For a lattice Γ⊂Rn\Gamma\subset\mathbb{R}^n and parameters t∈Cnt\in\mathbb{C}^n, the sums ∑γ∈Γ∏j=1nIxj+γj(tj)\sum_{\gamma\in\Gamma}\prod_{j=1}^n I_{x_j+\gamma_j}(t_j) of modified Bessel functions of the first kind are discrete analogues of theta functions. Hasegawa, Saigo, Saito and Sugiyama (2024) proved Poisson-summation identities for them, including character-twisted versions that yield transformation formulas of Riemann theta functions and of the Dedekind eta function, identities for lattice sums defined by linear codes, and explicit solutions of heat equations on lattices, but only under restrictive hypotheses: the shift xx must lie in Zn\mathbb{Z}^n, the lattice must be a sublattice of Zn\mathbb{Z}^n, in the twisted case every entry of the lattice generator matrix must be divisible by the character modulus qq, the Poisson summation formula requires a continuous periodization and an absolutely convergent dual series, and the continuum-limit boundedness estimates hold only at one specific scaling. The open problem is to remove these restrictions wherever they are not structurally necessary: classify the pairs (Γ,x)(\Gamma,x) for which the plain I-Bessel lattice sum converges absolutely or conditionally with sharp necessary and sufficient conditions, for general lattices Γ⊂Rn\Gamma\subset\mathbb{R}^n; extend the trivial and twisted identities to real shifts, non-divisible lattices, and imprimitive characters—the lattice remaining a sublattice of Zn\mathbb{Z}^n, through which the character twist χ(γA−1)\chi(\gamma A^{-1}) is defined—or determine the corrected form and the precise obstruction; establish Poisson summation for discontinuous periodizations and conditionally convergent dual series; and justify continuum limits at general scalings.

Background

A lattice Γ⊂Rn\Gamma\subset\mathbb{R}^n is a discrete subgroup of the form Γ=ZnA\Gamma=\mathbb{Z}^nA with A∈GLn(R)A\in\mathrm{GL}_n(\mathbb{R}); its dual lattice is Γ∗={γ∗∈Rn∣⟨γ,γ∗⟩∈Z for all γ∈Γ}\Gamma^{\ast}=\{\gamma^{\ast}\in\mathbb{R}^n\mid\langle\gamma,\gamma^{\ast}\rangle\in\mathbb{Z}\ \text{for all}\ \gamma\in\Gamma\}, where ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle is the standard inner product. Lattice sums ∑γ∈Γf(γ)\sum_{\gamma\in\Gamma}f(\gamma) are ubiquitous in number theory and physics (theta functions; Epstein zeta functions, the spectral zeta functions of a positive-definite quadratic form on a lattice; Madelung constants, the electrostatic lattice energies of ionic crystals). The modified Bessel function of the first kind Iν(t)I_\nu(t) is defined by the series Iν(t)=∑m≥0(t/2)ν+2m/(m! Γ(m+ν+1))I_\nu(t)=\sum_{m\ge0}(t/2)^{\nu+2m}/(m!\,\Gamma(m+\nu+1)); for real order the source paper also uses I~x(t)=1π∫0πetcos⁡θcos⁡(xθ) dθ\tilde{I}_x(t)=\frac{1}{\pi}\int_0^\pi e^{t\cos\theta}\cos(x\theta)\,d\theta, which equals Ix(t)I_x(t) for integer xx and whose large-order behavior is I~a(t)=e−tsin⁡πaπa+O(a−2)\tilde{I}_a(t)=\frac{e^{-t}\sin\pi a}{\pi a}+O(a^{-2}) as ∣a∣→∞|a|\to\infty. The function e−tIx−y(t)e^{-t}I_{x-y}(t) is a discrete analogue of the Gaussian heat kernel 14πte−(x−y)2/4t\frac{1}{\sqrt{4\pi t}}e^{-(x-y)^2/4t} on R\mathbb{R} (Karlsson and Neuhauser 2006; Chinta, Jorgenson and Karlsson 2010), which is why I-Bessel lattice sums behave like discrete theta functions. For example, for Γ=mZ\Gamma=m\mathbb{Z} and integer xx one has ∑γ∈mZIx+γ(t)=1m∑j=0m−1exp⁡(tcos⁡2πjm)e2πixj/m\sum_{\gamma\in m\mathbb{Z}}I_{x+\gamma}(t)=\frac{1}{m}\sum_{j=0}^{m-1}\exp(t\cos\frac{2\pi j}{m})e^{2\pi ixj/m}, and multiplying by me−tme^{-t}, replacing tt with 2m2t2m^2t and letting m→∞m\to\infty recovers the classical theta transformation formula 14πt∑r∈Ze−r2/4t=∑j∈Ze−4π2j2t\frac{1}{\sqrt{4\pi t}}\sum_{r\in\mathbb{Z}}e^{-r^2/4t}=\sum_{j\in\mathbb{Z}}e^{-4\pi^2j^2t}, a manipulation called a continuum limit.

Hasegawa, Saigo, Saito and Sugiyama (2024) proved such identities for any sublattice Γ\Gamma of Zn\mathbb{Z}^n, twisted by Dirichlet characters, by systematically generalizing the Poisson summation formula (PSF), the identity ∑γ∈Γf(x+γ)=1vol(Rn/Γ)∑γ∗∈Γ∗f^(γ∗)e2πi⟨x,γ∗⟩\sum_{\gamma\in\Gamma}f(x+\gamma)=\frac{1}{\mathrm{vol}(\mathbb{R}^n/\Gamma)}\sum_{\gamma^{\ast}\in\Gamma^{\ast}}\hat f(\gamma^{\ast})e^{2\pi i\langle x,\gamma^{\ast}\rangle} relating a periodization of a test function ff to samples of its Fourier transform f^\hat f, valid classically for sufficiently regular ff. A Dirichlet character modulo qq is a completely multiplicative function on Z\mathbb{Z}, periodic of period qq, vanishing on integers not coprime to qq; it is primitive if it is not induced by a character of smaller modulus; its Gauss sum is G(χ)=∑a=0q−1χ(a)e2πia/q\mathcal{G}(\chi)=\sum_{a=0}^{q-1}\chi(a)e^{2\pi ia/q}. Their Theorem 2.1 extends the PSF to possibly discontinuous L1L^1 test functions but still requires (i) the periodized series ∑γ∈Γf(x+γ)\sum_{\gamma\in\Gamma}f(x+\gamma) to be absolutely convergent and continuous as a function of xx, and (ii) absolute convergence of the dual series; the Bessel test functions are admitted through a rectangular function 1[−1/2,1/2]\mathbf{1}_{[-1/2,1/2]} that takes the value 1/21/2 at the endpoints, chosen precisely so that its periodization is continuous. Their Theorem 2.6 gives the identity for trivial characters for Γ⊂Zn\Gamma\subset\mathbb{Z}^n and integer shifts x∈Znx\in\mathbb{Z}^n; Theorem 1.1, the twisted identity for χ=∏jχj\chi=\prod_j\chi_j with each χj\chi_j a primitive Dirichlet character modulo qq (qq arbitrary, including composite), additionally hypothesizes that every entry of AA is divisible by qq when Γ=ZnA\Gamma=\mathbb{Z}^nA. The consequences include transformation formulas of Riemann theta functions (the lattice theta functions, generalizing the one-dimensional Jacobi theta function, attached to a lattice and its dual) and of the Dedekind eta function η(τ)\eta(\tau) (the modular form ∏m≥1(1−e2πimτ)\prod_{m\ge1}(1-e^{2\pi im\tau}) with a root factor), identities expressing Bessel lattice sums defined by linear codes over Z/mZ\mathbb{Z}/m\mathbb{Z} (a linear code is a submodule of (Z/mZ)n(\mathbb{Z}/m\mathbb{Z})^n; its complete weight enumerator is the polynomial counting how many coordinates of each codeword lie in each residue class) through complete weight enumerators, and explicit solutions of semidiscrete heat equations ∂tu=Δu\partial_tu=\Delta u on lattices, where Δ\Delta is the graph Laplacian, (Δu)(x)=∑y∼x(u(y)−u(x))(\Delta u)(x)=\sum_{y\sim x}(u(y)-u(x)) with the sum running over the nearest neighbors yy of xx.

These restrictions are not cosmetic. Lemma 2.4 of the source shows that the modified sum with the subtraction term I~xj+γj(tj)−e−tjsin⁡π(xj+γj)π(xj+γj)\tilde{I}_{x_j+\gamma_j}(t_j)-\frac{e^{-t_j}\sin\pi(x_j+\gamma_j)}{\pi(x_j+\gamma_j)} converges absolutely and locally uniformly for every lattice in Rn\mathbb{R}^n and every real shift, while the plain sum ∑γ∈Γ∏j=1nIxj+γj(tj)\sum_{\gamma\in\Gamma}\prod_{j=1}^n I_{x_j+\gamma_j}(t_j) is proven absolutely convergent only in the integer-order setting (Γ⊂Zn\Gamma\subset\mathbb{Z}^n, x∈Znx\in\mathbb{Z}^n); Remark 2.5 records that for a general lattice and general real shift the plain I-Bessel lattice sum is not absolutely convergent. In the proof of Theorem 1.1, the divisibility q∣Aq\mid A is used exactly to keep the Bessel orders xj+(kA)j/qx_j+(kA)_j/q integral so that this convergence lemma applies. Lemma 3.1 proves the uniform boundedness t Le−L2∣t∣∑k∈ZI(mk+b)L(L2t)\sqrt{t}\,Le^{-L^2|t|}\sum_{k\in\mathbb{Z}}I_{(mk+b)L}(L^2t) needed for the continuum limits only at the specific scaling order →L⋅\to L\cdotorder, t→L2tt\to L^2t. The unresolved target is the removal of all four restrictions.

Problem Statement

Remove the restrictive hypotheses under which the Poisson-summation identities for I-Bessel lattice sums are currently proved, so that the framework holds in its natural generality. Let Γ=ZnA\Gamma=\mathbb{Z}^nA be a lattice in Rn\mathbb{R}^n, x∈Rnx\in\mathbb{R}^n a shift, t=(t1,…,tn)∈Cnt=(t_1,\dots,t_n)\in\mathbb{C}^n, and let χj\chi_j be Dirichlet characters modulo qq. The problem has four jointly linked components.

(1) Convergence classification. Give sharp necessary and sufficient conditions on the pair (Γ,x)(\Gamma,x), expressed through the interaction of Γ\Gamma and xx with the integer grid, under which the plain I-Bessel lattice sum ∑γ∈Γ∏j=1nI~xj+γj(tj)\sum_{\gamma\in\Gamma}\prod_{j=1}^n\tilde{I}_{x_j+\gamma_j}(t_j) converges absolutely for each fixed t∈Cnt\in\mathbb{C}^n; and characterize the pairs for which it converges conditionally, specifying the summation convention. Since the leading asymptotic prefactor e−te^{-t} in I~a(t)=e−tsin⁡πaπa+O(a−2)\tilde{I}_a(t)=\frac{e^{-t}\sin\pi a}{\pi a}+O(a^{-2}) never vanishes, the answer should also state whether the resulting conditions on (Γ,x)(\Gamma,x) are independent of tt, and whether convergence is locally uniform in tt on compact sets as in Lemma 2.4 of the source. What is currently known covers only the integer-order case Γ⊂Zn\Gamma\subset\mathbb{Z}^n with x∈Znx\in\mathbb{Z}^n (absolute convergence) and the modified sum with the sinc-type subtraction terms of Lemma 2.4, which converges absolutely for arbitrary (Γ,x)(\Gamma,x); beyond Remark 2.5's general negative statement, no description exists.

(2) Identities beyond integrality and divisibility. Prove the summation identities of Theorems 2.6 and 1.1 of the source for real shifts x∈Rnx\in\mathbb{R}^n, for sublattices Γ=ZnA\Gamma=\mathbb{Z}^nA of Zn\mathbb{Z}^n whose generator matrix does not satisfy 'every entry of AA is divisible by qq', and for imprimitive characters χj\chi_j, with the Bessel factors at non-integral orders understood as the integral-representation function I~\tilde{I} (defined for all (x,t)∈C2(x,t)\in\mathbb{C}^2 and equal to Ix(t)I_x(t) for integer xx, so that the source's identities are recovered in the integer-order case), including the twisted identity

∑γ∈Γχ(γA−1){∏j=1nI~xj+γj/q(tj)}e2πi⟨y,x+γ/q⟩=∏j=1nG(χj)vol(Rn/Γ)∑γ∗∈Γ∗χˉ(γ∗ tA){∏j=1n1[−1/2,1/2](yj−γj∗) etjcos⁡2π(yj−γj∗)}e2πi⟨x,γ∗⟩\sum_{\gamma\in\Gamma}\chi(\gamma A^{-1})\Bigl\{\prod_{j=1}^n \tilde{I}_{x_j+\gamma_j/q}(t_j)\Bigr\}e^{2\pi i\langle y,x+\gamma/q\rangle}=\frac{\prod_{j=1}^n\mathcal{G}(\chi_j)}{\mathrm{vol}(\mathbb{R}^n/\Gamma)}\sum_{\gamma^{\ast}\in\Gamma^{\ast}}\bar\chi(\gamma^{\ast}\,{}^tA)\Bigl\{\prod_{j=1}^n\mathbf{1}_{[-1/2,1/2]}(y_j-\gamma_j^{\ast})\,e^{t_j\cos 2\pi(y_j-\gamma_j^{\ast})}\Bigr\}e^{2\pi i\langle x,\gamma^{\ast}\rangle}

without those hypotheses, or determine the precise obstruction and the corrected form of the statement (for example, additional boundary or correction terms, or genuine divergence for specific pairs (Γ,x)(\Gamma,x)).

(3) Poisson summation under weakened hypotheses. Establish the Poisson summation identity for the relevant test-function class when the periodized sum ∑γ∈Γf(x+γ)\sum_{\gamma\in\Gamma}f(x+\gamma) is discontinuous or the dual Fourier series is merely conditionally convergent, for instance in a distributional sense or under an explicit summation method, so that Bessel-type test functions are admissible without endpoint workarounds, with pointwise validity including at the jump points; the value required at a jump point of the periodization follows the source's midpoint convention, the value 1/21/2 assigned to 1[−1/2,1/2]\mathbf{1}_{[-1/2,1/2]} at the endpoints (Remark 2.2 of the source).

(4) Continuum limits at general scalings. Provide the uniform boundedness estimates that justify continuum-limit derivations for scaling regimes other than order →L⋅\to L\cdotorder with t→L2tt\to L^2t (beyond Lemma 3.1 of the source), and determine which modular or spectral objects arise as such limits (in the classical scaling regime the source's continuum limits recover transformation formulas of Riemann theta functions and of the Dedekind eta function).

A complete answer resolves all four components, which are technically interlocked: the integrality and divisibility hypotheses exist precisely because the convergence theory of component (1) is missing, while the primitivity hypothesis is character-theoretic rather than a convergence condition, and the continuum-limit estimates of component (4) inherit the same restrictions. Partial progress on individual components is meaningful but does not by itself complete the problem.

References

  1. Takehiro Hasegawa, Hayato Saigo, Seiken Saito, Shingo Sugiyama, Lattice sums of I-Bessel functions, theta functions, linear codes and heat equations, Research in the Mathematical Sciences 11 (2024), no. 4, Article 62, DOI: 10.1007/s40687-024-00472-9, arXiv:2311.06489, https://arxiv.org/abs/2311.06489
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