

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 Poincar\'e series, one of which is the Niebur-Poincar\'e 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 $\Gamma$ 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 $\Gamma$ is the full modular group $\text{\rm PSL}(2,\ZZ)$, thus reproving the theorems from \cite{BK19}; and second when $\Gamma$ is an Atkin-Lehner group $\Gamma_{0}(N)^+$, where explicit computations are given for certain genus zero, one and two levels.

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved a type of Kronecker’s limit formula associated to any divisor D on any smooth Kähler manifold X, assuming that D is smooth in codimension one. In the present article, it is shown how the aforementioned analogue of Kronecker’s limit formula applies to reprove and generalize Weil reciprocity. More precisely, we extend Weil reciprocity to (suitably normalized) meromorphic modular forms of even weight on a smooth, compact Riemann surface, and present a variant of Weil reciprocity for a class of harmonic functions with special types of singularities on a finite volume quotient of a symmetric space or a compact, smooth projective Kähler variety. We also prove an integral version of Weil reciprocity for a compact, smooth projective Kähler variety.

In this article we develop a general method by which one can explicitly evaluate certain sums of nth powers of products of d≥1 elementary trigonometric functions evaluated at m=(m1,…,md)-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 Gm which depends on m. The sums in question are then related to the nth step of a Markov chain on Gm. 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 Zd which covers Gm.

We derive an explicit formula for the fundamental solution KTq+1 (x, x(0); t) to the discrete-time diffusion equation on the (q + 1)- regular tree Tq+1 in terms of the discrete I-Bessel function. We then use the formula to derive an explicit expression for the fundamen-tal 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 applica-tions. 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 probabil-ity 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 uni-form 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 ver-tices 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 Tq+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.

Abstract In this paper we develop the parametrix approach for constructing the heat kernelon a graph G. In particular, we highlight two specific cases. First, we considerthe case when G is embedded in a Eulidean domain or manifold \Omega, andwe use a heat kernel associated to \Omega to obtain a formula for the heat kernelon G. Second, we consider when G is a possibly infinite 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.

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $$\chi $$ denote a finite dimensional unitary representation of the fundamental group of M. Let $$\Delta $$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with $$\chi $$ . From the spectral theory of $$\Delta $$ , there are three distinct sequences of numbers: the first coming from the eigenvalues of $$L^{2}$$ eigenfunctions, the second coming from resonances associated with 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, $$\mathcal {Z}_-(s,z)$$ and $$\mathcal {Z}_+(s,z)$$ that encode the spectrum of $$\Delta $$ in such a way that they can be used to define the regularized determinant of $$\Delta -z(1-z)I$$ . The resulting formula for the regularized determinant of $$\Delta -z(1-z)I$$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $$z\leftrightarrow 1-z$$ .