On Dirichlet eigenvalues of regular polygons

We prove that the first Dirichlet eigenvalue of a regular $N$-gon of area $\pi$ has an asymptotic expansion of the form $\lambda_1(1+\sum_{n\ge3}C_n(\lambda_1)N^{-n})$ as $N\to\infty$, where $\lambda_1$ is the first Dirichlet eigenvalue of the unit disk and $C_n$ are polynomials whose coefficients belong to the space of multiple zeta values of weight $n$. We also explicitly compute these polynomials for all $n\le14$.

One can consider many aspects regarding the asymptotics of Dirichlet eigenvalues and Dirichlet eigenfunctions, a large portion of which could be regarded as "classical" and being intensely studied: without aiming at being thorough, we just mention, for example, various forms of Weyl laws prescribing the asymptotics of large eigenvalues λ k in terms of the underlying geometric data; concentration phenomena for eigenfunctions and level set distribution; behaviour of eigenvalues with respect to domain perturbation; etc.For a very thorough and accessible overview we refer to the treatments in [8], [28].
In this note we are interested in the behavior of the eigenvalues with respect to domain perturbations in the special case of regular polygons.Let P N be a regular polygon of area π with N ≥ 3 sides.We study the behavior of λ k (P N ) as N goes to infinity.More precisely, we are interested in computing the coefficients C k,n of the asymptotic series (1) λ k (P N ) where D is the unit disk.
The above problem has been considered in several previous works.As an outcome of the works [17], [11], [12] the first four coefficients C 1,i were computed and, to a certain surprise, expressed as integer multiples of the Riemann zeta function.Roughly speaking, the corresponding methods (Calculus of Moving Surfaces) require one to consider an explicit deformation of the polygon into a disk and study the evolution of the corresponding eigenfunction.
Later on, the next two coefficients C 1,5 , C 1,6 were also computed and given in a similar form (cf. [4]).Recently, in [14] an expression for the next two coefficients C 1,7 , C 1,8 was proposed as a result of high-precision numerics and certain linear regression methods.
To summarize the above results, assuming that the polygons under consideration are normalized so that Area(P N ) = π, the asymptotic expansion (with proposed seventh and eighth terms being conjectural) is the following Here λ 1 = λ 1 (D) is the first Dirichlet eigenvalue of the unit disk.Recall that λ 1 = j 2 0,1 , where j 0,1 is the smallest positive zero of the Bessel function of the first kind J 0 .Formula (2) might lead one to suspect that all higher coefficients of the asymptotic expansion can be expressed as polynomials in λ 1 with coefficients that are polynomials in odd zeta values ζ(2m + 1), m ≥ 1.As we will show below, this is indeed the case for the first 10 coefficients of the expansion, but, assuming some widely believed algebraic independence results, the 11-th coefficient is no longer of this form.
As further motivation we note a couple of related problems.Drawing inspiration from the Faber-Krahn inequality and a conjecture of Pólya and Szegö (which states that among all n-gons with the same area, the regular n-gon has the smallest first Dirichlet eigenvalue), it was conjectured in [1] that for all N ≥ 3 and Area(P N ) = π, the first Dirichlet eigenvalues are monotonically decreasing in N, i.e., So far the monotonicity has been confirmed by numerical experiments (cf.[1]).Note, however, that since ζ(3) > 0, equation (2) implies this inequality for all sufficiently large N.For further results along the theme of Faber-Krahn and eigenvalue optimization via regular polygons under the presence of various constraints (in-and circumradius normalization, etc.) we refer to [19], [21] and the references therein.
A further intriguing application of the above eigenvalue asymptotics can be found in [20], where the Casimir energy of a scalar field on P N (and further generalized to P N × R k ) has been studied.For background we refer to [20] and the accompanying references.
Before describing our main results, we briefly recall the definition of multiple zeta values.Multiple zeta values (MZVs) are real numbers defined by where m 1 , . . ., m r are positive integers and m r > We denote the Q-linear span of all multiple zeta values of weight n by Z n .Our main result is the following theorem.
Theorem 1.There exists a sequence of polynomials , where Z n is the space of multiple zeta values of weight n, such that whenever λ k is a radially-symmetric Dirichlet eigenvalue of the unit disk.
Here radially-symmetric eigenvalues are λ k 's for which the corresponding eigenfunction is radially-symmetric.Explicitly, the theorem applies whenever λ k = j 2 0,m , where j 0,m is the m-th root of the Bessel function J 0 (x).In particular, the theorem applies in the case k = 1.
Our proof of Theorem 1 is based on an asymptotic version of the "method of particular solutions" (see, e.g., [10], [18]) and it provides an explicit procedure for producing increasingly better approximations (at least when N → ∞) to both the eigenvalues and the eigenfunctions (as long as they correspond to the radially symmetric eigenfunctions on the unit disk).We find that not only the eigenvalues, but the eigenfunctions themselves have asymptotic expansions in powers of 1/N with interesting coefficients (these coefficients turn out to be multiple polylogarithms, see Table 1.Coefficients of the asymptotic expansion for n ≤ 14 Our approach gives an explicit symbolic algorithm for computing the polynomials C n .We have calculated C n for n ≤ 12 using an implementation of this algorithm in the computer algebra system SAGE [22] and for n ≤ 14 using an optimized parallel implementation in Julia [15].As an immediate corollary we confirm the 7-th and 8-th terms in the asymptotic expansion (1) that were conjectured in [14].We collect the results of our calculations in Table 1.The initial expressions in terms of MZVs that we get by directly applying our algorithm are rather unwieldy and to obtain the simpler expressions given in the table we have used the MZV Datamine [3].In the table we use the notation for the first few nontrivial single-valued MZVs.The space of single-valued multiple zeta values of weight n is an important subspace of Z n that was introduced by Brown [6].Singlevalued MZVs appear, for example, in computation of string amplitudes, and as coefficients of Deligne's associator (for other examples, see the references in [6]).
Conjecture 1.The polynomial C n (λ) belongs to Z sv n [λ] for all n ≥ 1, where Z sv n denotes the space of single-valued multiple zeta values of weight n.
The results given in Table 1 confirm this conjecture for n ≤ 14, and in [2] we give strong numerical evidence in its support also for n = 15 and n = 16.If true, Conjecture 1 would also explain the curious fact that when P N is normalized to have area π (as opposed to, say, normalizing P N to have circumradius 1), the low order coefficients of the resulting asymptotic expansion do not involve even zeta values (see [4, p. 125]).
By analyzing the general recursion for C n (λ) obtained in the proof of Theorem 1 we obtain a formula for the generating function of the first two coefficients of the polynomials C n (λ).
Using this formula we also prove the following.
Theorem 3. The coefficients C n (0) and C ′ n (0) are polynomials with rational coefficients in odd zeta values of homogeneous weight n.
The proof is based on a hypergeometric identity (27) of Ramanujan-Dougall type that could be of independent interest.Theorem 3 confirms Conjecture 1 for the first two coefficients C n (0) and C ′ n (0).Note that the expression for C 11 from Table 1 shows that C ′′ 11 (0) involves ζ sv 3,5,3 , and thus the claim of Theorem 3 in general fails for C ′′ n (0), assuming the (widely believed) algebraic independence of ζ sv 3,5,3 and ζ(2m + 1), m ≥ 1 (see [6, p. 35]).As a final remark, we note that the normalizing factor that appears in several of our formulas is a specialization of Virasoro's closed bosonic string amplitude [24].We do not know if this is a simple coincidence, or if there is some conceptual explanation for this.

Multiple polylogarithms and multiple zeta values
In this section we very briefly recall some basic properties of multiple polylogarithms and multiple zeta values.For a much more detailed introduction (including the algebraic structure and interpretation of MZVs as periods of mixed Tate motives) we refer the reader to [25], [7].
The numbers ζ(m 1 , . . ., m r ) are called multiple zeta values (MZVs).We denote by Z the Q-linear span of all multiple zeta values and by Z k the Q-linear span of all multiple zeta values of weight k, i.e., the linear span of ζ(m 1 , . . ., m r ) over all r-tuples (m 1 , . . ., m r ) satisfying m 1 + • • • + m r = k.The Q-vector space Z forms an algebra (see (9) below), and multiplication respects weight, i.e., Z k • Z l ⊆ Z k+l .Zagier [27] has conjectured that there are no rational linear relations between elements of Z k for different k (that is, that Z = k≥0 Z k ) and that dim Z k = d k , where d k are defined by the generating series The upper bound dim Z k ≤ d k has been proved independently by Goncharov and Terasoma, but no nontrivial lower bounds for dim Z k are presently known.
For our purposes it is more convenient to index multiple polylogarithms by words in two letters X = {x 0 , x 1 }, reflecting their structure as iterated integrals as opposed to the definition as an infinite sum (5).In our treatement we mainly follow Brown [5] (see also [13]).Let X × be the free noncommutative monoid generated by X, i.e., the set of all words in x 0 , x 1 equipped with the concatenation product.Then {Li w } w is a family of analytic functions on the cut plane C ((−∞, 0] ∪ [1, ∞)) satisfying the recursive relations together with the following initial conditions: Li e (z) = 1, Li x n 0 (z) = 1 n! log n (z), and lim z→0 Li w (z) = 0 for all w ∈ X × not of the form x n 0 .Here e ∈ X × denotes the empty word.These conditions uniquely determine Li w (z) and for m 1 , . . ., m r ≥ 1 one has (8) Li . We also extend the notation Li w (z) by linearity to the elements of the monoid ring C X , i.e., for any formal combination i a i w i we set Li i a i w i (z) = i a i Li w i (z).Note that the algebra C X is graded by word length, and we denote by C X n the n-the graded piece.We will also write |w| for the length of w ∈ X × and we will say that the function Li w (z) has weight |w|.(Since the functions Li w (z) are linearly independent over C(z), see [5], this notion of weight is well-defined.) An important property of the space of multiple polylogarithms is that it is closed under mutliplication.More precisely, one has (9) Li w (z)Li w ′ (z) = Li w¡w ′ (z) .
Here ¡: C X × C X → C X denotes the shuffle product that is defined on words by where σ runs over all permutations satisfying σ −1 (1) Note that for w ∈ X × x 1 the function Li w (z) extends analytically to C [1, ∞), and for w ∈ x 0 X × x 1 it is moreover continuous on D. We will call the words w ∈ x 0 X × x 1 convergent and we will also call convergent any formal linear combination of convergent words in C X (in other words, all elements of x 0 C X x 1 are convergent).An important corollary of (9), is that for any w ∈ X × x 1 there exists a unique collection of convergent elements w 0 , w 1 , . . ., This allows one to define the multiple zeta value ζ(w) = Li w (1) in cases when the series diverges by setting Li w (1) := Li w 0 (1) for all w ∈ X × x 1 .
First, let us note the following simple corollaries of the definition of Li w .
Lemma 1.For all w ∈ X × x 1 and k ≥ 0 we have Proof.This follows trivially from ( 7) by induction on k.
Lemma 2. For all w ∈ X × x 1 we have Moreover, if u and v are convergent, then α(u, v) and β(u, v) are also convergent.
Proof.We will prove the statement by induction on k for all u, v ∈ X × x 1 ⊔ {e}.For the base of induction, when either u = e or v = e the identity becomes trivial if we set A u,e = A e,v = 0, α(u, e) = u, α(e, v) = 0, and β(u, e) = 0, β(e, v) = v.Note that from (7) it follows that for all w ∈ X × we have In view of this we recursively define α(au, bv) = aα(u, bv) + bα(au, v) where x 0 = −x 0 , x 1 = x 0 + x 1 , and δ is defined by δ(x 1 ) = 1, δ(x 0 ) = −1.Then by induction we obtain that If u, v, α, and β are all convergent, then we may simply take z = 1 to get (1).Otherwise we take z = e 2πix and take the limit x → 0+, which corresponds to a regularization of Li w (1) given by Li w (e ±2πi0 ) : where Li w (z) = k j=0 Li w j (z)Li j 1 (z) with all w j in x 0 C X x 1 .Thus A u,v ∈ Z k and by induction we get also that α(u, v) and To verify the last claim let us consider u = x 0 u ′ , v = x 0 v ′ .The recursive definition ( 12) with a = b = x 0 shows that α(u, v) and β(u, v) would be convergent if we can show that for some w, w ′ ∈ C X x 1 and hence, by Lemma 2, we obtain The proof shows that we may take β(u, v) = α(v, u).Note also that if we extend the definition of α(u, v), β(u, v), and A u,v to bilinear functionals on C X x 1 × C X x 1 , the identity (10) remains true for all u, v ∈ C X x 1 .
As a corollary of the above proposition we have the following curious fact.
Corollary 1.For all u, v ∈ X × x 1 we have Proof.It follows from (10) and Lemma 2 that and the claim then follows from Proposition 1.
As a further corollary, note that when u and v are both convergent equation (10) implies This formula thus gives a purely algebraic solution of the Dirichlet boundary value problem u(z) = ϕ(z) for |z| = 1 where ϕ is of the form ϕ(z) = Re Li u (z)Li v (z) and u is sought to be harmonic in D. It is exactly in this form that we will use Proposition 1 in the proof of Theorem 1.

Proof of Theorem 1
Let us fix the polygon P N ⊂ C to be the convex hull of {cζ j } 0≤j<N , where ζ is a primitive n-th root of unity, and c > 0 is chosen so that Area(P N ) = π.We will utilize the classical Schwarz-Christoffel map, f : D → P N , which maps the unit disk D conformally onto P N .It is given by any of the following equivalent expressions ( 13) where the constant ( 14) is determined by the condition that Area(P N ) = π.Here 2 F 1 is the ordinary Gauss hypergeometric function where (x) n := x(x + 1) . . .(x + n − 1) denotes the rising Pochhammer symbol.
The first fact that we will need is that the function F N (x) := 2 F 1 (2/N, 1/N, 1 + 1/N; x) can be expanded as a power series in 1/N (convergent for N ≥ 3) whose coefficients are multiple polylogarithms.Let us recall the definition of Nielsen polylogarithms (see [16], [9]) Lemma 3.For all N ≥ 3 and |x| ≤ 1 we have where Proof.This follows by expanding in powers of 1/N the right hand side of dt t and using the definition (15).
We will also need a formula for the asymptotic expansion of the Bessel function J 0 (x) around its zero.Recall that J 0 (x) satisfies xJ ′′ 0 (x) + J ′ 0 (x) + xJ 0 (x) = 0 and can be defined by the Taylor series where ).Let R n (x) denote the difference between the left hand side and the right hand side in the above equation.A simple calculation shows that R n (x) → 0 as x → ∞, and since R n ∈ Q(x), it is enough to show that it has no poles.The only potential singularities are at (Here j =k denotes the product over 1 ≤ j ≤ n, j = k.)Then the coefficient in front ε −2 vanishes since n k 2 = j =k ( k j − 1) −2 , and for ε −1 the vanishing is equivalent to which is again easy to verify.(ii) Let us denote the left hand side of (17) by f (x) and the right hand side by g(x).From the differential equation xJ ′′ 0 (x) + J ′ 0 (x) + xJ 0 (x) together with J ′ 0 (x) = −J 1 (x), we obtain that f ′′ (x) + α 2 e 2x f (x) = 0 and f (0) = 0, f ′ (0) = 1.Thus, it is enough to check that g(x) satisfies the same differential equation and initial conditions.The conditions g(0) = 0, g ′ (0) = 1 follow from part (i).Using the easily checked identity we get that g ′′ (x) + α 2 e 2x g(x).
Proposition 3. Let Ω be a bounded domain in R 2 , and let f : Ω → R be a function in Proof.This is a special case of [18,Theorem 1].
We are now ready to prove our main result.
By the general theory developed by Vekua [23, (13.5), p. 58] any function ϕ that satisfies ∆ϕ + λ (N ) ϕ = 0 in P N can be represented as (19) ϕ where a 0 ∈ R and U : P N → C is some holomorphic function.Since by assumption ϕ(z) is dihedrally-symmetric, we may write U(f (t)) = U (t N )/t, where U (0) = 0 and the Taylor series of U at 0 has real coefficients (we recall that f is defined by ( 13)).Then After replacing z and t by z 1/N and t 1/N respectively and setting ψ(z) := ϕ(f (z 1/N )) and where we set ρ := c 2 N λ (N ) and Now we make an ansatz that where V j : D → C are holomorphic and V j (0) = 0.In view of Proposition 2 (ii) it is convenient to set a 0 = c N λ 1/2 J 1 (λ 1/2 ) .We will expand (20) as an asymptotic series in powers of 1/N and then recursively compute κ j and V j using the boundary condition ψ(z) = 0, |z| = 1.Note that by (16) F N (x) = 1 + O(N −2 ) and using (16) and the expansion (t/z) 1/N = n≥0 where γ u,v,w,m are coefficients that depend on κ i and the summation is over words u, v, w ∈ x 0 X × x 1 and m ≥ 0 satisfying |u| + |v| + |w| + m ≤ r.We get a similar expression (involving only products Li u (z)Li v (z)) after expanding a 0 J 0 (ρ 1/2 |F N (z)|) using (17).We claim that κ i and V i (z) can be calculated inductively by comparing the coefficients of the 1/N-expansion.Indeed, comparing the coefficients of 1/N we see that X x 1 is of total weight i (we define the total weight of λ a zw, where z ∈ Z k to be k + |w|).Note that by Lemma 1 where w ′ = v ¡x m+1 0 w.Thus, when comparing the coefficients of N −k−1 , we need to ensure an identity of the form where the terms in the sum only depend on already computed quantities κ 1 , . . ., κ k and V 0 (z), . . ., V k−1 (z) and all have total weight k + 1 (where we define the total weight of γLi By Proposition 1 this amounts to setting This indeed can be done since by assumption the elements u and v are convergent and hence α(u, v), β(u, v) ∈ x 0 C X x 1 .This gives an explicit algebraic recursion for κ n and V n (z) that shows, in particular, that κ n ∈ Z n [λ].In Table 2 we list the functions V n (z) for n ≤ 4. We also note that the coefficients κ n vanish for n ≤ 4. We still need to verify that the ansatz (21) indeed gives an asymptotic expansion for λ (N ) .To see this, note that plugging a truncated solution for the boundary condition ψ Table 2.The functions V n (z) for n ≤ 4 |z| = 1 back into (19) we obtain a sequence of functions ϕ N,r : P N → R, and numbers λ N,r .The numbers λ N,r converge to λ k as N → ∞ for each fixed r and the functions ϕ N,r satisfy ∆ϕ N,r (z) + λ N,r ϕ N,r (z) = 0 , z ∈ P N , together with ϕ N,r 2 ≫ 1 and ϕ N,r | ∂P N ∞ ≪ r N −r−1 .Therefore, applying Proposition 3 shows that |λ N,r − λ k (P N )| ≪ r N −r−1 , so that λ N,r indeed give an asymptotic expansion for λ k (P N ).
Finally, expanding G N (t) as a power series in t and integrating the above identity term-byterm we obtain Finally, let us prove Theorem 3.For this we will need the following lemma to evaluate the right hand side of (24) in terms of gamma function and its derivatives.
Proof.We will prove this identity by induction on m, the case m = 0 being trivial.