The Lerch zeta function IV. Hecke operators
 Jeffrey C. Lagarias^{1}Email author and
 WenChing Winnie Li^{2}
https://doi.org/10.1186/s4068701600829
© The Author(s) 2016
Received: 16 December 2015
Accepted: 17 August 2016
Published: 12 December 2016
Abstract
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of twovariable Hecke operators \(\{ \mathrm{T}_m: \, m \ge 1\}\) given by \(\mathrm{T}_m(f)(a, c) = \frac{1}{m} \sum _{k=0}^{m1} f(\frac{a+k}{m}, mc)\) acting on certain spaces of realanalytic functions, including Lerch zeta functions for various parameter values. The actions of various related operators on these function spaces are determined. It is shown that, for each \(s \in {\mathbb C}\), there is a twodimensional vector space spanned by linear combinations of Lerch zeta functions characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This is an analog of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the (a, c)variables having the Lerch zeta function as an eigenfunction.
Keywords
Mathematics Subject Classification
1 Background
 (1)It is an eigenfunction of a linear partial differential operatorIt satisfies$$\begin{aligned} D_L := \frac{1}{2 \pi i} \frac{\partial }{\partial a} \frac{\partial }{\partial c} + c \frac{\partial }{\partial c}. \end{aligned}$$(2)This property was noted in Parts II and III ([29, 30]). When restricting to real variables, we regard \(\frac{\partial }{\partial a}\) and \(\frac{\partial }{\partial c}\) as real differential operators. (They were treated as complex differential operators in [29] and [30].)$$\begin{aligned} (D_L \zeta )(s, a, c) = s \zeta (s, a,c). \end{aligned}$$(3)
 (2)In particular, it satisfies two fourterm functional equations encoding a discrete symmetry under (s, a, c) to \((1s, 1c, a)\). These functional equations were noted by Weil [39] under the restriction \(0<a <1\) and \(0<c<1\) and were studied in Part I ([28]). To state the functional equations, letand define the completed functions$$\begin{aligned} L^{\pm }(s, a, c) :=\zeta (s, a, c) \pm e^{2 \pi i a} \zeta (s, 1a, 1c) \end{aligned}$$where \(\epsilon := \frac{1}{2}(1  (\pm 1))\) takes values 0 or 1. Then the functional equations are$$\begin{aligned} \hat{L}^{\pm } (s, a , c) :=\pi ^{ \frac{s+ \epsilon }{2} } \Gamma ( \frac{s+\epsilon }{2} ) L^{\pm }(s, a, c), \end{aligned}$$and$$\begin{aligned} \hat{L}^{+}(s, a, c) = e^{2 \pi i a c} \hat{L}^{+}(1s, 1c , a) \end{aligned}$$(4)$$\begin{aligned} \hat{L}^{}(s, a,c) = i\,e^{2 \pi i ac} \hat{L}^{}(1s, 1c , a). \end{aligned}$$(5)
The question this paper considers is that of obtaining an extension of the operators above inside suitable function spaces, in which the Lerch zeta function will be a simultaneous eigenfunction for all \(s \in {\mathbb C}\). We will show there is such an extension, and study the action of all these operators on the ensuing function spaces. We then obtain a characterization of simultaneous eigenfunction solutions of these operators in these function spaces, in the spirit of Milnor [32].
1.1 Hecke operators and function spaces
 (1)
analytic continuation will be required in the svariable to cover all \(s \in {\mathbb C}\);
 (2)
the functional equations do not leave the domain \({\mathcal D}_{++}\) invariant;
 (3)
the analytically continued Lerch zeta function has discontinuities at parameter values (a, c) where a is an integer or c is a nonpositive integer.
 (1)The realanalytic extension obeys twistedperiodicity conditions in the a and c variables:$$\begin{aligned} \zeta _{*}(s, a+1, c)= & {} \zeta _{*}(s, a, c),\\ \zeta _{*}(s, a, c+1)= & {} e^{2\pi ia}\zeta _{*}(s, a, c), \end{aligned}$$
 (2)
The extended function satisfies two symmetrized fourterm functional equations. This fact for \(0<a<1, 0< c<1\) was originally noted by A. Weil [39, pp. 54–58]. These functional equations generalize that of the Riemann zeta function and are given in Sect. 5.1.
 (3)
The function \(\zeta _{*}(s, a,c)\) has discontinuities in the a and c variables at integer values of a, c, for various ranges of s. These discontinuities encode information about the nature of the singularities of these functions in the (a, c)variables at integer values.
To obtain results that apply to the four families and for all complex s, we introduce function spaces of piecewise continuous functions, allowing discontinuities. The restriction to piecewise continuous functions, without completing the function space, is made because the singularities of the Lerch zeta function at the boundary of the unit square become large as \(\mathfrak {R}(s) \rightarrow \infty \). For values of s inside the critical strip \(0< \mathfrak {R}(s)<1\), we are able to work inside various Banach spaces. Of particular interest is the Banach space \(L^1( \Box , da\, dc)\), which is relevant since the functions \(L^{\pm }(s, a, c)\) belong to this space inside the critical strip ([28, Theorem 2.4]). More generally, one may consider the spaces \(L^p(\Box , da \, dc)\) for \(1 \le p \le 2\). Note that for twistedperiodic functions F(a, c) the \(L^p\)norm of F(a, c) is invariant under measurement in any unit square \(x_0 \le a \le x_0+1, y_0 \le c \le y_0 +1\), allowing real values \(x_0, y_0\).
The Hilbert space \({\mathcal H}= L^2({\Box }, da \, dc)\) is also of great interest; however, the Lerch zeta function \(\zeta (s, a, c)\) for \((a, c) \in {\Box }\) does not belong to this space for any \(s \in {\mathbb C}\); for \(\mathfrak {R}(s) >1\), this follows from the single term \(c^{s}\) not being in \({\mathcal H}\), while for other s it is more subtle, see the proof of [28, Theorem 2.4]. The operators \(\mathrm{R}\) and the four families of Hecke operators induce welldefined actions of bounded operators on this space and together generate an interesting noncommutative algebra of operators that is a \(\star \)algebra in \({\mathcal B}({\mathcal H})\) which seems worthy of further study.
To obtain function spaces that allow an action of differential operators, we restrict to suitable smaller spaces of piecewise smooth functions that allow discontinuities located at a fixed lattice of axisparallel vertical lines and horizontal lines, with relevant coordinate vector (horizontal and/or vertical) contained in a lattice \(\frac{1}{d}{\mathbb Z}\) for a fixed d depending on the function. We can start this construction with such functions defined on the unit square with specified discontinuities produced by twistedperiodicity having relative coordinate vectors in the same lattice \(\frac{1}{d} {\mathbb Z}\).
1.2 Motivation and results
 (1)
We introduce a set of auxiliary operators acting on twistedperiodic function spaces, a linear partial differential operator \(\mathrm{D}_L\), a unitary operator \(\mathrm{R}\) and the family of twovariable Hecke operators \(\mathrm{T}_m\), viewed as acting on the unit square \({\Box }\) through the use of twistedperiodic function spaces. We determine commutation relations between all these operators on these function spaces.
 (2)For each \(s \in {\mathbb C}\), we define a twodimensional vector space \({\mathcal E}_s\), the Lerch eigenspace, consisting of twistedperiodic realanalytic functions defined on \({\mathbb R}^2{\smallsetminus }{\mathbb Z}^2\) (but sometimes discontinuous on the grid \(({\mathbb Z}\times {\mathbb R}) \cup ({\mathbb R}\times {\mathbb Z})\)) which satisfy the eigenvalue identitysimultaneously for all \(m \ge 1\). The spaces \({\mathcal E}_s\) are preserved by the \(\mathrm{R}\)operator, and those at s and \(1s\) are related under the symmetries of the functional equation.$$\begin{aligned} \mathrm{T}_m (\zeta _{*})(s, a, c) = m^{s} \zeta _{*} (s, a, c) \end{aligned}$$(10)
 (3)
We give a simultaneous Hecke eigenfunction interpretation of the Lerch zeta function, in the spirit of Milnor’s [32] characterization of Kubert functions, including the Hurwitz zeta function. In Sect. 6, we show that for each \(s \in {\mathbb C}\) there is a twodimensional vector space \({\mathcal E}_s\) of simultaneous eigenfunctions, the Lerch eigenspace, satisfying suitable integrability side conditions. This is a generalization of Milnor’s converse result characterizing the Hurwitz zeta function and Kubert functions.
1.3 Related work
We have already mentioned the definition of twovariable Hecke operators in 1989 by Sun [37] in connection with covering systems, as well as the work of Porubsky [34] noting the eigenfunction property (7) of the Lerch zeta function. See also a survey paper of Porubsky and Schönheim [35] on Erdős’s work on covering systems.
In 1983, Milnor [32] considered operators of form (11), calling them Kubert operators. General operators of the type (11) acting on an abelian group (e.g., \({\mathbb R}/ {\mathbb Z}\)) had previously been studied by Kubert and Lang [24] and Kubert [23]. Milnor characterized simultaneous eigenfunction solutions of such operators, in the space of continuous functions on the open interval (0, 1). In Sect. 6.1, we review Milnor’s work.
In 1999, BáezDuarte [4] noted that this family of dilation operators relates to the realvariables approach to the Riemann hypothesis due to Nyman [33] and Beurling [11], see also BáezDuarte [4] and Burnol [14, 15]. Convergence of series composed out of dilations of functions has been much studied; they include Fourier series as a special case, see Gaposkin [19], Aisletiner et al. [1, 2], Berkes and Weber [7] and Weber [38].
2 Main results
In this paper, we study actions of the twovariable Hecke operators on different function spaces and determine their commutation relations relative to various other operators appearing in parts I–III.
2.1 Twovariable Hecke operators and \(\mathrm{R}\)operator
Theorem 1
 (1)
The four sets of twovariable Hecke operators \(\{ \mathrm{T}_m, \mathrm{S}_m, \mathrm{T}_m^{\vee }, \mathrm{S}_m^{\vee }:\, m \ge 1\}\) continuously extend to bounded operators on each Banach space \(L^{p}(\Box , da \, dc)\) for \(1 \le p \le \infty \), by viewing these (almost everywhere defined) functions of \(\Box \) as extended to \({\mathbb R}\times {\mathbb R}\) via the twistedperiodicity relations \( f(a+1, c) = f(a, c)\) and \(f(a, c+1) = e^{2\pi i a} f(a, c)\). These operators satisfy \(\mathrm{T}_m = \mathrm{T}_m^{\vee }\), \(\mathrm{S}_m = \mathrm{S}_m^{\vee }\) and \(\mathrm{S}_m = \frac{1}{m}(\mathrm{T}_m)^{1}\) for all \(m \ge 1\).
 (2)
The \(\mathbb C\)algebra \({\mathcal A}_{0}^{p}\) of operators on \(L^p({\Box }, da \, dc)\) generated by these four sets of operators under addition and operator multiplication is commutative.
 (3)
On \(L^2(\Box , da \, dc)\) the adjoint Hecke operator \((\mathrm{T}_m)^{*} = \mathrm{S}_m\), and \((\mathrm{S}_m)^{*}= \mathrm{T}_m\). In particular, the \({\mathbb C}\)algebra \({\mathcal A}_0^2\) is a \(\star \)algebra. In addition each of \(\sqrt{m}\mathrm{T}_m, \sqrt{m}\mathrm{S}_m\) is a unitary operator on \(L^2(\Box , da \, dc)\).
For all \(p \ge 1\), the operator \(\mathrm{R}\) defines an isometry \(\mathrm{R}(f)_p = f_p\) on \(L^p({\Box }, da \, dc)\). In consequence, we obtain an extended algebra \({\mathcal A}^p := {\mathcal A}_0^p [ \,\mathrm{R}\,]\) of operators, by adjoining the operator \(\mathrm{R}\) to \({\mathcal A}_0^p\). The algebra \({\mathcal A}^p\) is a noncommutative algebra. The algebra \({\mathcal A}^2\) is also a \(\star \)algebra, which follows using \((\mathrm{R})^{\star } = \mathrm{R}^{1}= \mathrm{R}^3.\)
2.2 Twodimensional Lerch eigenspace
Theorem 2
 (i)(Lerch differential operator eigenfunctions) Each \(f \in {\mathcal E}_s\) is an eigenfunction of the Lerch differential operator \(\mathrm{D}_L = \frac{1}{2 \pi i} \frac{\partial }{\partial a} \frac{\partial }{\partial c} + c \frac{\partial }{\partial c}\) with eigenvalue \(s\), namelyholds at all \((a, c) \in {\mathbb R}\times {\mathbb R}\), with both a and c nonintegers.$$\begin{aligned} (\mathrm{D}_L f )(s, a, c) = sf (s, a, c) \end{aligned}$$
 (ii)(Simultaneous Hecke operator eigenfunctions) Each \(f \in {\mathcal E}_s\) is a simultaneous eigenfunction with eigenvalue \(m^{s}\) of all twovariable Hecke operatorsin the sense that, for each \(m \ge 1\),$$\begin{aligned} \mathrm{T}_m (f)(a,c) = \frac{1}{m} \sum _{k=0}^{m1} f \left( \frac{a+k}{m}, mc \right) \end{aligned}$$holds on the domain \(({\mathbb R}{\smallsetminus } \frac{1}{m}{\mathbb Z}) \times ({\mathbb R}{\smallsetminus } {\mathbb Z})\).$$\begin{aligned} \mathrm{T}_m f = m^{s} f \end{aligned}$$
 (iii)(\(\mathrm{J}\)operator eigenfunctions) The space \({\mathcal E}_s\) admits the involutionunder which it decomposes into onedimensional eigenspaces \({\mathcal E}_s = {\mathcal E}_s^+ \oplus {\mathcal E}_s^\) with eigenvalues \(\pm 1\), that is, \({\mathcal E}_s^{\pm } = <F_{s}^{\pm }>\) and$$\begin{aligned} \mathrm{J}f (a,c) : = e^{2 \pi ia} f(1a, 1c), \end{aligned}$$$$\begin{aligned} \mathrm{J}(F_s^{\pm }) = \pm F_s^{\pm }. \end{aligned}$$
 (iv)(\(\mathrm{R}\)operator action) The \(\mathrm{R}\)operator \(\mathrm{R}(F) (a, c) = e^{2\pi i a c} F(1c, a)\) acts by \(\mathrm{R}({\mathcal E}_s)= {\mathcal E}_{1s}\) withwhere \(w_{+}= 1\), \(w_{}=i\), \(\gamma ^+(s) = \Gamma _{{\mathbb R}}(s)/\Gamma _{{\mathbb R}}(1s)\), \(\gamma ^(s) = \gamma ^+(s+1)\) and \(\Gamma _{{\mathbb R}}(s) = \pi ^{s/2}\Gamma (s/2)\).$$\begin{aligned} \mathrm{R}(L^{\pm }(s, a, c) ) = w_{\pm } ^{1}{\gamma }^{\pm }(1s) L^{\pm }(1s, a, c), \end{aligned}$$
The members of the Lerch eigenspaces are shown to have the following analytic properties (Theorem 6).
Theorem 3
 (i)(TwistedPeriodicity Property) All functions F(a, c) in \({\mathcal E}_{s}\) satisfy the twistedperiodicity functional equations$$\begin{aligned} F(a+1,~c~)= & {} ~~~~~~~~ F(a,c), \\ F(~a~, c+1)= & {} e^{2\pi i a} F(a, c). \end{aligned}$$
 (ii)(Integrability Properties)
 (a)If \(\mathfrak {R}(s) >0\), then for each noninteger c all functions in \({\mathcal E}_{s}\) have \(f_c(a) :=F(a, c) \in L^{1}[(0,1), da],\) and all their Fourier coefficientsare continuous functions of c on \(0<c<1\).$$\begin{aligned} f_n(c) := \int _{0}^1 F(a,c) e^{2\pi i n a} da, \quad n \in {\mathbb Z}, \end{aligned}$$
 (b)If \(\mathfrak {R}(s)<1\), then for each noninteger a all functions in \({\mathcal E}_{{s}}\) have \( g_a(c):=e^{2 \pi i a c}F(a, c) \in L^{1}[(0,1), dc],\) and all Fourier coefficientsare continuous functions of a on \(0< a<1\).$$\begin{aligned} g_n(a) := \int _{0}^1 e^{2\pi i a c}F(a,c) e^{2\pi i n c} dc, \quad n \in {\mathbb Z}, \end{aligned}$$
 (c)
If \(0< \mathfrak {R}(s) < 1\), then all functions in \({\mathcal E}_{s}\) belong to \(L^{1}[\Box , da dc]\). In this range, the vector space \({\mathcal E}_s\) is invariant under the action of all four sets \( \mathrm{T}_m, \mathrm{S}_m, \mathrm{T}_m^{\vee }, \mathrm{S}_m^{\vee } ~( m \ge 1)\) of twovariable Hecke operators.
 (a)
2.3 Eigenfunction characterization of Lerch eigenspace
In Sect. 6, we first review Milnor’s simultaneous eigenfunction characterization of Kubert functions (Theorem 7). We then prove the following characterization for the Lerch eigenspace \({\mathcal E}_s\) (Theorem 8), the main result of this paper, which can be viewed as a generalization to two dimensions of Milnor’s result. It may also be regarded as a converse theorem to Theorem 3.
Theorem 4
 (1)(TwistedPeriodicity Condition) For \((a,c) \in ({\mathbb R}{\smallsetminus } {\mathbb Z}) \times ({\mathbb R}{\smallsetminus } {\mathbb Z})\),$$\begin{aligned} F(a+1, ~c~)= & {} F(a,c) \\ F(a, ~c+1)= & {} e^{2\pi i a} F(a,c). \end{aligned}$$
 (2)(Integrability Condition) At least one of the following two conditions (2a) or (2c) holds. (2a) The svariable has \(\mathfrak {R}(s) > 0\). For \(0<c<1\) each function \(f_c(a):= F(a, c) \in L^1[(0,1), da]\), and all the Fourier coefficientsare continuous functions of c. (2c) The svariable has \(\mathfrak {R}(s) < 1\). For \(0<a<1\) each function \(g_a(c):= e^{2 \pi i a c}F(a,c) \in L^1[(0,1), dc]\), and all the Fourier coefficients$$\begin{aligned} f_n(c) := \int _{0}^{1} f_c(a)e^{2\pi i n a} da= \int _{0}^1 F(a,c) e^{2\pi i n a}da,\quad n \in {\mathbb Z}, \end{aligned}$$are continuous functions of a.$$\begin{aligned} g_n(a) := \int _{0}^{1} g_a(c)e^{2\pi i n c} dc= \int _{0}^1 e^{2\pi i a c}F(a,c) e^{2\pi i n c}dc,~~~n \in {\mathbb Z}, \end{aligned}$$
 (3)(Hecke Eigenfunction Condition) For all \(m \ge 1\),holds for \((a, c) \in ({\mathbb R}{\smallsetminus } {\mathbb Z})\times ({\mathbb R}{\smallsetminus } \frac{1}{m}{\mathbb Z})\).$$\begin{aligned} \mathrm{T}_{m}(F)(a, c) = m^{{s}}F(a, c) \end{aligned}$$
This generalization imposes extra integrability conditions in order to control the dilation property of the twovariable Hecke operators in the ccoordinate. The twistedperiodicity hypothesis plays an essential role in the proof of Theorem 4, in giving identities that the Fourier series coefficients of such functions must satisfy, see (84).
2.4 Further extensions
Subsequent work of the first author [27] gives a representationtheoretic interpretation of the “realvariables” version of the Lerch zeta function treated in this paper, related to function spaces associated with the Heisenberg group. Under it, various combinations of Lerch zeta function and variants twisted by Dirichlet characters are found to play the role of Eisenstein series with respect to the operator \(\Delta _L=D_L + \frac{1}{2} \mathbf{I }\), where \(\mathbf{I }\) is the identity operator, which plays the role of a Laplacian. The Lerch zeta functions treated in this paper correspond to the trivial Dirichlet character. In this Eisenstein series interpretation, the Lerch functions \(L^{\pm }(s, a, c)\) on the line \(\mathfrak {R}(s)= \frac{1}{2}\) parametrize a pure continuous spectrum of the operator \(\Delta _L\) on a suitable Hilbert space \({\mathcal H}\). The realanalytic Hecke operators \(\mathrm{T}_m\) described in this paper then correspond to dilation operators similar to \(\tilde{\mathrm{T}}_m\) in Sect. 1.3.
In another sequel paper, the first author studies a “complex variables” twovariable Hecke operator action associated to the Lerch zeta functions. This framework again includes the Lerch zeta function for \(\mathfrak {R}(s)>1\), with twovariable Hecke operators initially viewed as acting on a suitable domain of holomorphic functions. It then studies these Hecke operators acting on spaces of (multivalued) holomorphic functions on various domains. These are quite different function spaces than the ones treated here and the resulting Hecke action has new features. In particular, members of different families of Hecke operators do not commute on these function spaces.
Notation Much of the analysis of this paper concerns functions with \(s \in {\mathbb C}\) regarded as a parameter. We let \({\Box }= \{ (a, c) \in [0, 1]\times [0, 1]\}\), and its interior \( {\Box ^{\circ }}= \{ (a,c) \in (0,1) \times (0,1)\}\).
3 Roperator and differential operators
We introduce various operators with respect to which the Lerch zeta function has invariance properties. We first restate operators acting on functions of two real variables (a, c) defined almost everywhere on the unit square \(\Box ^{\circ }\). Then we consider operators defined on larger domains of \({\mathbb R}\times {\mathbb R}\). We start with a nonlocal operator \(\mathrm{R}\) mentioned before, associated with the Fourier transform, and certain linear partial differential operators, which we term realanalytic Lerch differential operators. In Sect. 3.1, we define the \(\mathrm{R}\)operator and reformulate the functional equations for the Lerch zeta function in part I, using this operator. In Sect. 3.2, we define Lerch differential operators, and in Sect. 3.3, we determine their commutation relations on suitable function spaces, together with the \(\mathrm{R}\)operator.
3.1 Local operators: Realvariable differential operators
Lemma 1
The operator \(\mathrm{D}_L\) takes a piecewise \(C^{1,1}\)function that is twistedperiodic on \({\mathbb R}\times {\mathbb R}\) to a twistedperiodic function on \({\mathbb R}\times {\mathbb R}\).
Proof
3.2 Nonlocal operator: \(\mathrm{R}\)operator
These operators have welldefined actions on continuous functions \(C^{0}( (0,1)^2)\) defined in the open unit square. They extend by closure to an operator action on (almost everywhere defined) \(L^p\)functions on the unit square, for any \(p \ge 1\). In particular, the extended action on \(L^2 ( \Box , da dc)\) is unitary.
Lemma 2
The operator \(\mathrm{R}\) preserves the property of being (almost everywhere) twistedperiodic on \({\mathbb R}\times {\mathbb R}\).
Proof
Remark 1
The \(\mathrm{R}\)operator can be extended to act pointwise on functions defined almost everywhere on \(\Box ^{\circ }\), with the same rule (20). That is, we can allow singularities on a finite set of horizontal and vertical lines. It can also be extended to the domain \({\mathbb R}\times {\mathbb R}\), with the same rule (20), acting on functions satisfying a twistedperiodic relation, see Lemma 4.
3.3 Commutation relations
Lemma 3
 (i)The operators \({D_L^{+}}= \frac{\partial }{\partial c}\) and \({D_L^{}}= \frac{1}{2 \pi i} \frac{\partial }{\partial a} + c\) satisfy the commutation relationwhere I denotes the identity operator.$$\begin{aligned} {D_L^{+}}{D_L^{}} {D_L^{}}{D_L^{+}}= I ~, \end{aligned}$$(24)
 (ii)The operator \(\mathrm{R}f (a,c) := e^{ 2 \pi i ac} f(1c,a)\) leaves \(\mathrm{C}^{1,1} (\Box ^{\circ })\) invariant and satisfies$$\begin{aligned} {D_L^{+}}\mathrm{R}= & {}  2 \pi i ~\mathrm{R}{D_L^{}}~, \end{aligned}$$(25)$$\begin{aligned} {D_L^{}}\mathrm{R}= & {} \frac{1}{2 \pi i} \mathrm{R}{D_L^{+}}. \end{aligned}$$(26)
 (iii)The operator \(\mathrm{D}_L = {D_L^{}}{D_L^{+}}\) satisfiesIn particular, \(\mathrm{D}_L\) and \(\mathrm{R}^2\) commute:$$\begin{aligned} \mathrm{D}_L \mathrm{R}+ \mathrm{R}\mathrm{D}_L =  \mathrm{R}~ . \end{aligned}$$(27)$$\begin{aligned} \mathrm{D}_L \mathrm{R}^2 = \mathrm{R}^2 \mathrm{D}_L. \end{aligned}$$(28)
Proof
4 Twovariable Hecke operators
The viewpoint of this paper is start with functions defined (almost everywhere) on the unit square \(\Box = [0,1] \times [0,1]\), extend them to functions defined on the plane \({\mathbb R}\times {\mathbb R}\) by imposing twistedperiodicity conditions: \(f(a+1, c) = f(a, c)\) and \(f(a, c+1) = e^{2 \pi i a} f(a, c)\), and use the extended functions to define an action of the twovariable Hecke operators on functions on the unit square; the resulting action is linear and preserves twistedperiodicity conditions on the functions.
We would like to accommodate functions with discontinuities of the type that naturally appear in connection with the Lerch zeta function in the realanalytic framework, which occur for integer values of a and/or c. We must also deal with the problem that the action of \(\mathrm{T}_m\) changes the location of the discontinuities. We will introduce appropriate function spaces that are closed under the Hecke operator action allowing a finite (but variable) number of discontinuities in horizontal and vertical directions. These operators also make sense on the Banach space \(L^1( \Box , da\, dc)\) and on the Hilbert space \(L^2(\Box , da\,dc)\).
4.1 Twistedperiodic function spaces
In the realvariables framework treated in this paper, we take as a function space the set of piecewise continuous functions with a finite number of pieces defined in the open unit square \(\Box ^{\circ }\), allowing discontinuities on horizontal and vertical lines and permitting the number of discontinuities to depend on the function.
Definition 1
We allow “functions” in \({\mathcal P}({\mathbb R}\times {\mathbb R})\) to be undefined on the set of horizontal lines with coordinates \(c = \frac{j}{d}\) for all \(j \in {\mathbb Z}\) and do not compare their values at such points. Note that a function having admissible denominator d can be regarded also a function with admissible denominator kd for any integer \(k \ge 1\). Aside from this ambiguity, any function in \({\mathcal P}({\mathbb R}\times {\mathbb R})\) is completely determined by its values in \({\mathcal P}(\Box ^{\circ })\), using the twistedperiodicity conditions. More precisely, the equivalence relation on “functions” in \({\mathcal P}({\mathbb R}\times {\mathbb R})\) calls functions equivalent whose values agree outside of a set of measure zero; the space \({\mathcal P}({\mathbb R}\times {\mathbb R})\) has a welldefined vector space structure on equivalence classes.
Proposition 1
Proof
If \(f(a, c) \in { {\mathcal P}({\mathbb R}\times {\mathbb R})}\), with admissible denominator d, then each \(f( \frac{a+k}{m}, mc) \in { {\mathcal P}({\mathbb R}\times {\mathbb R})}\) with admissible denominator md, hence \(\mathrm{T}_m(f) \in { {\mathcal P}({\mathbb R}\times {\mathbb R})}\) with admissible denominator md.
Remark 2
One may also study twovariable Hecke operators \(\mathrm{T}_m\) and \(\mathrm{R}\) acting as bounded operators on the function spaces \(L^1(\Box , da\, dc)\) or \(L^2(\Box , da \,dc)\). Here, we view the functions as extended to (most of) \({\mathbb R}\times {\mathbb R}\) by twistedperiodicity to define the Hecke operator action. We note that the twistedperiodicity conditions in Definition 1 preserve both the \(L^1\)norm and the \(L^2\)norm of functions on unit squares having integer lattice points as corners.
4.2 \(\mathrm{R}\)operator conjugates of Hecke operators
In order to have a suitable domain inside \({\mathbb R}\times {\mathbb R}\) on which all four families of operators are simultaneously defined, we must use essentially all of \({\mathbb R}\times {\mathbb R}\), because between them the operators are expansive in the positive and negative adirections and cdirections. We make the following definition, allowing functions with discontinuities, for the reasons given above.
Definition 2

along horizontal lines at rational coordinates \(c= \frac{j}{d}\), \(0 \le j < d\),

along vertical lines at rational coordinates \(a= \frac{k}{d}\), \(0 \le k < d\).
 (i)These functions are extended to functions \(f: ({\mathbb R}\smallsetminus {\mathbb Z}) \times ({\mathbb R}\smallsetminus {\mathbb Z}) \rightarrow {\mathbb C}\), by imposing the twistedperiodicity conditions$$\begin{aligned} f(a+1, c)= & {} f(a, c)\\ f(a~~, c+1)= & {} e^{2 \pi i a} f(a, c). \end{aligned}$$
 (ii)Two functions with admissible denominators d, \(d'\) are considered equivalent, if their values coincide on the setWe denote the set of such equivalence classes of functions by \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\).$$\begin{aligned} {\mathcal S}_{dd'} := \left\{ (a, c) \in ({\mathbb R}\smallsetminus \frac{1}{dd'} {\mathbb Z}) \times ({\mathbb R}\smallsetminus \frac{1}{d d'}{\mathbb Z})\right\} . \end{aligned}$$
 (iii)
Given a function in an equivalence class in \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\), any integer \(d \ge 1\) for which all discontinuities of some function equivalent to f(a, c) are on the lattice \(\frac{1}{d}{\mathbb Z}^2\) will be called a support lattice of the function. The minimal value d will be called the conductor of the function.^{3}
The individual operators \(f(a, c) \mapsto \frac{1}{m} f(\frac{a+k}{m}, mc)\) making up the twovariable Hecke operator \(\mathrm{T}_m\) leave the space \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\) invariant, and map a function in \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\) with conductor d to one with conductor dividing md.
One can extend the “realvariables” framework to all four of these families of Hecke operators, acting on the full twistedperiodic function space, using conjugation by \(\mathrm{R}^j\). This assertion is justified by the following result, in which we regard the operator \(\mathrm{R}\) as acting on functions with domain values \((a, c) \in {\mathbb R}\times {\mathbb R}\) by (20).
Lemma 4
The extended twistedperiodic function space \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\) is invariant under the action of the operator \(\mathrm{R}\).
Proof
This follows from Lemma 2.
4.3 Commutation relations of twovariable Hecke operators
We now show some surprising commutation relations; on suitable function spaces all four sets of Hecke operators mutually commute, despite their noncommutativity with the \(\mathrm{R}\)operator.
Lemma 5
 (1)For \(m \ge 1\),$$\begin{aligned} \mathrm{S}_m \circ \mathrm{T}_{dm} (f)(a, c) = \frac{1}{m} \mathrm{T}_d( f)(a, c). \end{aligned}$$
 (2)For \(d=1\) and all \(m \ge 1\),Here \(\mathrm{T}_m\) is invertible and \(\mathrm{S}_m = \frac{1}{m}(\mathrm{T}_m)^{1}\), and \(S_m\) and \(T_m\) commute.$$\begin{aligned} \mathrm{S}_m \circ \mathrm{T}_m(f) (a, c) { ~= \mathrm{T}_m \circ \mathrm{S}_m(f) (a, c) =\frac{1}{m} f(a,c) }. \end{aligned}$$
Proof
We then have \(\mathrm{S}_m= \frac{1}{m}(\mathrm{T}_m)^{1}\) on this domain. \(\square \)
Lemma 6
On the extended twistedperiodic function vector space \({\mathcal P}^{*}({\mathbb R}\times {\mathbb R})\), the operators \(\mathrm{S}_m\) and \(\mathrm{T}_{\ell }\) commute for all \(\ell \ge 1, m \ge 1.\)
Proof
Lemma 7
Proof
We obtain \(\mathrm{S}_m^{\vee }= \mathrm{S}_m\) by an analogous calculation; details are omitted.
4.4 \(L^p\)spaces: Proof of Theorem 1
Lemma 8
Proof
Proof of Theorem 1
(2) The pairwise commutativity of members of all four sets of these operators on \(L^p(\Box , da \, dc)\) extends by continuity from their pairwise commutativity on the domain \({\mathcal D}_p\), which they preserve by (1), which itself follows from the commutativity result established in Lemma 6.
5 Lerch Eigenspaces \({\mathcal E}_s\) and their properties
We construct a twodimensional space \({\mathcal E}_s\) of simultaneous eigenfunctions for all the twovariable Hecke operators, built out of the Lerch zeta function. We make use of the functional equations for the Lerch zeta function.
5.1 Lerch zeta function: functional equations
Definition 3
Proposition 2
Proof
For \(0< a, c < 1\), this result is shown in part I [28, Theorem 2.1], where we have restated that result using the \(\mathrm{R}\)operator. The extension to the boundary cases then uses the definition of \(\zeta _{*}(s, a,c)\) given in part I. It applies on the domain \((a, c) \in {\mathbb R}\times {\mathbb R}\), given in [28, Theorem 2.2], including integer values of a and c.
5.2 Lerch eigenspaces \({\mathcal E}_s\)
Definition 4
The vector space \({\mathcal E}_s\) is at most twodimensional, since, as noted above, \(L_s^{+}(a,c)\) and \(R_s^{+}(a,c)\) (resp. \(L_s^{}(a,c)\) and \(R_s^{}(a,c)\)) are linearly dependent, with dependency relations given by the functional equation (53).
Lemma 9
For each \(s \in {\mathbb C}\), the Lerch eigenspace \({\mathcal E}_s\) is a twodimensional space all of whose members are realanalytic in (a, c) on the domain \(\Box ^{\circ }\).
Proof
To see that \({\mathcal E}_s\) is exactly twodimensional for all \(s \in {\mathbb C}\), note for fixed \(\mathfrak {R}(s) >0\) one has \({\mathcal E}_s= \langle L_s^{\pm }(a,c)\rangle \), with both functions of (a, c) being linearly independent. For fixed \(\mathfrak {R}(s) < 1\), one has \({\mathcal E}_s = \langle R_s^{\pm }(a, c)\rangle \), with both functions of (a, c) being linearly independent. \(\square \)
The reason for introducing four functions in (54) is that at integer values of s at least one of the four functions \(L_s^{\pm }(a, c), R_s^{\pm }(a, c)\) is identically zero, corresponding to poles in the term \(\gamma ^{\pm }(1s)\) in (49).
5.3 Operator properties of Lerch eigenspaces \({\mathcal E}_s\)
We show that the functions in the Lerch eigenspace are simultaneous eigenfunctions of the twovariable Hecke operators, the Lerch differential operator \(\mathrm{D}_L\) and the involution operator \(\mathrm{J}= \mathrm{R}^2\). We also show the vector spaces \({\mathcal E}_s\) are permuted by related operators.
Theorem 5
 (i)(Lerch differential operator eigenfunctions) Each \(f \in {\mathcal E}_s\) is an eigenfunction of the Lerch differential operator \(\mathrm{D}_L = \frac{1}{2 \pi i} \frac{\partial }{\partial a} \frac{\partial }{\partial c} + c \frac{\partial }{\partial c}\) with eigenvalue \(s\), i.e.,holds at all \((a, c) \in {\mathbb R}\times {\mathbb R}\), with both a and c nonintegers.$$\begin{aligned} \mathrm{D}_L f (s, a, c) = sf (s, a, c) \end{aligned}$$(57)
 (ii)(Simultaneous Hecke operator eigenfunctions) Each \(f \in {\mathcal E}_s\) is a simultaneous eigenfunction of all twovariable Hecke operators \(\{ \mathrm{T}_m : m \ge 1 \}\) with eigenvalue \(m^{s}\), in the sense thatholds on the domain \(({\mathbb R}\smallsetminus \frac{1}{m}{\mathbb Z}) \times ({\mathbb R}\smallsetminus {\mathbb Z})\).$$\begin{aligned} \mathrm{T}_m f = m^{s} f \end{aligned}$$(58)
 (iii)(\(\mathrm{J}\)operator eigenfunctions) The space \({\mathcal E}_s\) admits the involution \(\mathrm{J}f (a,c) = e^{2 \pi ia} f(1a, 1c)\), under which it decomposes into onedimensional eigenspaces \({\mathcal E}_s = {\mathcal E}_s^+ \oplus {\mathcal E}_s^\) with eigenvalues \(\pm 1\), namely \({\mathcal E}_s^{\pm } = < F_{s}^{\pm }>\) and$$\begin{aligned} \mathrm{J}(F_s^{\pm }) = \pm F_s^{\pm }. \end{aligned}$$(59)
 (iv)(\(\mathrm{R}\)operator action) The \(\mathrm{R}\)operator acts by \(\mathrm{R}({\mathcal E}_s)= {\mathcal E}_{1s}\) withwhere \(w_{+}= 1\) and \(w_{}=i\).$$\begin{aligned} \mathrm{R}(L_s^{\pm } ) = w_{\pm }^{1} {\gamma }^{\pm }(1s) L_{1s}^{\pm }, \end{aligned}$$(60)
This result yields the following consequence.
Corollary 1
(Invariance of \({\mathcal E}_s\) under Hecke operator families) For each \(s \in {\mathbb C}\), the Lerch eigenspace \({\mathcal E}_s\) is invariant under all four families of realanalytic twovariable Hecke operators \(\{ \mathrm{T}_m: m \ge 1\},\) \(\{ \mathrm{T}_m^{\vee }: m \ge 1\}\), \(\{\mathrm{S}_m: m \ge 1\},\) and \(\{\mathrm{S}_m^{\vee }: m \ge 1\}.\)
Proof
These four sets of operators are \(\{ \mathrm{R}^j \mathrm{T}_m \mathrm{R}^{j} \}\) for \(j=0,1 ,2, 3\). The case \(j=0\) is covered by (ii). By (iv) the effect of \(\mathrm{R}\) is to map the four generating functions for \({\mathcal E}_s\) to a permutation of them spanning \({\mathcal E}_{1s}\). Since the \(\mathrm{R}\) operator is applied an even number of times for \(j=1, 2, 3\), the final image is always \({\mathcal E}_s\).
To prove Theorem 5, we first determine the action of \({D_L^{+}}\) and \({D_L^{}}\) on the eigenspace \({\mathcal E}_s\).
Lemma 10
 (i)For each \(s \in {\mathbb C}\), the operator \({D_L^{}}= \frac{1}{2 \pi i} \frac{\partial }{\partial a} +c\) hasso that it takes \({D_L^{}}( {\mathcal E}_s ) = {\mathcal E}_{s1}\).$$\begin{aligned} ({D_L^{}}L_s^{\pm } )(s, a,c) ) = L_{s1}^{\mp } (s, a,c ), \end{aligned}$$(61)
 (ii)For each \(s \in {\mathbb C}\), the operator \({D_L^{+}}= \frac{\partial }{\partial c}\) hasso that it takes \({D_L^{+}}({\mathcal E}_s ) = {\mathcal E}_{s+1}\).$$\begin{aligned} ({D_L^{+}}L_s^{\pm }) (s, a,c) =  s L_{s+1}^{\mp }(s, a,c), \end{aligned}$$(62)
Proof
For \(\mathfrak {R}(s) < 0\) (resp. \(\mathfrak {R}(s) <1\)), we apply similar reasoning to the basis \(R_s^{\pm }(a,c)\) for \({\mathcal E}_s,\) which is possible since these functions have absolutely (resp. conditionally) convergent Dirichlet series expansion in this range. \(\square \)
Proof
If \(s \in {\mathbb C}\smallsetminus {\mathbb Z}\), then the coefficient on the right side of (68) is in \({\mathbb C}\smallsetminus \{0\}\) for both choices of sign ± , which exhibits an explicit isomorphism \(\mathrm{R}: {\mathcal E}_s \rightarrow {\mathcal E}_{1s}\). If \(s=s_0 \in {\mathbb Z}\), then exactly one of \(\Gamma _{{\mathbb R}}^\pm (s)\) and \(\Gamma _{{\mathbb R}}^\pm (1s)\) has a pole at \(s_0\), and (68) is not well defined. What happens is that one of \(L_s^{\pm }(a,c)\) or \(L_{1s}^{\pm } (a,c)\) is identically zero. This can be seen by studying the functional equation (68) as s varies, approaching the value \(s_0\). One side of the equation is defined and finite at \(s_0\); the other side must, for each fixed \((a,c) \in (0,1) \times (0,1)\), have the corresponding \(F_s^{\pm }(a,c) \rightarrow 0\) as \(s \rightarrow s_0\), since it is an entire function of s. However, we can still see that \(\mathrm{R}({\mathcal E}_{s_0} ) = {\mathcal E}_{1s_0}\), by noting that \(\mathrm{R}({\mathcal E}_s ) \subseteq {\mathcal E}_{1s}\) for \(s \not \in {\mathbb Z}\), and letting \(s \rightarrow s_0\) we obtain \(\mathrm{R}({\mathcal E}_{s_0} ) \subseteq {\mathcal E}_{1s_0} \), by analytic continuation in s. The image is necessarily twodimensional, since \(\mathrm{R}\) is invertible, so \(\mathrm{R}({\mathcal E}_{s_0} ) = {\mathcal E}_{1s_0}\) holds for \(s=s_0\). \(\square \)
5.4 Analytic properties of Lerch eigenspaces \({\mathcal E}_s\)
We establish the following analytic properties of members of \({\mathcal E}_s\), which we deduce from results in part I.
Theorem 6
 (i)(TwistedPeriodicity Property) All functions F(a, c) in \({\mathcal E}_{s}\) satisfy the twistedperiodic functional equations$$\begin{aligned} F(a+1,~c~)= & {} F(a,c), \end{aligned}$$(69)$$\begin{aligned} F(~a~, c+1)= & {} e^{2\pi i a} F(a, c). \end{aligned}$$(70)
 (ii)(Integrability Properties)
 (a)If \(\mathfrak {R}(s) >0\), then for each noninteger c all functions in \({\mathcal E}_{s}\) have \(f_c(a) :=F(a, c) \in L^{1}[(0,1), da],\) and all their Fourier coefficientsare continuous functions of c on \(0<c<1\).$$\begin{aligned} f_n(c) := \int _{0}^1 F(a,c) e^{2\pi i n a} da, \quad n \in {\mathbb Z}, \end{aligned}$$(71)
 (b)If \(\mathfrak {R}(s)<1\), then for each noninteger a all functions in \({\mathcal E}_{{s}}\) have \( g_a(c):=e^{2 \pi i a c}F(a, c) \in L^{1}[(0,1), dc],\) and all Fourier coefficientsare continuous functions of a on \(0< a<1\).$$\begin{aligned} g_n(a) := \int _{0}^1 e^{2\pi i a c}F(a,c) e^{2\pi i n c} dc, \quad n \in {\mathbb Z}, \end{aligned}$$(72)
 (c)
If \(0< \mathfrak {R}(s) < 1\) then all functions in \({\mathcal E}_{s}\) belong to \(L^{1}[\Box , da dc]\).
 (a)
Proof
 (i)
Theorem 2.2 of part I [28] established the twistedperiodicity functional equations for \(\zeta _{*}(s, a, c)\). It follows by repeated applications of \(\mathrm{R}\) that these functional equations also hold for \( e^{2\pi i a c} \zeta (1s, 1c , a)\), \(e^{2\pi i a} \zeta (s, 1a, 1c)\), \(e^{2\pi i a( c+1)} \zeta (1s, c, 1a).\) These four functions span the twodimensional vector space \({\mathcal E}_{s}\) for every \(s \in {\mathbb C}\).
 (ii)Part I [28, Theorem 6.1] shows for \(s \in {\mathbb C}\smallsetminus {\mathbb Z}\), that subtracting off suitable members of the four basis functionsfrom the two functions \(L^{\pm }(s, a, c)\) yields functions \(\tilde{L}^{\pm }( s,a,c)\) that are continuous on the closed unit square \(\Box = [0,1]\times [0,1]\), and which therefore belong to \(L^2[ \Box , da dc].\) Since$$\begin{aligned} c^{s}, e^{2\pi i a}(1c)^{s}, e^{2\pi i (1a) c} (1a)^{s1}, e^{2\pi i a c} a^{s1} \end{aligned}$$(73)it also has a continuous extension to the closed unit square after subtracting off suitable multiples of these four functions. In fact, only three of the four basis functions are needed in the subtraction, for \(\zeta _{*}(s, a, c)\) the function \(e^{2\pi i a}(1c)^{s}\) is omitted, see [28, Theorems 5.1 and 5.2]. At the integer values of s excluded, some of the terms subtracted off have poles.$$\begin{aligned} \zeta _{*}(s, a, c)= L^{+}( s, a, c) + L^{}( s, a, c), \end{aligned}$$
We next establish the remaining cases of properties (iia) and (iib), which are (iia) for integer \(s \ge 1\) and (iib) for integer \(s \le 1\). For (iia), we directly use the Fourier series expansion (8). After removing the term \(c^{s}\), which is clearly in \(L^1[(0,1), da]\), the remaining Fourier series for fixed \(0< c< 1\) is absolutely integrable at all integers \(s=n \ge 2\), and its Fourier coefficients are continuous in c by inspection. A similar property holds for \(\zeta _{*}(s, 1a, 1c)\), after removing the term \(e^{2\pi i a}(1c)^{s}\). Since for \(\mathfrak {R}(s)>0\) these functions span \({\mathcal E}_s\), property (iia) holds in these cases. For the remaining case \(s=1\), Rohrlich showed (see Milnor [32, Lemma 4]) that both the functions \(\zeta _{*}(1, a, c)  c^{1} = \sum _{n=1}^{\infty } \frac{e^{2\pi i na}}{n+c}\) and \(\zeta _{*}(1, 1a, 1c)  \frac{e^{2\pi i a}}{(1c)}\) are in \(L^1[(0,1), da]\) with the given Fourier coefficients, completing this case. The corresponding property (iib) for \(s=n \le 1\) is established similarly.
Property (iic) is shown in part I [28, Theorem 2.4]. \(\square \)
6 Hecke Eigenfunction characterization of Lerch Eigenspaces \({\mathcal E}_s\)
In this section, we characterize the Lerch eigenspace as being the complete set of simultaneous eigenfunctions of the family of twovariable Hecke operators \(\{ \mathrm{T}_m: m \ge 1\}\) that satisfy some auxiliary integrability and continuity conditions on the function. This can be viewed as a generalizing Milnor’s characterization of space of Hurwitz zeta functions, which we first explain in §6.1. Our characterization theorem is given in §6.2. We note that this characterization theorem does not impose any eigenfunction condition with respect to the differential operator \(D_L\).
6.1 Milnor’s theorem for Kubert functions
Milnor studied the family of operators \(\mathrm{T}_m : \mathrm{C}^0 ((0,1)) \rightarrow \mathrm{C}^0 ((0,1))\) given by (74). These operators form a commuting family of operators on \(\mathrm{C}^0 ((0,1))\).
Theorem 7
Proof
This is proved in [32, Theorem 1].
Milnor observes that \(\frac{\partial }{\partial x}\) maps \({\mathcal K}_s\) to \({\mathcal K}_{s1}\), acting as a “lowering operator.” Because the individual operators inside the sum on the right side of (74) are contracting, this “lowering operator” suffices in his proof.
In §3 we observed, in the twovariable context, that \(\frac{1}{2 \pi i} \frac{\partial }{\partial a} + c : {\mathcal E}_s \rightarrow {\mathcal E}_{s1}\) is a “lowering” operator, while \(\frac{\partial }{\partial c} : {\mathcal E}_s \rightarrow {\mathcal E}_{s+1}\) is a “raising” operator. Property (i) of Theorem 5 is derived using these properties. Milnor’s theorem formally corresponds to setting \(a=x\) and \(c=0\) in Theorem 5, except that \(c=0\) falls outside the domain of definition of the functions we consider.
6.2 Characterization of Lerch eigenspaces \({\mathcal E}_s\)
Milnor’s proof of Theorem 7 used in an essential way the property that for “Kubert operators” \(\mathrm{T}_m\) all terms on the right side of (74) are contracting operators on the domain \(x \in (0,1)\).
In contrast, the twovariable Hecke operators (6) are expanding in the cdirection. To deal with the expanding property, we impose extra analytic conditions on the function in the whole plane \({\mathbb R}\times {\mathbb R}\), in order to obtain a characterization of \({\mathcal E}_s\) as being simultaneous eigenfunctions of twovariable Hecke operators.
Our main result shows that the twistedperiodicity and integrabilities properties of Theorem 6 yield such a characterization.
Theorem 8
 (1)(TwistedPeriodicity Condition) For \((a,c) \in ({\mathbb R}\smallsetminus {\mathbb Z}) \times ({\mathbb R}\smallsetminus {\mathbb Z})\),$$\begin{aligned} F(a+1, ~c~)= & {} F(a,c), \end{aligned}$$(79)$$\begin{aligned} F(~a~, c+1)= & {} e^{2\pi i a} F(a,c) . \end{aligned}$$(80)
 (2)(Integrability Condition) At least one of the following two conditions (2a) or (2c) holds. (2a) The svariable has \(\mathfrak {R}(s) > 0\). For \(0<c<1\) each function \(f_c(a):= F(a, c) \in L^1[(0,1), da]\), and all the Fourier coefficientsare continuous functions of c. (2c) The svariable has \(\mathfrak {R}(s) < 1\). For \(0<a<1\) each function \(g_a(c):= e^{2 \pi i a c}F(a,c) \in L^1[(0,1), dc]\), and all the Fourier coefficients$$\begin{aligned} f_n(c) := \int _{0}^{1} f_c(a)e^{2\pi i n a} da= \int _{0}^1 F(a,c) e^{2\pi i n a}da,\quad n \in {\mathbb Z}, \end{aligned}$$are continuous functions of a.$$\begin{aligned} g_n(a) := \int _{0}^{1} g_a(c)e^{2\pi i n c} dc= \int _{0}^1 e^{2\pi i a c}F(a,c) e^{2\pi i n c}dc,\quad n \in {\mathbb Z}, \end{aligned}$$
 (3)(Hecke Eigenfunction Condition) For all \(m \ge 1\),holds on the domain \(\{ (a, c) \in ({\mathbb R}\smallsetminus {\mathbb Z})\times ({\mathbb R}\smallsetminus \frac{1}{m}{\mathbb Z})\}\).$$\begin{aligned} \mathrm{T}_{m}(F)(a, c) = m^{{s}}F(a, c) \end{aligned}$$(81)
Remarks. (i) Theorem 6 shows that all functions in \({\mathcal E}_s\) for \(\mathfrak {R}(s) >0\) satisfy conditions (1), (2a) and (3) and all functions in \({\mathcal E}_s\) for \(\mathfrak {R}(s) < 1\) satisfy conditions (1), (2c) and (3) above. Conditions (2a), (2c) between them cover all \(s \in {\mathbb C}\), and they hold simultaneously inside the critical strip \(0< \mathfrak {R}(s) <1\).
(ii) The function \(F(a, c) := c^{s}\) satisfies properties (2a) and (2c) and also the eigenvalue property (3). However, it fails to satisfy the twistedperiodicity property (1).
Proof
7 Concluding remarks
 (1)
At \(a=0\), the operator \(\mathrm{T}_m\) degenerates to the dilation operator \(\tilde{\mathrm{T}}_m (f) (c) = f(mc)\). In 1999, BáezDuarte [4] noted that this family of dilation operators relates to the realvariables approach to the Riemann hypothesis due to Nyman [33] and Beurling [11], see also BáezDuarte [5], Burnol [14, 15] and Bagchi [6]. Therefore, one may ask whether there is a Riemann hypothesis criterion directly formulable in terms of the twovariable Hecke operators \(\mathrm{T}_m\).
 (2)
The Lerch zeta function at \(a=0\) reduces for \(\mathfrak {R}(s)>1\) to the Hurwitz zeta function. The Hurwitz zeta function inherits the discontinuities of the Lerch zeta function at integer values of c. Milnor [32, p. 281] noted that at the value \(s=1\) the space \({\mathcal K}_s\) includes on (0, 1) the odd function \( c  \frac{1}{2} \) which, due to the discontinuities, extends to the periodic function \(\beta _1(c):= \{ c \frac{1}{2}\}\), the first Bernoulli polynomial, a fractional part function. The fractional part function appears in the various realvariables forms of the Riemann hypothesis above. The discontinuities of the Lerch zeta function at integer values of a or c (for some values of s) represent an important feature of these functions, worthy of further study in this context.
 (1)It is natural to consider the symmetrized Lerch differential operatorThis operator has many features of a Laplacian operator. It is formally skewsymmetric and satisfies$$\begin{aligned} \Delta _L := \frac{1}{2} \left( D_L^{+}D_L^{} + D_L^{} D_L^{+}\right) = \frac{1}{2 \pi i} \frac{\partial }{\partial a}\frac{\partial }{\partial c} + c \frac{\partial }{\partial c} + \frac{1}{2} \mathbf{I }. \end{aligned}$$(101)so that the line of skew symmetry is the critical line \(Re(s) = \frac{1}{2}\). This operator has the “xp” form suggested by Berry and Keating [8, 9], as the appropriate form for a “Hilbert–Polya” operator encoding the zeta zeros as eigenvalues.$$\begin{aligned} ( \Delta _L\zeta )(s, a, c) = \left( s \frac{1}{2}\right) \zeta (s,a , c), \end{aligned}$$
 (2)
The operator \(\Delta _L\) commutes with all the \(\mathrm{T}_m\) on the twodimensional Lerch eigenspace \({\mathcal E}_s\), which is, however, not contained in \(L^2({\Box }, da \, dc)\). It formally commutes with the \(\mathrm{T}_m\), but its commutativity depends on the specified domain of the unbounded operator \(\Delta _L\), viewed inside \(L^2({\Box }, da \, dc)\). Such a domain is specified in [27, Sect. 9.2], for which the resulting operator \(\Delta _L\) has purely continuous spectrum.
 (3)
In order to view \(\Delta _L\) as a suitable Hilbert–Polya operator for zeta zeros along these lines, it may be that one must instead find a scattering on \({\mathcal H}\) and a small closed subspace of \({\mathcal H}\) carrying the operator \(\Delta _L\). Related viewpoints on Hilbert–Polya operators have been proposed by Connes [16, 17], Burnol [12] and the first author [26].
Acknowledgements
The authors thank Paul Federbush for helpful remarks regarding Fourier expansions in Theorem 8. The authors thank the two reviewers for many helpful comments and corrections. In particular, we thank one of them for the observation (41) which strengthened Theorem 2.1 (3). This project was initiated at AT&T LabsResearch when the first author worked there and the second author consulted there; they thank AT&T for support. The first author received support from the Mathematics Research Center at Stanford University in 2009–2010. The second author received support from the National Center for Theoretical Sciences and National Tsing Hua University in Taiwan in 2009–2014. To these institutions the authors express their gratitude. The research of the first author was supported by NSF Grants DMS0801029, DMS1101373 and DMS1401224, that of the second author by NSF Grant DMS1101368 and Simons Foundation Grant #355798.
There is no need to take the absolute value in this formula, but it becomes useful in further generalizations.
The operator \(A_n\) appears for prime p as \(A_p= T_p^{*}  U_p^{*}\). However, Atkin and Lehner apply \(T_p^{*}\) only for \(p \not \mid N\), the level, and \(U_p^{*}\) for p  N, but the definitions of each operator make sense for all p, so we may relate them in the identity above.
One can define a refined notion which allows discontinuities with denominator \(d_1\) in the cdirection and \(d_2\) in the adirection. Here d is the least common multiple \(d= [ d_1, d_2]\). One can also define a refined notation \((d_1, d_2)\) of conductor in the variables separately.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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