Pólya’s conjecture for Euclidean balls

The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds. Our proofs are purely analytic, except for a rigorous computer-assisted argument needed to cover the short interval of values of the spectral parameter in the case of the Neumann problem in the disk.

( . ) It is well known that the spectrum of ( . ) is discrete and consists of isolated eigenvalues of nite multiplicity accumulating to +∞, which we enumerate with account of multiplicities.
Similarly, assuming additionally that ∂Ω is Lipschitz, consider the Neumann eigenvalue problem where ∂ n u = 〈∇u, n〉| ∂Ω denotes the normal derivative of u with respect to the exterior unit normal n on the boundary.The spectrum of ( . ) again consists of isolated eigenvalues of nite multiplicity accumulating to +∞, enumerated with account of multiplicities.Let, for λ ∈ R, denote the counting functions of the Dirichlet and Neumann eigenvalue problems on Ω.It follows from the variational principles for ( . ) and ( . ) that for any λ ≥ 0.
Under the assumptions stated above, the leading term asymptotics of the counting functions is given by Weyl's law [Wey ], Strictly speaking, we are counting the number of eigenvalues less than or equal to a given λ 2 , but such normalisation will be convenient to us throughout.
One can also de ne the counting functions using strict inequalities; this does not a ect any of the results below.This in fact can be improved to N D where N Ω (λ) denotes either N D Ω (λ) or N N Ω (λ), | • | d denotes the d -dimensional volume, R(λ) = o λ d as λ → +∞, and is the so-called Weyl constant.We refer to [SafVas ] for a historical review, as well as numerous generalisations and improvements.H. Weyl himself conjectured [Wey ] a sharper version of ( . ) taking into account the boundary conditions: for Ω ⊂ R d with a piecewise smooth boundary, where the minus sign is taken for the Dirichlet boundary conditions and the plus sign for the Neumann ones, and We note that for planar domains ( . ) takes the particularly simple form ( . ) The two-term Weyl's law ( . ) remains open in full generality.It has been proved by V. Ivrii [Ivr ] under the condition that the set of periodic billiard trajectories in Ω has measure zero.While this condition is conjectured to be satis ed for all Euclidean domains, it has been veri ed only for a few classes, such as convex analytic domains and polygons, see [SafVas ] and references therein.Speci cally for a disk, it was proved by N. Kuznetsov and B. Fedosov in [KuzFed ].
Assuming that the two-term Weyl's asymptotics ( . ) holds for a domain Ω ⊂ R d , we immediately obtain that for λ above some sufficiently large but unspecified value Λ 1 we have ( . ) We refer also to [Mel ] for results of the same kind in the Riemannian setting.In , G. Pólya [Pól ] conjectured that the inequalities ( . ) hold for all λ ≥ 0. He later proved this conjecture in [Pól ] for tiling domains Ω: that is, domains such that R d can be covered, up to a set of measure zero, by a disjoint union of copies of Ω.In fact, in the Neumann case, some additional assumptions were imposed in [Pól ] that have been removed in [Kel ].It has been also shown that Pólya's conjecture in the Dirichlet case holds for a Cartesian product [Lap , Theorem . ].For general domains, somewhat weakened versions of ( . ) are known to hold as a consequence of the so-called Berezin-Li-Yau inequalities: we have for all λ ≥ 0, see [LiYau ], [Krö ], and [Lap ].We refer also to [Lin ], [KLS ], [FLP ], and [FreSal ] for some recent results on Pólya's conjecture and further interesting links to other problems in spectral geometry.
In fact, Pólya's original conjecture was only for planar domains, and in a slightly di erent form.

Pólya's conjecture for Euclidean balls
Remark . .Pólya's conjecture ( . ) can be equivalently restated as the inequalities for the eigenvalues (instead of the counting functions), for all n ≥ 1.It is known that inequalities ( . ) hold for any domain in any dimension for n = 1, 2. In particular, for n = 1 this follows from the celebrated Faber-Krahn and Szegő-Weinberger inequalities, and for n = 2 in the Dirichlet case from the Krahn-Szego inequality, see [Hen ].For n = 2 in the Neumann case, we refer to [GNP ], [BucHen ].These are the only eigenvalues for which it is known in full generality.We refer also to [Fre ] for further results on the validity of the Dirichlet Pólya's conjecture for low eigenvalues in higher dimensions.
Remarkably, since balls do not tile the space, Pólya's conjecture has so far remained open for Euclidean balls, including planar disks.Although all the eigenvalues of the Dirichlet and Neumann Laplacians on the unit disk are explicitly known in terms of zeros of the Bessel functions or their derivatives, see § below, in each case the spectrum is given by a two-parametric family, and rearranging it into a single monotone sequence appears to be an unfeasible task.
. Therefore the leading Weyl's term in ( . ) for B d becomes The main results of this paper address the validity of Pólya's conjecture for disks and balls.Namely, we prove the following results.
Theorem . .The Dirichlet Pólya's conjecture for the unit ball holds in any dimension d ≥ 2, that is we have for all λ > 0.
Our results in the Neumann case are restricted to the case d = 2. Higher-dimensional Neumann problems are harder, and we intend to treat them in a subsequent paper.
We rst state where Proof.Taking the span of {1, x, y} as a test space in the Rayleigh quotient for the Neumann Laplacian on D gives µ 3 (D) ≤ 4. Therefore, As stated in [Lap , p. ]: "Remarkably this conjecture still remains open even for such a simple domain as the disc, where the eigenvalues of the Dirichlet Laplacians could be calculated via the roots of Bessel functions."See also [FLW , p. ] and [Lau , p. ].
( . ) We note that Λ 0 > 3 and Λ 1 < 14, so we already have the validity of the Neumann Pólya's conjecture for the disk for all λ outside the interval (3, 14).
The proof of Theorem . is rigorous but computer-assisted.More speci cally, it is based on a realisation of an algorithm which satis es two fundamental principles.
Principle .The algorithm should complete in a nite number of steps.
Principle .The algorithm should operate only with integer or rational numbers, thus avoiding any use of oating-point arithmetic and any rounding errors.
The combination of Lemma .and Theorems .and .ensures that the Neumann Pólya conjecture for the disk is valid for all λ > 0, that is we have 4 for all λ > 0.
Remark . .Since Pólya's conjecture is scale-invariant, its validity for a unit ball immediately implies that it is valid for any ball of the same dimension.
We additionally have the following generalisation of Pólya's result for tiling domains: we show that Pólya's conjecture holds not only for domains which tile Euclidean space, but also for domains which tile another domain for which it is known to be true.
Theorem . .Let Ω ⊂ R d be a domain for which either the Dirichlet or the Neumann Pólya's conjecture holds, and let Ω be a domain which tiles Ω.Then the same Pólya's conjecture also holds for Ω .
Proof.Assume that Ω can be tiled by ≥ 2 congruent copies of Ω , so that |Ω| d = |Ω | d .We have, by bracketing and since the eigenvalues of all the congruent copies coincide with those of Ω , Assuming now ( . ) for all λ ≥ 0, we get and the result follows by cancelling .
Remark . .If the inequalities in Pólya's conjecture ( . ) for Ω are strict, they are also strict for Ω .
Theorem .immediately implies the following Corollary . .Let Ω ⊂ S d −1 be a spherical domain which tiles S d −1 .Then the Dirichlet Pólya's conjecture holds for the spherical cone in R d with the base Ω and the vertex at the origin.

Pólya's conjecture for Euclidean balls
We refer also to [FreSal ] for an alternative proof of Corollary .for su ciently large (but unspeci ed) .
We can in fact extend the result of Corollary .to arbitrary sectors.

Plan of the paper
In the next section we describe two lattice counting problems ( . ) and ( .), variants of which were originally introduced by N. Kuznetsov and B. Fedosov in [KuzFed ], and which are closely linked to the Dirichlet and Neumann eigenvalue counting problems in the ball.The key novel tool is Theorem ., originally obtained in part in [She ], which gives a uniform bound between the eigenvalue and the lattice counting functions, as opposed to asymptotic relations that were previously known.We provide an independent proof of this result in § .In § we state the results on the lattice counting functions which are su cient for proving Pólya's conjecture for balls.The bulk of the paper, § § -, is devoted to the proofs of these results.Theorem . is proved in § .

§ . Dirichlet and Neumann eigenvalues of the ball and lattice counting problems
Throughout this paper, with ν ≥ 0, and k ∈ N, let J ν (z) be the Bessel functions of order ν, let j ν,k be the kth positive zero of J ν , and let j ν,k be the kth positive zero of its derivative J ν , with the exception of J 0 for which j 0,1 = 0.
It is well known that the eigenvalues of the Dirichlet Laplacian in the unit ball are given by the squares of the zeros of the cylindrical Bessel functions.Namely, considering the Dirichlet Laplacian in B d , we have the simple eigenvalues that is of multiplicity and the eigenvalues Remark . .We note that the numbers κ d ,m , d ≥ 2, m ∈ N 0 = N ∪ {0}, coincide with the multiplicity of the eigenvalue m(m +d −2) of the Laplace-Beltrami operator on the unit sphere S d −1 , or alternatively with the dimension of the space of homogeneous harmonic polynomials of degree m in R d .In the planar case we have We therefore have ( . ) Remark . .The sum in ( . ) is in fact nite: we have This is due to the fact that j ν,1 > ν [DLMF, Eq. . .].Note that in ( . ) and further on, any sum in which the lower limit exceeds the upper limit is assumed to be zero, which immediately gives In the planar case the expression ( . ) simpli es to Similarly, the eigenvalues of the Neumann Laplacian in the unit disk D are given by the squares of the zeros of the derivatives of Bessel functions.We have the simple eigenvalues and the double eigenvalues We therefore have where the sum is again nite since j ν,1 ≥ ν [DLMF, Eq. . .].
For illustrative purposes only, we show the graphs of the Dirichlet and Neumann eigenvalue counting functions for the disk in Figure .
We will be comparing the counting functions N D B d (λ) and N N D (λ) with some weighted lattice counting functions.Let and let P be a planar region under the graph of h(x), Here and further on, x = max{k ∈ Z : k ≤ x} denotes the integer part of x ∈ R, and x = min{k ∈ Z : k ≥ x} denotes its ceiling.
The plot is produced using the oating-point evaluation of zeros of the Bessel functions and their derivatives.If we were to assume (contrary to the philosophy of this paper) the validity of oating-point arithmetic, this plot would have presented a numerically assisted (as opposed to computer-assisted) "proof" of Pólya's conjecture for the disk for ( . ) and let P λ be a dilation of P with coe cient λ with respect to the origin, that is, the region under the graph of G λ (z).Let be the sets of shifted integer lattice points which lie in P λ , see Figure .The de nitions of the two sets for d = 2 di er by a vertical shift.The reason for choosing this particular notation will become evident later.
We now introduce the weighted lattice point counting functions and ( . ) It is immediately seen from the de nitions ( .)-( .) that with ℵ ∈ {D, N} we have where The region P λ , and the sets of shifted lattice points It is well known that as λ → +∞, the asymptotics of the lattice point counting function is intricately linked to the asymptotics of the eigenvalue counting function N D B d (λ).This was rst shown in the planar case in [KuzFed ] and later re-discovered in [CdV ], see also [Gra ].Namely, in some appropriate sense, This observation, together with asymptotic bounds on the di erence between the two functions, has been used to great e ect to estimate the remainder in Weyl's law for the unit ball.In particular, for the Dirichlet problem in the disk the two-term Weyl asymptotics ( . ) holds with an improved remainder estimate O λ 131/208 (log λ) 18627/8320 , see [GMWW ] (the remainder estimate O λ 2/3 was already obtained in [KuzFed ], [CdV ]).Similar improved remainder estimates are also known in the Dirichlet case for higher-dimensional balls [Guo ] and in the planar Neumann case [GWW ].
As has been recently found in [She ] in the Dirichlet case, there is a further simple non-asymptotic relation between the lattice point and the eigenvalue counting functions, which lies at the cornerstone of our proofs of Theorems .and . .Theorem . .For any d ≥ 2 and any λ ≥ 0, we have We also have, for any λ ≥ 0, We start by introducing some additional notation.Set, for ν ≥ 0, λ ≥ 0, and ℵ ∈ {D, N}, where G λ is de ned by ( . ) and s ℵ is de ned by ( .).Some typical graphs of the functions A ℵ ν (λ) are shown in Figure .The crucial step in the proof of Theorem .comes from the following bounds on the number of zeros of Bessel functions and their derivatives below a given number. and ( . ) Remark . .For λ ∈ [0, ν], the inequalities ( . ) and ( . ) become the trivial identities 0 = s ℵ = 0.
Proof of Proposition . .We recall the representations of the Bessel functions of the rst and second kind, J ν and Y ν , and their derivatives in terms of the so-called modulus functions M ν and N ν and the phase functions θ ν and φ ν , (see [DLMF, Eqs. . . -]).We will be using various properties of the phase functions below; for a review of these properties see [Hor ].We will be only considering the cases ν ≥ 0 and x ≥ 0 for which the moduli M ν (x) and N ν (x) are both positive.
The bound ( . ) can be also proved without relying on the properties of the Bessel phase function.Instead, one uses the known asymptotics of the Bessel zeros and the Sturm comparison theorem.We present this alternative argument below for an interested reader.The bound ( . ) can be proved in the same manner; we omit the details.
Proof of Lemma . .The function U ν (x) := x J ν (x) satis es the di erential equation ( . ) Consider the function Therefore, and On the other hand the asymptotics is well known, see for example [DLMF, Eq. . .]. Suppose that b > 0. Then there exists K ∈ N such that The coe cient in front of U ν in ( . ) is greater than the coe cient in front of V ν in ( .): By the Sturm comparison theorem there is a zero of , and by induction j ν,k ≤ v ν,k (b) for all k ≥ k 0 which contradicts ( .).Therefore, ( . ) holds for all natural k.
Similarly, the Neumann Pólya's conjecture for D would follow immediately if we can prove that for all λ ∈ (Λ 0 , +∞); we note that we have already dealt with λ ≤ Λ 0 by Lemma . .We establish ( . ) in the following cases, which will be dealt with separately.
Theorem .will be proved in § .Together with Theorem ., it implies Theorem . in the planar case.
In the Neumann case, the situation is more delicate, as we cannot expect ( . ) to hold for all values of λ ∈ (0, +∞) since P N 2 (λ) is identically zero for λ < π 4 , see Figure .We prove the following results.
Theorem .will be proved in § .Together with Theorem ., it implies Theorem . .We are further able to eliminate the remaining gap in the Neumann case.
The proof of this result, presented in § , is computer-assisted.Theorem .implies Theorem . .
In all cases, we deal with estimating a (weighted) count of (shifted) lattice points under the graph of a particular function G λ .Such problems have been extensively studied in number theory, going back to the Gauss circle problem.Important contributions in the general case can be traced through the works of van der Corput [vdC ] and Krätzel [Krä ] to some very recent results of Laugesen and Liu [Liu , LauLiu a].In particular, [LauLiu b, Proposition ] is directly applicable (with account of the fact that Laugesen and Liu do not count the points on the vertical axis and do not double-count the points inside) to our shifted lattice point count P D 2 (λ), yielding the bound Unfortunately, since the coe cient in front of λ in this formula is positive, this bound is weaker than our required bound ( .).We need therefore to obtain sharper lattice point count bounds than those available generally, and to do so we additionally use some properties of the derivative of the function G λ in addition to the properties of the function itself, see Theorems .and ., and also Remarks .and .for an informal explanation.
For future use, we summarise below some elementary properties of the function G λ .
The rst lemma is checked by a direct calculation.

Lemma . . The function G
We can therefore de ne the inverse function G −1 λ : 0, λ π → [0, λ] which is also monotone decreasing and convex.Sometimes, it will be also convenient for us to consider G λ on the interval [0, λ ] by extending it by zero to (λ, λ ]: the resulting function, which we for simplicity denote by the same symbol, remains monotone decreasing, convex, and C 1 .
In particular, ( . ) Proof.In fact, the identity can be checked using computer algebra software, but we include a proof for the sake of completeness.After a change of variables z = λ cos τ, we obtain for any ρ, σ ≥ 0. Therefore, , and so .
Substituting this into ( . ) we get the result.
We rst state the following

( . )
The equality is possible only if g is identically zero on [0, b].
Remark . .We explain here, very informally, the ideas behind the proof of Theorem . .The area under the graph of the function g on the interval [m, m + 1] is approximately equal to the area under the straight line passing through the points (m, g (m)) and (m + 1, g (m + 1)), so Summing up these equalities over m we obtain If a number x is chosen randomly then x ≈ x − 1 2 on average.Thus, g (m) + 1 In order to prove Theorem .we require the following Lemma . .Let i , j ∈ Z, i < j .Let g be a decreasing convex function on [i , j + 1] satisfying ( . ) for all z, w ∈ [i , j + 1].Assume additionally that for some n ∈ Z. Then ( . ) Proof of Lemma . .The validity of the claim does not change if we add a constant integer number to the function g .So, without loss of generality we can assume n = 0, so that ( . ) becomes ( . Additionally, ( . ) implies that g (i ) and consider four cases.
Case K = 0.The left-hand side of ( . ) is zero, and the right-hand side is non-negative by ( .).
Case K = 1.Here and the left-hand side of ( . ) is equal to 1 2 .The assumption ( . ) yields g i + 1 2 ≥ 1 2 , therefore by non-negativity and convexity of g on [i , j ], Here and the left-hand side of ( . ) equals 3 2 .By ( . ) we have j ≥ i + 2, and therefore by nonnegativity and convexity of g , Case K ≥ 3.Here The left-hand side of ( . ) is equal to K − 1 2 .By convexity of g , and therefore as K ≥ 3.
Remark . .One can easily see from the proof that the equality in ( . ) is attained in the following three cases only: We can now proceed to the proof of Theorem .proper.

Figure :
The numbers L k , see ( . ) and also Remark . .and applying Lemma .with i = L k+1 , j = L k , and n = k yields g (z) dz.
( . ) Substituting ( . ) into ( . ) gives as required, where in the last inequality we used non-negativity of g .
Finally, assume that we have the equality in ( .).Due to Remark ., the function g is linear on each interval [L k+1 , L k ], and either dist(g for all k, then in particular g (L 0 ) = g ( B ) ≥ 1 4 , and the last inequality in ( . ) is strict.Therefore, the equality in ( . ) requires g ≡ 0 on the whole interval [0, b].
Remark . .If g is strictly monotone on [0, b], then the inverse function g −1 is well-de ned on [0, g (0)], and the de nitions ( . ) may be equivalently rewritten as We nally use Theorem .to prove Theorem . .
Proof of Theorem . .We apply ( . ) with b = λ and g = G λ (which we can do since Lemma .ensures that ( . ) holds in this case), and use ( .), giving the bound ( . ) and therefore con rming the validity of the Dirichlet Pólya's conjecture for the disk.§ .Proof of Theorem .
We start by stating Theorem . .Let b > 0, and let g be a non-negative decreasing convex function on [0, b] such that Remark . .If g is strictly monotone on [0, b], then the inverse function g −1 is well-de ned on [0, g (0)], and cf. Remark . .
Remark . .Once more, we rst outline a very informal plan of proving Theorem . .As we have argued in Remark ., we have g (m) + 3 4 ≈ g (m) + 1 4 , which should in principle ensure the correct inequality sign in ( .). "Bad" points are now the points with n ≤ g (m) ≤ n + 1 4 .So, we divide the graph of g by the horizontal lines at y = n + 1 4 , where n = 0, 1, . . ., g (0) + 3 4 .Again, this guarantees that the number of "bad" points in each resulting interval is less than half the total number of points there.This still leaves an unresolved issue of points m lying under the tail of the graph of g , where 0 ≤ g (m) < 1 4 .Such points make no contribution to the left-hand side of ( .), but the tail does contribute to the integral: consider, for example, a toy case of a function g (z) = 10−z 80 on the interval [0, 10].To account for that, we subtract an additional term in the right-hand side of ( .).
Before proceeding to the proof of Theorem ., we require for some n ∈ Z. Then ( . )
We can now proceed to the proof of Theorem . .We start with Proposition . .Let λ ≥ 2. Then where Proof.We have, with account of G λ (0) = λ π , We apply Theorem .to the sum in the right-hand side, with g = G λ , b = λ, and ).Given that λ ≥ 2, we take into account the bound ( . ) (which ensures that M G λ ,0 ≤ λ), and the value of the integral from Lemma ., leading to Finally, we use to obtain ( .).
. We have the following "dimension reduction" formula.
Theorem . .Let d ≥ 3. Then ) where for r ∈ [0, λ] we denote by the "standard" two-dimensional weighted shifted lattice point count under the graph of the function

Pólya's conjecture for Euclidean balls
We remark that in comparison to our original de nition . ), where the weights κ d ,m are attached at each individual abscissa m, the formula in the righthand side of ( . ) attaches weights n+d −3 d −3 to the whole counts P D n+ d 2 −1 (λ), which we will later estimate using the previously proven Theorem . .Note also that for λ < d 2 −1, the equation ( . ) becomes the trivial identity 0 = 0 by our notational convention for sums, see Remark . .Proof of Theorem . .With account of ( .), the right-hand side of ( . ) reads 4 in this expression appears with the factor where we have used the standard identity [DLMF, Eq. . .] ) and another identity [DLMF, Eq. . .], Thus, the contributions of 4 in both sides of ( . ) coincide.
Before proceeding to the proof of Theorem .we will introduce some additional notation and state some auxiliary facts which will be required later.Let, for x ≥ 0, The function Π n (x) is closely related to Pochhammer's symbol, or the rising factorial (x) n := x • • • (x + n − 1) (for which numerous other notation is also used) in the sense that Π n (x) = (x + 1) n .We also have Let us introduce, for d ≥ 3, a piecewise-constant function and let In what follows we will require an upper polynomial bound on where we used ( . ) to evaluate the sum of binomial coe cients.
To establish a bound on F d (z), we apply the AM-GM inequality with l = d − 1 and Collecting together the terms with Π d −2 (m) and substituting the resulting bound into the right-hand side of ( .), we deduce that ( . ) We will require an auxiliary Lemma . .Let f be a locally integrable function on [0, +∞), and let F ∈ C 1 [0, +∞) with F (0) = 0 be such that for all z ≥ 0.
We now proceed to the proof of Theorem .proper.
and the veri cation is again reduced to elementary operations on rationals.In the same manner, π = 3 arccos 1 2 and π = 3 arccos 1 2 provide veri ed rational approximations for π.
To nish describing our process, we need also to rationalise the square root appearing in the denition of δ(λ): we e ectively replace e(λ) by a smaller number e(λ) ) and also replace δ(λ) by a smaller number where a veri cation is again by taking squares.
Remark . .In practice, we use the following process to nd lower and upper rational approximations of a number x ∈ R (which may be a square root, or an arccosine).Throughout, we x a relatively small number ε (say, ε = 10 −3 ) as an accuracy parameter.We nd numerically some approximation x 0 of x (which may be above or below x) with some better accuracy.Then, we de ne x as the rational number in the interval [x 0 − 3ε, x 0 − ε] with the smallest possible denominator, and x as the rational number in the interval [x 0 + ε, x 0 + 3ε] with the smallest possible denominator, using a modi cation of a fast algorithm for traversing the Stern-Brocot tree [For ].As we always verify the resulting approximations using the procedures described above, we do not in fact depend on the quality of an original numerical "guess" x 0 as long as |x 0 − x| < ε.
Thus, our main algorithms work as follows.In order to prove ( . ) for λ ∈ [Λ 0 , Λ 1 ], we move upwards: set λ = Λ 0 , compute the margin e(λ) from ( .), set λ new = λ+δ (λ) using ( .), and continue the process.If the margins are positive on each step, the process will stop successfully if after a nite number of steps we reach λ > Λ 1 , see Figure .The algorithm works extremely fast (when implemented in Mathematica, see the footnote on the title page), thus proving Theorem .: in principle, with enough patience the whole implementation can be done by hand.We summarise its outcomes in Table .where the both sums are nite since j ν,1 > j ν,1 ≥ ν.
§ .Weyl's law and Pólya's conjectureLet Ω ⊂ R d be a bounded domain.Consider the Dirichlet eigenvalue problem for the Laplacian

Figure :
Figure : The Dirichlet eigenvalue counting function N D D (λ) (blue), the Neumann eigenvalue counting function N N D (λ) (red), and the leading Weyl's term W d(λ) = λ 2 4 (black) in dimension d = 2.The plot is produced using the oating-point evaluation of zeros of the Bessel functions and their derivatives.If we were to assume (contrary to the philosophy of this paper) the validity of oating-point arithmetic, this plot would have presented a numerically assisted (as opposed to computer-assisted) "proof" of Pólya's conjecture for the disk for

Theorem . .
Let b > 0, and let g be a non-negative decreasing convex function on [0, b] such that g (b) = 0 and g (z) − g (w)

#
Figure :The basic algorithm.
Then g λ,α is a monotone decreasing convex function on [0, b] with g (b) = 0; moreover, g λ,α (t ) assumption α ≥ π.With this notation, the right-hand side of ( . ) becomes estimate it from above directly by Theorem .with g = g λ,α , giving Substituting this into ( . ) proves the Dirichlet Pólya's conjecture for sectors containing a half-disk.We now turn to the Neumann problem in S α , still assuming that α ∈ [π, 2π].Following the same argument as in Lemma ., we conclude that the Neumann Pólya's conjecture for S α holds for λ ≤ function g λ,α and parameter b as above.
We now apply the second statement of Lemma .with σ = π 3 which guarantees that b −3M 0 < 0 for λ this nishes the proof of Theorem .for sectors S α of aperture α ∈ [π, 2π].To complete the proof of Theorem .we now need to consider the case α ∈ (0, π).