Average Number of Lattice Points in a Disk

Transcription

1 Average Number of Laice Poins in a Disk Sujay Jayakar Rober S. Sricharz Absrac The difference beween he number of laice poins in a disk of radius /π and he area of he disk /4π is equal o he error in he Weyl asympoic esimae for he eigenvalue couning funcion of he Laplacian on he sandard fla orus. We give a sharp asympoic expression for he average value of he difference over he inerval R. We obain similar resuls for families of ellipses. We also obain relaions o he eigenvalue couning funcion for he Klein bole and projecive plane. The simples case Consider he sandard fla orus [, ] [, ] wih boundaries idenified. The eigenfuncions of he Laplacian are e πin x for n Z wih eigenvalues (π) n, so he eigenvalue couning funcion is { N() = # n Z d : n } /π, (.) he number of laice poins inside he disk B /π of radius /π abou he origin. To firs approximaion N() is he area of he disk /4π, and his is exacly he Weyl asympoic law. The problem of esimaing he difference D() = N() 4π (.) is nooriously difficul (conjecured o be O( /4+ɛ ) for every ɛ > ). Here Research suppored in par by NSF gran #DMS-6544 Mahemaics Subjec Classificaion. Primary 35J5; Primary 4B99. Key words and phrases: laice poins, Weyl asympoics, Bessel funcion

2 D() /4 D() Figure : D() Figure : /4 D() A(). /4 A() Figure 3: A() Figure 4: /4 A() we sudy he simpler problem of approximaing he average value A(R) = R D() d. (.3) Noe ha we are no aking he absolue value of D() in he average, so we may exploi he cancellaion from regions where N() is greaer han and less han /4π. We will show ha A(R) = O(R /4 ) as R, and more precisely A(R) = g(r / )R /4 + O(R 3/4 ) as R (.4) where g(r) is an explici uniformly almos periodic funcion of mean value zero. Somewha differen bu relaed ideas are given in Bleher [, 3]. The

3 following Lemma is well-known (see [4], p. 74), bu we include he proof for he convenience of he reader. Lemma. We have A(R) = n π n J ( n R) (.5) where n = (n, n ) is a variable in Z, and J denoes he Bessel funcion. The series in (.5) converges uniformly and absoluely. Proof. Le χ denoe he characerisic funcion of he ball B /π. I is well-known ha { π z ˆχ (z) = J ( z ) z (.6) z = 4π Following sandard mehods (see [7] or [4]) we apply he Poisson summaion formula o F R,δ = R χ ψ δ d, (.7) where ψ δ is a smooh approximae ideniy. The ψ δ convoluion makes F R,δ smooh, bu evenually we will le δ. Noe ha R N() d = lim F R,δ (n). (.8) δ n Z The Poisson summaion formula gives F R,δ (n) = ˆFR,δ (n) (.9) n Z n Z = R n Z = R ˆχ (n) ˆψ(δn) d 4π d + n by (.6). Combining (.8) and (.9) yields A(R) = lim δ n R R R R π n J ( n ) d ˆψ(δn) π n J ( n ) d ˆψ(δn). (.) 3

4 Now we use he propery of Bessel funcions ([6]) s α+ J α (s) ds = R α+ J α+ (R) (.) for α =, ogeher wih he change of variables s = n, o evaluae he inegral in (.) R R π n J ( n ) d = n R R = π n J ( n R), and subsiue his ino (.) o obain s π n 4 J (s) ds (.) A(R) = lim δ π n J ( n R) ˆψ(δn). (.3) n The esimae J ( n R) = O( n / R /4 ) shows he convergence of he sum in (.3) wihou he erm ˆψ(δn), so we can ake he limi in (.3) and obain (.5). Theorem. Consider he uniformly almos periodic funcion wih mean value zero ( g(x) = n 5/ cos n x π ). (.4) π 3/ 4 We have n A(R) = g(r / )R /4 + O(R 3/4 ) as R. (.5) More generally, here exiss a sequence of uniformly almos periodic funcions g, g,... wih g = g such ha for any n, A(R) = n g j (R / )R 4 j + O(R 4 n ). (.6) j= Proof. We use he well-known asympoic expression for Bessel funcions ( J α (x) = π x / cos x απ π ) + O(x 3/ ) as x. (.7) 4 4

5 ..5.4 g( ) /4 A() g( ) Figure 5: g( ) Figure 6: /4 A() g( ) When α = his is ( J (x) = π x / cos x π ) + O(x 3/ ), (.8) 4 and we subsiue his ino (.5) wih x = n R o obain A(R) = g(r / )R /4 + n n O((n R) 3/ ). (.9) I is easy o see ha he remainder erm in (.9) is O(R 3/4 ), so (.9) yields (.5). To obain he more refined asympoic expression (.6) we use he known more refined asympoic expansion for Bessel funcions (see [6]). In paricular we noe ha i is possible o obain explici series expansions of he funcions g j ; for example, g (x) = 5 ( n 7/ sin n x π ). (.) 8π 3/ 4 n I is also reasonable o consider he funcion N((πr) ) ha couns he number of laice poins inside he ball B r of radius r, he difference D((πr) ) = N((πr) ) πr, and he average wih respec o he radius variable Ã(R) = R 5 D((πr) ) dr. (.)

6 3.4 ( /4 A() g( )) Ã() Figure 7: ( /4 A() g( )) Figure 8: Ã()..4.5 A(π )/ π... A(π )/ π Ã() Figure 9: π A(π ) Figure : π A(π ) Ã() This is a differen average, bu a change of variable shows ha ( Ã(R) = ) R πr D() d. (.) πr / Since mos of he conribuion o he inegral occurs for values of near πr, we see ha Ã(R) has he same asympoics as π A(πR ). In Figure we show he graph of D() and in Figure he graph of /4 D(). This illusraes he rough /4 growh rae of D(). In Figure 3 we show he graph of A(), and Figure 4 he graph of /4 A(). Figure 5 shows he graph of g( ), which is almos idenical o Figure 4 for large. Figure 6 shows he difference of /4 A() and g( ), and Figure 7 shows his 6

7 difference muliplied by /. Figure 8 shows he graph of Ã(). Figure 9 shows he graph of π A(π ), which agrees wih Figure 8 for large, and Figure shows he difference. For more daa see he websie [5]. The general case Consider he general fla -dimensional orus, R /L for some laice L. The eigenfuncions of he Laplacian (resricion of he sandard R Laplacian) have he form e πix ξ for ξ in he dual laice L, wih eigenvalues (π) ξ ξ. By diagonalizing he quadraic form ξ ξ on L we can find an orhonormal basis v, v in R and posiive consans a, a, such ha he eigenvalues are (π) ( (n v a ) ( ) ) n v + a for n Z. Thus he eigenvalue couning funcion is ( (n ) ( ) ) / N() = # n v n v Z : + /π. (.) a In place of disks we consider he family of ellipses { ( (x ) ( ) ) } E = x R : (π) v x v +. (.) a a Of course N() is jus he number of laice poins in E, and he volume of E is a a. Again we wrie D() = N() a a for he difference and define 4π 4π he average A(R) by (.3). The analog of (.6) is { a a ˆχ E (z) = π (a J ( (a z v ) + (a z v ) ) z z v ) +(a z v ) a a z = 4π (.3) Lemma 3. We have A(R) = a a π[(a n v ) + (a n v ) ] J ( (a n v ) + (a n v ) R), n (.4) he series converging uniformly and absoluely. 7 a

8 g( ) /4 A() g( ) Figure : g( ), a =, a = / Figure : /4 A() g( ) Proof. The same proof as for Lemma, wih (.3) used in place of (.6). Theorem 4. The asympoic expansions (.5) and (.6) hold, where now g(x) = π a ( a 3/ (a n v ) + (a n v ) ) 5/4 n ( ((a cos n v ) + (a n v ) ) / π ) x. (.5) 4 Proof. Same as for Theorem, using Lemma 3 in place of Lemma. See Figure for g( ) wih a = and a = / and Figure for he difference /4 A() g( ) for he same family of ellipses. Plos for differen families of ellipses are available on he websie [5]. 3 The Klein bole and projecive plane If we idenify he verical boundaries of he square direcly, and he horizonal boundaries wih reflecion, we obain he sandard fla Klein bole KB. In erms of funcions defined on he square, we are imposing he boundary condiions u(, y) = u(, y) and u(x, ) = u( x, ) in order o have a funcion on KB. We may cover KB by he recangular orus [, ] [, ] wih he ideniies { u(x +, y) = u(x, y) (3.) u( x, y + ) = u(x, y) 8

9 describing he lifs of funcions on KB o R. The eigenfuncions of he Laplacian on KB lif o eigenfuncions on he recangular orus, and so are linear combinaions of funcions of he form e πi(jx+ k y) wih eigenvalue (π) (j + ( k ) ). Now we observe ha e πi(j( x)+ k (y+)) = ( ) k e πi( jk+ k y). Thus here are wo families of eigenfuncions e πi k y for k even (corresponding o j = ), and (3.) e πi(jx+ k y) + ( ) k e πi( jx+ k y) for j >. (3.3) We can herefore see ha he eigenvalue funcion N KB is close o one half he couning funcion N T, for he [, ] [, ] orus. Theorem 5. N KB () = N T, () ±. Proof. N T, () couns all inegers j, k such ha j + ( k ). When (π) j he pair ±j conribues jus a single eigenvalue o N KB (). When j = we coun all k such ha k in N π T, (), bu jus he even values of k in N KB (), and #{k even : k } = {k : k } ±. π π I is ineresing o compare he Klein bole wih he projecive plane (PP) obained from [, ] [, ] by idenifying boh ses of boundary edges wih reflecions. Funcions on PP lif o R wih he ideniies { u( x, y + ) = u(x, y) (3.4) u(x +, y) = u(x, y) and he orus [, ] [, ] is a four-fold covering of PP. However, while i is possible o pull back he sandard Laplacian o PP, he pairs {(, ), (, )} and {(, ), (, )} of idenified poins on PP are singulariies (cone poins wih oal angle π) wih respec o he oherwise fla meric. Reasoning as in he KB example, we know ha eigenfuncions of he Laplacian on PP mus be linear combinaions of he funcions e πi( j x+ k y) wih eigenvalue (π) (( j ) + ( k ) ). Imposing he condiions (3.4) leads o 9

10 four families of eigenfuncions: consans (corresponding o j = and k = ) (3.5) e ıi k y + e πi k y for k > even (corresponding o j = bu k ) (3.6) e πi j x + e πi j x for j > even (corresponding o k = bu j ) (3.7) e πi( j x+ k y) + e πi( j x k y) + ( ) j+k ( e πi( j x+ k y) + e πi( j x k y)) This leads o he ideniy for j > and k >. (3.8) N PP () = 4 N T, () + 4 ± = 4 N T, (4) + 4 ±. (3.9) Similar resuls hold for KB and PP consruced from he ori considered in secion. Relaed quesions in he conex of fracal Laplacians wih he Sierpinski carpe replacing he square are discussed in []. References [] M. Begué, T. Kalloniais, and R. Sricharz. Harmonic funcions and he specrum of he Laplacian on he Sierpinski carpe. Preprin,. [] P.M. Bleher. On he disribuion of he number of laice poins inside a family of convex ovals. Duke Mah J. 67, pages 46 48, 99. [3] P.M. Bleher. Disribuion of he error erm in he Weyl asympoics for he Laplace operaor on a wo-dimensional orus and relaed laice problems. Duke Mah J. 7, pages , 993. [4] H. Iwaniec and E. Kowalski. Analyic Number Theory. AMS Colloq. Publ. vol 53, 4. [5] S. Jayakar and R. Sricharz. Average number of laice poins in a disk. hp:// June. [6] N.N Lebedev. Special funcions and heir applicaions. Dover Publicaions, New York, 965.

11 [7] E.M. Sein and R. Shakarchi. Funcional Analysis. Princeon Univ. Press,. 563 Malo Hall, Cornell Universiy, Ihaca, NY 4853, USA addresses: