Slavutsky, I. Sh.

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumI. Sh. SlavutskyOn representations of numbers by sums of squares and quadratic fieldsArchivum Mathematicum, Vol. 13 (1977), No. 1, 29--40Persistent URL: http://dml.cz/dmlcz/106953Terms of use:© Masaryk University, 1977Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCH. MATH. 1, SCRIPTA FAC SCI. NAT. UJEP BRUNENSIS XIII: 29—40, 1977 ON REPRESENTATIONS OF NUMBERS BY SUMS OF SQUARES AND QUADRATIC FIELDS I. SH. SLAVUTSKY, Leningrad (Received August 16, 1976) This article is concerned with the well-known arithmetical function — the number of representations of naturals by sums of an odd number of integer squares. The famous results are summarized and some new arithmetical properties of this function, its connections with other number — theoretic functions, are proved. 1°. Let be N, Z, Q, R and C correspondingly the set of natural, integer, rational, real and complex numbers, a prime p _- 3, s = 2a -f 1, a eN, (1-1) rs(n)= £ '• 1 xi2 + ...+xs2-n is the number of representations of a number n e N by the sums of squares of the numbers x f eZ, / = 1,...,^. The classic investigations of the function rs(ri) (cf. [14] for historical remarks on this topic) was continued by Lomadze ([14] —[16]). Following to the results of Hardy and Estermann with the help of only elements of the theory of the functions of complex variables, Lomadze proved MordelPs identity (simultaneously for even and odd numbers s > 8) in the form (1.2) 93(01 T) = 1 + f Qs(n) fT + i xs(k) « r 8k(01 T) 9**(01 T) »?(01 T), where the theta-functions *o(- | t )= I (-l)m/T2exp(2m/z), m = — oo 3 2 ( Z | T ) = t /3(m+*)2exp((2m + l)iz), m = —oo 9 3 ( - | T ) = £ /T2exp(2m.z), so that эцoiтЬi + LЧOO/Г', п = i 29 and ft = exp (/TIT), T = x 4- iy, x, y e R, y > 0. The number e e N is determined uni-quely from the condition (for s > 8) 8e < s S 8(e + 1). The arithmetical properties of the constants ccs(k) are considered below and the main member Qs(n), containing the singular series, in the formulae for the quantity of representations (1.3) has the form fs(n) = Qs(n) + ðs(n) -»(«) = - ^ Ż - . ľ. _. . z i w а д җ a , r,t), K°(22°-l)\B2a\ where w = / 2 * = 2?u, f is squarefree number, (2, u) = 1, rj = (—!)*, v = «q~a, if (v, q) = 1 and q is a prime odd number, (— ) — symbol of Kronecker with j for (d, u) = b > 1, (1.4) '•(4)--?(».i0- I fr-V"' w-nri-f-fV-lnd-í1-^"1 n (i-«<1-2o,£ )̂ 91/L \Q / J*l» i l^^a •»•«• i - ( f )« -r r . i j , « Ҷi)-. and as usual the empty product is supposed to be equal to one, (1-5) Za(и) = 1 + _d ± _1(2 - 21 "2 f f) 2U " 2 f f ) ^ ~ l T _ ± - ( 2 - 2 , - 2 0 2 ( 1 " 1 ' ) í 1 T _1 ± dí l " - 1 - 21-<,)2<1"2ff)"V", 1 + A + A(2" - 1 - 21-°)2il~2*)1^~, ľ = l , 2 ^ ľ , rçu = 3 (mod 4), 21 y, rçu = 5 (mod 8), 217, цu = 1 (mod 8), 21 y, where _d = (2° — 21 ""ff)~1 and the upper signs are taken when a = 1; 2 and the lower ones when a == 3; 0 (mod 4). Further, starting from the famous investigations of Hardy ([5], [6]), Suetuna ([24]) and Bateman ([2]), we conclude that the representation (1.3) is true for every odd s ^ 5> 3, if we assume Ss(n) = 0 for s = 3, 5, 7. 30 Let below d = I **' nX = 1 ^mod 4^ \A\\t, r[t == 2 ; 3(mod 4), Bk be Bernoulli numbers, satisfying symbolic identity Bk + k = (B + l)fc, k = 2, 3,.. j5f0 = 1, Bk(x) = £ (*) Bfcx*"1 be Bernoulli polynomials (for the identity character ° 1(4MT5TH - • • are supposed) and Bm(z) = ^m - 1 £x(" )B« U = l be generalized Bernoulli numbers belonging to a primitive residue character x modulo # \ Then in the force (̂a,̂ ) = [i-(jy-«]n°,d), n^d) = i(^y-', and (̂ff)d) = (-l)[i]+, J l - B ^ where x(x) — (— I is a symbol of Kronecker ([13]), we obtain finally I"" l + i rfirflo-l 2ff-x (2> (1-6) QM -(-D----F1 ( p . , ^ , ^ , | ^ r ^ > T > > W with 1,2 | d, 0,2fd. Here as usual [Z] is the integral part of Z. If we propose that n is squarefree so that all representations are proper: in (1.1) (xx,..., xs) = 1; then in this case Ta(ri) = 1, as it follows from (1.4), and therefore by (1.6) we obtain (1.7) Qs(n) = ( - lJxl+ ' 2° . J! "~ *ff x2(n)B,(A Since equation (1.3) for s = 3, 5, 7 is transformed to rs(n) = Qs(n) we obtain from (1.7) (or from corresponding results of Bateman, Suetuna or Lomadze with s = 3) Gauss formulae for the number of a proper representations of naturals by the sum of three square for s = 3 (note that for n = 7 (mod 8) the vanishing of rs(n) takes place because of Xi(n) = 0) and Smith —Minkowski formulas for s = 5 and 7. 31 2°. As it is known, Gauss determined the connections between the number rs(n) of the representations of natural n by the sum of three square and the class number H(df2) of binary quadratic forms with the determinant df2 which is some multiple of this natural. The investigations in this way were continued for squarefree naturals by Kiselev ([9]) and Kiselev, Slavutsky ([10], [11]). More general connections of this kind are given below and new facts about the arithmetical structure of the remainder term in (1.3) are received. Lemma 1. ([10], [11]). If H(df2) is the class number of equivalent binary quadratic forms (positive for d < 0) of the type ax2 + bxy + cy2 where (a9 b, c) = 1, with the determinant df29 then (2.D 'W>*& - -«T„v Jj[l - ( { ) | ] ^ ( - ^ ' ) for d = np > 0, n _• 1, x mod n; (2.2, «w>, -/n[> - (i)±]iija. (1Md/> for d = — np < —4, n }> 1, x m°d n; (2.3) mtftZp- -^ . ,n [ . - ({)!]*^(«d/> for d > 0, p X df, x mod d; (2.4) » ( d / ^ i = -*r(/),n[i - (A)±]%&>(modp') ford> 0,pXd9(p,fo) = l,f = p%,ceN, ^modd; (2,, [l-(i)]lW>- -/Jj[. -(i)±]-^if) ,modrt for d < — 4, p /f d, # mod | d |, í > ~ ! j - i wAere m = —-— p , x(«) & coincide with the corresponding symbol of Kronecker, x = 1 for NOE^) = — 1 a«d ?< = 2 for N(Ki) = 1, NC^i) is the norm of the main unit Et = Tt 4- t/1 V^ i« the quadratic field Q(>/d), 2^, Ut is the least positive solution of the equation T2 — dU2 = ±1 in integers or halfintegers, with the same sup-positions T(f)tl9 U(f)ti is the solution of T2 - df2U2 = ±1 , E(f)l = T{f)a + + uif)jS~E?f)\9 Eif)tl = r(/M + t7(/Mfvd = 4/M T ^ \ Remark. The properties of the generalized Bernoulli numbers Bk(x) introduced by Leopoldt ([13]) were considered by many of the authors ([3], [23] etc.). In particular, 32 it is known that k^ ' is p-integer number for the character x m°d ^» (/*> &) =* I x and & > 1. At last, going to thep-adic limit in (2.1)-(2.5) we obtain the local analog of the formulas of Dirichlet for the class number of quadratic fields (in the narrow sense) in terms of p-adic ^-functions of Kubota —Leopoldt ([12], [8], [7], [19], [22]). Lemma 2. IfeeN is fixed by the prime p > 1 so that 8e < p S 8(e + 1), k g e, k eN, then the rationals ocp(k) determined by the identity (1.2) satisfy the condition (xp(k) = 0 (mod p). First of all from (1.6) => Qs(n) e Q, then also ds(n) e Q. Further dr8k(0|T)34fc(0|t)94fc(0|t) = = (1 + 2p + 2^ + . . .r8 f c(l - 2p + 2/?4 - ...)4fc(l + P2 + ...)*k2upk = (2.6) =Zfc*(s,k)/? fc+f, i = 0 where A,(s, k) e Z for i, s e N u {0} and s > 8, 1 £ k < —, h0(s, k) = 24fc, so that o the identity (L2) may be written as i r,(n) r = t Qs(n) P" + f a.(fe) £ fc((s, fc) /?+' n = l n = l fc=l i = 0 or (2.7) £rXn)j8n = £eX«)/3n + i : rZ« s(fc) ' ' n-^fc)+ E /?" £ a.(fc)/tB_t(S> fc). 11=1 11=1 11 = 1 fc=l « = e + l *=-l Therefore from the three-cornered system (2.8) t asW h-k(s> k) = 5,(n), n = 1,..., e, *=i about as(k) we conclude that <xs(k) 6 Q. Let further s = 2 ( 7 + l = p b e a prime. Since rp(n) s 0 (mod p), then the con-gruence (2.9) 8p(n) = 0(modp) is the consequence of gp(n) = 0 (mod/?), as this follows from (1.3). To prove the lemma it is sufficient to verify the congruence (2.9) for n £ e only, where Se < p £ S(e + 1). First of all we turn to (1.4) and show that TJji) is p-adic integer for a = -£~r—. Indeed, as n < p, then qp~~2 =£ 1 (mod p) for q \ n. Further so I — ) = ( — ] in this 33 case when 2\a, qa\n9 but q**1 \n, then every multiplier from the denominator in (1.4) is found to be equal to that of the first product. Now consider Xi(n)- Here a = -^-~—, 1 — 2a =- - ( p - 2), so 2±a = I — 1, 2*~2ff == 2(modp) and the upper signs in (1.5) are corresponded to the cases p s 3; 5 (mod 8) but the lower ones top = 1; 7 (mod 8). Then from (1.5) in the case y Jj 1, 2 ^ 7 we obtain , v 1 - 21-2* T 2~ff ± 2"a(2 - 21-2*) . 2 ( 1 " 2 f f ) V X2(n) = 1 - 2 1 - 2 ' (210) = (1 ± 2 - ) {l T 2 ^ T 2 - ± (1 ± 2~a) . 2(1"2<T)I21} 1 -2 1 - 2 * and by making use of the congruences 1 ± 2" == 1 ± ( — 1 = 1 - 1 — J = 0 (mod p) finally we have Xi(n) = 0(modp). By analogy in the case 2 | y we get X2(n) s 0 (modp), rju =. 3 (mod 4), Za(n) s T Fl + (jJ ~ 2 1 . 2HT (mod p), ^u = 5 (mod 8), X2(n) = ± |"l - (j\ - 2 1 . 2 ^ (mod p), ^u = 1 (mod 8). Hence, always Xi(n) — 0 (modp), except the cases rju = 5 (mod 8),p = 3; 5 (mod 8), 2 I y or f/w == 1 (mod 8), p = 1; 7 (mod 8), 2 | y. However in these exceptional cases from 2 ) y => fe) = /--£-J, so that 2" = (—J (modp) implies 2* - x(2) = 0 (modp), where x is the character with the conductor t. Therefore in view (2.10) and the similar representations for Xi(n) 1a the other cases we observe that the multiplier of the numerator of (1.6), which is divided by p, is equal to 2a ± 1 and X2(n){2a - X(2)) 2la - \ is p-adic integer. Since the conductor f(x) of the character x is not exceed 4« and n S e < ~-then f(x) < p and consequently Bjfa) is the p-adic integer. At last, the theorem of 34 Staudt implies B^ = O(modp). So gp(n) = O(modp) provided n < -|- and then the congruence (2.9) is true for any prime p. To complete the demonstration of lemma we turn to the system (2.8), where s = p and as we had proved 5p(n) = 0 (modp) for n = 1,..., e, the numbers h0(p, k) = = 24k, hn„k(p, k) e Z. Consecutively surveying the system (n = 1,..., e) we convince that <xp(k) = 0 (mod p) for k = 1,..., e and a prime p > 1 (note that the remainder term in (1.3) for p g 7 equals zero). Remark. For the numbers 5p(\), and consequently aP(l), there may be received the evident expression with the help of (1.3) and (1.6). For example, if p = 5 (mod 8), then rp(\) = 2p and a = 2 (mod 4) imply P-I 2p = r 1 (2 2 ~ 2) Bp^ + 8p(\) \Bp.x\.2^ V PW or as dp(\) = 24aP(l) finally aP(l) = 2"3 (p - (f^ - l) I B;lx I. Bp^J Further as 2 2 ^ 1 (modp) andp | Bp„x \ = 1 (modp), then from the last equality it follows the equivalent of famous Ankeny-Artin-Chowla hypothesis ([1], [10], [17], [20], [21]) in the form (2.11) B p . t -£ 0(mod p)o ^Qt & 1 (mod p). _ p Analogous calculations for p = 1 (mod 8) lead to (2.11), too, so that (2.11) is true generally for any prime p = 1 (mod 4). 3°. Starting from above results it can be taken the arithmetical interpretation of the right parts of the congruences (2.1) —(2.5). Namely, the series of the congruences connecting gs(n) and H(df2) for the discriminant df2 (any multiple of n), and hence rs(n) and H(df2), may be derived from (1.6) and (2.1) —(2.5), if the quantities 5s(n) are calculated. Here a must be chosen as one of the forms m, m + 1, 2m, 2m + \, where m = —~—p '"1 , leN, p is an odd prime. Let for example p = 5 (mod 8), n = 1 (mod 8) be a squarefree integer and hence f = 1, n = t, T9(n) = 1, then choosing a = m, so that a = 2 (mod 4), rj = 1, u~n, X(2) = 1, a = 0, xi(n) = (1 + 2"*) (1 - 21"*), we obtain H(np) J ^ ^ 2 H T l 7 - ^ % - y p\B2ff\ (mod p<), (n, p) - 1, p 3 (P- I ) i> 35 if we take into account that g w o - 2(2,o "T— Ux) and H(np) - ^ V = ~xTi — - ( m o d jfy x mod n, p1 - d" as it follows from (1.7) and (2.1). So pBip^1)pi^ = p - l(modp') and hence in this case p | B(p„i)pl-t | = 1—/? (mod/?1) then finally (3.1) H(np) A = -f-*-T, ICa.+1(fi)(mod P1)-Further, if we are restricted by the calculations of Lomadze who found 5s(ri) for 5 <; 31, then it may be obtained 43 congruences for rs(ri) with the prime moduli p S 31 (and some of these powers forp = 3, 5). For squarefree integers n we indicate some examples. I. We set p = 3, / = 2, a = 3, n = 1 (mod 4) so that s = 7, rj = — 1, u = n. Since in this case tju == 3 (mod 4) and #(*) = I 1, then (1.5), (1.7) and (2.1) imply EM « - 1 - ^ - , so that finally from tf (12n) - ^ a -2T2 M l (mod 9), (3, n) = 1, we obtain (3.2) H(12n) ¥± = 4T2 - ^ - (mod 9), (3, n) = 1, n = 1 (mod 4). II. Fotp = 5, / = 1, <x = 3, n ss 2 (mod 4), that is s = 7, rj = - 1 and n = t = 2w, (1.5), (1.7) and (2.2) with *(*) = ( - = ^ L ) imply (3.3) H(-20n) a ^ ^ - ( m o d 5), (n, 5) = 1, n s 2(mod 4). III. Let p = 5, / = 1, a = 2, n s 1 (mod 8), so that s = 5, *f = 1, n = w and Then from (3.1), in particular, we obtain *->=0) (3.4) H(5n)l/1-.xT1-^-(mod5), (n, 5) - 1, nѕl(mod8). 36 IV. Let p = 17, / = 1, a = 9, n = 2(mod4), so that n « 2if, if » - 1 and %(x) = I J. Calculating Qig(n) and turning to (2.2) we conclude that (3.5) J/(-68n) ss 12<?19(«) (mod 17), (n, 17) = 1, n s 2 (mod 4), where in this case * - s w = 4* S 6 7 I (*i - 3*1*2) -4-> 8 ( > / x,*+... + x,i2 = " as it shows the calculations ([14]). V. Letp = 31, / = 1, a = 15, n = 1 (mod 4), so that s = 31, rj = - 1 , * = tf = « and x(x) = ( — — ) . Then (1.5), (1.7) and (2A) imply (3.6) H(124n) Ut = 47\ ^ & (mod 31), (n, 31) = 1, n s 1 (mod 4). VI. At last, if /> = 11, / = 1, a = 5, n = 7 (mod 8) so that s = 11, ̂  = - 1 and X(*) == ( )> then w e obtain analogously (3.7) H^l^U^^T.^^^odlll (n, l l) = l, W=s7(mod8), where JCi2+X22 + .«32 in the examined case. In the congruences (3.1)—(3.7), as it is known H(d) = 2h(d) provided N(Et) = +1 for d > 0 and #(d) == h(d) in other cases, so that H(d) coincides with the class number of quadratic field (in the narrow sense). Here h(d) is the class number of quadratic field Q(Vd). 4°. So far as (4.1) 3&m*»Mimodfl provided (p - \)p1"1 \ i, k > /, p Jff(x)> f(x) is the conductor of the character % ([10], [7]), then (1.6) implies the congruences connecting Qs(n) (and consequently rs(n) in those cases when Ss(n) was calculated) for the various values s =- i -F k and the fixed value of a number n. The natural s is chosen equal to ~r—pl~ *, -?—— x 37 xp' l + l,(p - l)p ' x and(p — l)p ' ' + 1, leN. So, for example, if« is square-free, then (1.7), (1.5) and (4.1) imply - ^ ^ - ^ ( m o d S ) , 3 ' " > «--7(mod8), ' J i t | - j r3(«) _ 2 gi3(n) Ł 5 - 5 13 - (mod 5), 5 ^ и, и = 7 (mod 8), I z W = - 4 - ^ ^ ( m o d 9), 3 X n9 n = 7 (mod 8). In last case S19(ri) = 0 (mod 81) (the value of Sl9(ri) was given in 3°, case IV). 5°. We fix now s = 2a + 1 and consider (in the previous notations) connections between Qs(t) and Qs(tf2), t is squarefree. As d and consequently %(x), a, Ba(x) are fully determinated by t and er, so (1.6) implies (5.D Qs(n) = ^ O /2 ' " 1 ^ ) - ^ . « = tf2, If (2,«) = 1 in particular, then f2 = 1 (mod 8) and t = n = u (mod 8), so that Xi(n) = Z2(0» a n d hence from (5.1) it follows (5.2) Q^-Q^f^-'TM^Xn. And if n = 22*,>, t = w is squarefree, / = 2k, then Fff(n) = 1 and the equality (5.1) implies (5-3) es(«) = e s(02*( 2 '- 1 ,-g00. The correspondences (5.1) —(5.3) contained the results which belong to different authors and had received as elementaly-combinatory methods as analytic ones. So, in particular, if n = 4f, t is squarefree, for s = 5, 7 we obtain the theorems of Cohen ([4]). Namely, the equality (5.3) implies so that from (1.5) finally f5r5(0, t s 2 ; 3 (mod 4), r5(4ř) = and 38 9r5(ř), t =. 1 (mod 8), 47 — r5(t), í з 5 (mod 8), r7(4í) = 3 2 ^ - r 7 ( 0 , so that r7(4.) = t = l;2(mod4), t = 3 (mod 8), t є 7 (mod 8). REFERENCES [1] Ankeny N. S., Artin E., Chowla S.: The class-numbers of real quadratic fields, Ann. of Math., 56, N 3, 1952, 479—493. [2] Batеman P.: On representations ofnumber as the sums of three squares, Тrans. Amеr. Math. Soc, 71, N 1, 1951, 70—101. [3] Carlitz L.: Arithmeticproperties of generalizedBernoullinumbers, Journ. für Math., 202,1959, 174—182. [4] Cohеn H.: Sommes de carrès, fonctions L et forms modulaires, Comptе Rеnd. Acad. Sci. Paris, 277, N 17, 1973, 827—830. [5] Hardy G. H.: On the representation ofa number as the sum ofany number of squares, and in particular offive, Тrans. Amеr. Math. Soc, 21, 1920, 255—284. [б] Hardy G. 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On representations of numbers by sums of squares and quadratic fields

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On representations of numbers by sums of squares and quadratic fields

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Institute of Mathematics AS CR, v. v. i.