Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

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1 Open Math. 207; 5: Open Mathematics Open Access Research Article Ebénézer Ntienjem* Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52 DOI 0.55/math Received August 30, 206; accepted February 28, 207. Abstract: The convolution sum, 0 lcˇ mdn.l/.m/, where ˇ D 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for ˇ D 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a.x 2 Cx2 2 Cx2 3 Cx2 4 /Cb.x2 5 Cx2 6 Cx2 7 Cx2 8 /, where.a; b/ D.; /;.; 3/. Keywords: Sums of Divisors function, Convolution Sums, Dedekind eta function, Modular Forms, Eisenstein Series, Cusp Forms, Octonary quadratic Forms, Number of Representations MSC: A25, E20, E25, F, F20, F27 Introduction Let in the sequel N, N 0, Z, Q, R and C denote the sets of positive integers, non-negative integers, integers, rational numbers, real numbers and complex numbers, respectively. Suppose that k; n 2 N. Then the sum of positive divisors of n to the power of k, k.n/, is defined by k.n/ D d k : () 0<djn We write.n/ as a synonym for.n/. For m N we set k.m/ D 0. Suppose now that ; ˇ 2 N are such that ˇ. Then the convolution sum, W. ;ˇ/.n/, is defined as follows: W. ;ˇ/.n/ D.l/.m/: (2) 0 lcˇ mdn We write Wˇ.n/ as a synonym for W.;ˇ/.n/. Given ; ˇ 2 N, if for all.l; m/ 2 N 2 0 it holds that l C ˇ m n then we set W. ;ˇ/.n/ D 0. For those convolution sums W. ;ˇ/.n/ that have so far been evaluated, the levels ˇ are given in Table. We discuss the evaluation of the convolution sums of level ˇ D 22; 44 and ˇ D 52, i.e.,. ; ˇ/ D.; 22/,.2; /,.; 44/,.4; /,.; 52/,.4; 3/. Convolution sums of these levels have not been evaluated yet as one can notice from Table. *Corresponding Author: Ebénézer Ntienjem: Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, 25 Colonel By Drive, Ottawa, Ontario, KS 5B6, Canada, ebenezer.ntienjem@carleton.ca, ntienjem@gmail.com 207 Ntienjem, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Download Date 2/8/8 5:2 M

2 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and Table. Known convolution sums W. ;ˇ/.n/ Level ˇ Authors References M. Besge, J. W. L. Glaisher, S. Ramanujan [ 3] 2, 3, 4 J. G. Huard & Z. M. Ou & B. K. Spearman & K. S. Williams [4] 5, 7 M. Lemire & K. S. Williams, S. Cooper &. C. Toh [5, 6] 6 S. Alaca & K. S. Williams [7] 8, 9 K. S. Williams [8, 9] 0,, 3, 4 E. Royer [0] 2, 6, 8, 24 A. Alaca & S. Alaca & K. S. Williams [ 4] 5 B. Ramakrishman & B. Sahu [5] 20, 0 S. Cooper & D. Ye [6] 23 H. H. Chan & S. Cooper [7] 25 E.. W. ia &. L. Tian & O.. M. Yao [8] 27, 32 S. Alaca & Y. Kesicio Lglu [9] 36 D. Ye [20] 4, 26, 28, 30 E. Ntienjem [2] As an application, convolution sums are used to determine explicit formulae for the number of representations of a positive integer n by the octonary quadratic forms a.x 2 C x2 2 C x2 3 C x2 4 / C b.x2 5 C x2 6 C x2 7 C x2 8 /; (3) and c.x 2 C x x 2 C x2 2 C x2 3 C x 3x 4 C x4 2 / C d.x2 5 C x 5x 6 C x6 2 C x2 7 C x 7x 8 C x8 2 /; (4) respectively, where a; b; c; d 2 N. So far known explicit formulae for the number of representations of n by the octonary form Equation 3 are referenced in Table 2. Table 2. Known representations of n by the form Equation 3.a; b/ Authors References (,2) K. S. Williams [8] (,4) A. Alaca & S. Alaca & K. S. Williams [2] (,5) S. Cooper & D. Ye [6] (,6) B. Ramakrishman & B. Sahu [5] (,8) S. Alaca & Y. Kesicio Lglu [9] (,7) E. Ntienjem [2] We determine formulae for the number of representations of a positive integer n by the octonary quadratic form Equation 3 for which.a; b/ D.; /;.; 3/. These formulae for the number of representations are also new according to Table 2. This paper is organized in the following way. In Section 2 we discuss modular forms, briefly define eta functions and convolution sums, and prove the generalization of the extraction of the convolution sum. Our main results on the evaluation of the convolution sums are discussed in Section 3. The determination of formulae for the number of representations of a positive integer n is discussed in Section 4. Software for symbolic scientific computation is used to obtain the results of this paper. This software comprises the open source software packages GiNaC, Maxima, REDUCE, SAGE and the commercial software package MALE. Download Date 2/8/8 5:2 M

3 448 E. Ntienjem 2 Modular forms and convolution sums Let H be the upper half-plane, that is H D fz 2 C j Im.z/ > 0g, and let G D SL 2.R/ be the group of 2 2-matrices a b c d such that a; b; c; d 2 R and ad bc D hold. Let furthermore D SL2.Z/ be the full modular group which is a subgroup of SL 2.R/. Let N 2 N. Then.N / D 2 SL2.Z/ j 0 0.mod N / a b c d is a subgroup of G and is called the principal congruence subgroup of level N. A subgroup H of G is called a congruence subgroup of level N if it contains.n /. Relevant for our purposes is the following congruence subgroup: 0.N / D a b c d a b c d 2 SL2.Z/ j c 0.mod N / : Let k; N 2 N and let 0 be a congruence subgroup of level N 2 N. Let k 2 Z; 2 SL 2.Z/ and f W H [ Q [ fg! C [ fg. We denote by f Œ k the function whose value at z is.cz C d/ k f..z//, i.e., f Œ k.z/ D.cz C d/ k f..z//. The following definition is based on the textbook by N. Koblitz [22, p. 08]. Definition 2.. Let N 2 N, k 2 Z, f be a meromorphic function on H and 0 a congruence subgroup of level N. (a) f is called a modular function of weight k for 0 if (a) for all 2 0 it holds that f Œ k D f. (a2) for any ı 2 it holds that f Œı k.z/ can be expressed in the form a n e 2izn N, wherein a n 0 for finitely n2z many n 2 Z such that n < 0. (b) f is called a modular form of weight k for 0 if (b) f is a modular function of weight k for 0, (b2) f is holomorphic on H, (b3) for all ı 2 and for all n 2 Z such that n < 0 it holds that a n D 0. (c) f is called a cusp form of weight k for 0 if (c) f is a modular form of weight k for 0, (c2) for all ı 2 it holds that a 0 D 0. For k; N 2 N, let M k. 0.N // be the space of modular forms of weight k for 0.N /, S k. 0.N // be the subspace of cusp forms of weight k for 0.N /, and E k. 0.N // be the subspace of Eisenstein forms of weight k for 0.N /. Then the decomposition of the space of modular forms as a direct sum of the space generated by the Eisenstein series and the space of cusp forms, i.e., M k. 0.N // D E k. 0.N // S k. 0.N //, is well-known; see for example W. A. Stein s book (online version) [23, p. 8]. As noted in Section 5.3 of W. A. Stein s book [23, p. 86] if the primitive Dirichlet characters are trivial and 2 k is even, then E k.q/ D 2k B k k.n/ q n, where B k are the Bernoulli numbers. For the purpose of this paper we only consider trivial Dirichlet characters and 2 k even. Theorems 5.8 and 5.9 in Section 5.3 of [23, p. 86] also hold for this special case. 2. Eta functions The Dedekind eta function,.z/, is defined on the upper half-plane H by.z/ D e 2iz 24 q D e 2iz. Then.z/ D q 24 Q. q n Q / D q 24 F.q/, where F.q/ D. q n /. Q. e 2inz /. We set M. Newman [24, 25] systematically used the Dedekind eta function to construct modular forms for 0.N /. M. Newman determined when a function f.z/ is a modular form for 0.N / by providing conditions (i)-(iv) in the following theorem. G. Ligozat [26] determined the order of vanishing of an eta function at the cusps of 0.N /, which is condition (v) or (v 0 ) in Theorem 2.2. Download Date 2/8/8 5:2 M

4 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and The following theorem is proved in L. J.. Kilford s book [27, p. 99] and G. Köhler s book [28, p. 37]; we will apply that theorem to determine eta quotients, f.z/, which belong to M k. 0.N //, and especially those eta quotients which are in S k. 0.N //. Theorem 2.2 (M. Newman and G. Ligozat ). Let N 2 N, D.N / be the set of all positive divisors of N, ı 2 D.N / and r ı 2 Z. Let furthermore f.z/ D Q r ı.ız/ be an -quotient. If the following five conditions are satisfied (i) (ii) (iii) (iv) 0 < Q ı r ı 0.mod 24/, N ı r ı 0.mod 24/, ı r ı is a square in Q, r ı 0.mod 4/ (v) for each d 2 D.N / it holds that then f.z/ 2 M k. 0.N //, where k D 2 gcd.ı;d/ 2 ı r ı 0, Moreover, the -quotient f.z/ belongs to S k. 0.N // if (v) is replaced by gcd.ı;d/ (v ) for each d 2 D.N / it holds that 2 r ı ı > 0. r ı. 2.2 Convolution sums W. ;ˇ/.n/ Recall that given ; ˇ 2 N such that ˇ, the convolution sum is defined by Equation 2. As observed by A. Alaca et al. [], we can assume that gcd. ; ˇ/ D. Let q 2 C be such that jqj <. Then the Eisenstein series L.q/ and M.q/ are defined as follows: L.q/ D E 2.q/ D 24 M.q/ D E 4.q/ D C 240.n/q n ; (5) 3.n/q n : (6) The following two relevant results are essential for the sequel of this work and are a generalization of the extraction of the convolution sum using Eisenstein forms of weight 4 for all pairs. ; ˇ/ 2 N 2. Their proofs are given by E. Ntienjem [2]. Lemma 2.3. Let ; ˇ 2 N. Then. L.q / ˇ L.qˇ// 2 2 M ˇ//: Theorem 2.4. Let ; ˇ 2 N be such that and ˇ are relatively prime and < ˇ. Then. L.q / ˇ L.qˇ// 2 D. ˇ/ 2 C n / C 240 ˇ2 3. nˇ / C 48.ˇ 6n/. / C 48 ˇ. 6n/. / n nˇ 52 ˇ W. ;ˇ/.n/ q n : (7) Download Date 2/8/8 5:2 M

5 450 E. Ntienjem 3 Evaluation of the convolution sums W. ;ˇ/.n/, where ˇ D 22; 44; 52 In this section, we give explicit formulae for the convolution sums W.;22/.n/, W.2;/.n/, W.;44/.n/, W.4;/.n/, W.;52/.n/ and W.4;3/.n/. 3. Bases for E ˇ// and S ˇ// with ˇ D 44; 52 We observe the following inclusion relations M 4. 0.// M // M // (8) M // M // M //: (9) Therefore, it suffices to correspondingly determine the basis of the spaces M // and M //, respectively. We use the dimension formulae for the space of Eisenstein forms and the space of cusp forms in T. Miyake s book [29, Thrm 2.5.2, p. 60] or W. A. Stein s book [23, rop. 6., p. 9] to deduce that dim.e /// D dim.e /// D 6, dim.s // D 5 and dim.s // D 8. Let D.44/ D f; 2; 4; ; 22; 44g and D.52/ D f; 2; 4; 3; 26; 52g be the sets of all positive divisors of 44 and 52, respectively. Theorem 3.. (a) The sets B E;44 D f M.q t / j t 2 D.44/ g and B E;52 D f M.q t / j t 2 D.52/ g are bases of E // and E //, respectively. (b) Let i 5 and j 8 be positive integers. Let ı 2 D.44/ and.r.i; ı // i;ı be the Table 3 of the powers of.ı z/. Let ı 2 2 D.52/ and.r.j; ı 2 // j;ı2 be the Table 4 of the powers of.ı 2 z/. Q Let furthermore A i.q/ D r.i;ı/ Q.ı z/ and B j.q/ D r.j;ı2/.ı 2 z/ be selected elements of ı 2D.44/ ı 2 2D.52/ S // and S //, respectively. Then the sets B S;44 D f A i.q/ j i 5 g and B S;52 D f B j.q/ j j 8 g are bases of S // and S //, repectively. (c) The sets B M;44 D B E;44 [ B S;44 and B M;52 D B E;52 [ B S;52 constitute bases of M // and M //, respectively. For i 5 and j 8 let in the sequel A i.q/ be expressed in the form a i.n/q n and B j.q/ be expressed in the form b j.n/q n. roof. We give the proof for the case ˇ D 44. The case ˇ D 52 is proved similarly. (a) By Theorem 5.8 in Section 5.3 of W. A. Stein [23, p. 86] M.q t / is in M 4. 0.t// for each t which is an element of D.44/. Since E // has a finite dimension, it suffices to show that M.q t / with t 2 D.44/ are linearly independent. Suppose that x t 2 C with t 2 D.44/. We prove this by induction on the elements of the set D.44/ which is assumed to be ascendantly ordered. The case t D 2 D.44/ is obvious since comparing the coefficients of q t on both sides of the equation x t M.q t / D 0 clearly gives x t D 0. Suppose now that the cardinality of the set D.44/ is greater than and that M.q t / are linearly independent for all tj44 and t t for a given t with < t < 44. Let C be the proper non-empty subset of D.44/ which contains all positive divisors of 44 less than or equal to t. Note that all positive divisors of t constitute a subset of C. Let Download Date 2/8/8 5:2 M

6 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and us consider the non-empty subset C [ ft 0 g of D.44/, wherein t 0 is the next ascendant element of D.44/ which is greater than t the greatest element of the set C. Then x t M.q t / C x t 0 M.q t 0 / D 0: t2c [ft 0 gx t M.q t / D t2c By the induction hypothesis it holds that x t D 0 for all t 2 C. So, we obtain from the above equation that x t 0 D 0 when we compare the coefficient of q t 0 on both sides of the equation. Hence, the solution is x t D 0 for all t such that t is a positive divisor of 44. Therefore, the set B E;44 is linearly independent. Hence, the set B E;44 is a basis of E //. (b) The A i.q/ with i 5 are obtained from an exhaustive search using Theorem 2.2.i/.v 0 /. Hence, each A i.q/ is an element of the space S //. Since the dimension of S // is 5, it suffices to show that the set f A i.q/ j i 5g is linearly independent. Suppose that x i 2 C and 5 x i A i.q/ D 0. Then 5 id id x i A i.q/ D 5. x i a i.n/ /q n D 0 id which gives the following homogeneous system of linear equations 5 id a i.n/ x i D 0; n 5: (0) A simple computation using software for symbolic scientific computation shows that the determinant of the matrix of this homogeneous system of linear equations is non-zero. So, x i D 0 for all i 5. Hence, the set f A i.q/ j i 5 g is linearly independent and therefore a basis of S //. (c) Since M // D E // S //, the result follows from (a) and (b). According to Equation 8 the basis elements A i.q/, where i 5, are contained in S //. The basis element A 2.q/ is the only element of the space S 4. 0.// that we are able to generate with the help of Theorem 2.2. Even though the basis element A 4.q/ looks like an element of S //, it cannot be generated at level 22 using Theorem 2.2. To evaluate the convolution sums W.;22/.n/ and W.2;/.n/, we determine two additional basis elements of S // which are A 0 6.q/ D.2z/3.z/ 5.22z/.z/ D a6 0.n/qn ; A 0 7.q/ D 9.2z/ 7.z/ 5.z/ 3.22z/ D a7 0.n/qn : Due to Equation 9 the basis elements B j.q/, where j 7 and j D 5; 7, belong to S //. We are unable to generate any elements of the space S // using Theorem 2.2. We note that B 2j.q/ D B j.q 2 /, where 4 j 7, B 6.q/ D B 5.q 2 / and B 8.q/ D B 7.q 2 /. Therefore, one can easily replicate the evaluation of the convolution sums W.;26/.n/ and W.2;3/.n/ shown by E. Ntienjem [2]. 3.2 Evaluation of W. ;ˇ/.n/ where ˇ D 22; 44; 52 Lemma 3.2. We have.l.q 2 / 22 L.q 22 // 2 D 4 C n/ C n 2 / Download Date 2/8/8 5:2 M

7 452 E. Ntienjem 0880 C n / C a 3.n/ C n 22 / C 2096 a.n/ C a 2.n/ a 4.n/ a 5.n/ C 2276 a6 0.n/ C 864 a0 7 q.n/ n ; ().2 L.q 2 / L.q // 2 D 8 C C C n / a 3.n/ C n/ C n 2 / 3. n 22 / C a 4.n/ C a.n/ C a 2.n/ a 5.n/ 2276 a 0 6.n/ 864 a0 7.n/ q n ; (2).L.q/ 44 L.q 44 // 2 D 849 C C C n/ n 2 / 3. n 4 / n / C n 22 / C n 44 / a n/ a 2.n/ a 3.n/ a 4.n/ a 5.n/ C a 6.n/ a 7.n/ C a 8.n/ a 9.n/ a 0.n/ C a.n/ a 2.n/ a 3.n/ C a 4.n/ a 5.n/ q n ; (3).4 L.q 4 / L.q // 2 D 49 C C 0880 C C n 4 / C a.n/ C a 5.n/ a 9.n/ C n/ C n / a 2.n/ C C n 2 / 3. n 22 / a 3.n/ C n 44 / a 4.n/ a 6.n/ C a 7.n/ a 8.n/ a 0.n/ a.n/ C a 2.n/ a 3.n/ a 4.n/ C a 5.n/ 9 95 q n ; (4).L.q/ 52 L.q 52 // 2 D 260 C C n/ C C n 4 / n 52 / b 3.n/ n 2 / 3. n 3 / b.n/ C b 4.n/ b 6.n/ C 7488 b 9.n/ C n 26 / b 2.n/ b 7.n/ C b 0.n/ C b.n/ b 8.n/ b 5.n/ Download Date 2/8/8 5:2 M

8 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and b 2.n/ C 7472 b 3.n/ C b 4.n/ b 5.n/ C b 6.n/ b 7.n/ C b 8.n/ q n ; (5).4 L.q 4 / 3 L.q 3 // 2 D 8 C C n/ C C n 4 / C n 52 / b 3.n/ b 6.n/ n 2 / 3. n 3 / b.n/ C b 4.n/ b 9.n/ C b 0.n/ b 2.n/ 7472 b 3.n/ b 5.n/ C b 6.n/ n 26 / b 2.n/ b 7.n/ C b.n/ b 8.n/ b 5.n/ b 4.n/ b 7.n/ 2304 b 8.n/ q n : (6) roof. We just prove the case.4 L.q 4 / L.q // 2. The other cases are proved similarly. It follows from Lemma 2.3 that.4 L.q 4 / L.q // 2 2 M //. Hence, by Theorem 3. (c), there exist ı ; Y j 2 C; j 5 and ı 2 D.44/, such that.4 L.q 4 / L.q // 2 D D ı2d.44/ ı2d.44/ ı M.q ı / C ı C 5 j D 240 Y j A j.q/ ı2d.44/ 3. n ı / ı C m S j D a j.n/ Y j q n : (7) We equate the right hand side of Equation 7 with that of Equation 7 when setting. ; ˇ/ D.4; / to obtain 240 ı2d.44/ 3. n ı / ı C 5 j D We now take the coefficients of q n for which n is in a j.n/ Y j q n D n 4 / C n / C92. 6 n/. n 4 / C n/. n / W.4;/.n/ f ; 2; 3; 4; 5; 6; 7; 8; 9; 0; ; 2; 3; 4; 5; 6; 7; 8; 20; 22; 44 g: This results in a system of linear equations whose unique solution determines the values of the unknown ı for all ı 2 D.44/ and the values of the unkown Y j for all j 5. Hence, we obtain the stated result. Our main result of this section is as follows. Theorem 3.3. Let n be a positive integer. Then W.;22/.n/ D n/ n 2 / C n / C n 22 / q n : Download Date 2/8/8 5:2 M

9 454 E. Ntienjem W.2;/.n/ D C a 3.n/ n/.n/ C n/. n 22 / a 4.n/ C 2 a 5.n/ n/ C n 2 / C n / C 39 C n/. n 2 / C n/. n / a 3.n/ a 4.n/ 2684 a 59.n/ 5368 a 2.n/ 7 8 a0 6.n/ 3 88 a0 7.n/; (8) n 22 / a 2.n/ 67 a.n/ a 5.n/ C 7 8 a0 6.n/ C 3 88 a0 7.n/; (9) W.;44/.n/ D n/ C n 2 / 3 3. n 4 / C n / n 22 / C n 44 / C. C. 24 W.4;/.n/ D n/ W.;52/.n/ D 76 n/.n/ 24 4 n/. n 44 / a.n/ C a 2.n/ C a 3.n/ C a 4.n/ C a 5.n/ C a n/ a 8.n/ C a 9.n/ C a 0.n/ a.n/ C a 2.n/ C a 3.n/ a 4.n/ a 6.n/ C a 5.n/; (20) n 2 / C n 4 / n / C n 22 / C n 44 / C. 24 C n/. n / a.n/ a 3.n/ a 7.n/ C a 8.n/ a 4.n/ 44 n/. n 4 / a 2.n/ a 5.n/ C a n/ a 0.n/ a 3.n/ C a 4.n/ a 6.n/ C a 60.n/ 90 a 2.n/ a 5.n/; (2) 97 3.n/ C n 2 / n 4 / C n 3 / C n 26 / n 52 / C n/.n/ C n/. n 52 / C b n/ b 2.n/ b 3.n/ C b 4.n/ C b 5.n/ C b 6.n/ C b 39 7.n/ 328 b 8.n/ 8 b 9.n/ 7 24 b 3.n/ b 0.n/ b 4.n/ C b 5.n/ 24 b.n/ C b 2.n/ Download Date 2/8/8 5:2 M

10 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and b 6.n/ C b 7.n/ b 8.n/; (22) W.4;3/.n/ D n/ C n 2 / C n 4 / C n 3 / C n 26 / n 52 / C. 24 C n/. n 3 / C b.n/ 52 n/. n 4 / b 2.n/ b 3.n/ C b 4.n/ C b 5.n/ C b 6.n/ C b 7.n/ C 8 b 9.n/ b 8.n/ b 0.n/ C 24 b.n/ C b 2.n/ C 7 24 b 3.n/ C b 4.n/ C b 5.n/ b 6.n/ C b 7.n/ C 066 b 8.n/: (23) roof. We prove the case W.4;3/.n/ as the other cases are proved similarly. We equate the right hand side of Equation 6 with that of Equation 7 when setting. ; ˇ/ D.4; 3/, namely n 4 / C n 3 / C n/. n 4 / C C n/. n 3 / W.4;3/.n/ n/ q n D 3. n 2 / 3. n 4 / C n 3 / n 26 / C n 52 / b.n/ C b 2.n/ C b 3.n/ b 4.n/ b 5.n/ b 6.n/ b 7.n/ C b 8.n/ 7488 b 9.n/ C b 0.n/ b.n/ b 2.n/ 7472 b 3.n/ b 4.n/ b 5.n/ C b 6.n/ b 7.n/ We then solve for W.4;3/.n/ to obtain the stated result b 8.n/ q n : 4 Number of representations of a positive integer n by the qctonary quadratic form using W. ;ˇ/.n/ when ˇ D 44; 52 Let n 2 N 0 and the number of representations of n by the quaternary quadratic form x 2 C x2 2 C x2 3 C x2 4 be denoted by r 4.n/. That means, r 4.n/ D card.f.x ; x 2 ; x 3 ; x 4 / 2 Z 4 j m D x 2 C x2 2 C x2 3 C x2 4 g/: Download Date 2/8/8 5:2 M

11 456 E. Ntienjem We set r 4.0/ D. For all n 2 N, the following Jacobi s identity is proved in K. S. Williams book [30, Thrm 9.5, p. 83] r 4.n/ D 8.n/ 32. n /: (24) 4 Let furthermore the number of representations of n by the octonary quadratic form be denoted by N.a;b/.n/. That means, a.x 2 C x2 2 C x2 3 C x2 4 / C b.x2 5 C x2 6 C x2 7 C x2 8 / N.a;b/.n/ D card.f.x ; x 2 ; x 3 ; x 4 ; x 5 ; x 6 ; x 7 ; x 8 / 2 Z 8 j n D a.x 2 C x2 2 C x2 3 C x2 4 / C b.x2 5 C x2 6 C x2 7 C x2 8 /g/: We infer the following result: Theorem 4.. Let n 2 N and.a; b/ D.; /;.; 3/. Then N.;/.n/ D8.n/ 32. n 4 / C 8. n / 32. n 44 / C 64 W.;/.n/ C 024 W.;/. n W 4 / 256.4;/.n/ C W.;44/.n/ ; N.;3/.n/ D8.n/ 32. n 4 / C 8. n 3 / 32. n 52 / C 64 W.;3/.n/ C 024 W.;3/. n W 4 / 256.4;3/.n/ C W.;52/.n/ : roof. We only prove N.;/.n/ since that for N.;3/.n/ is done similarly. It holds that N.;/.n/ D r 4.l/r 4.m/ D r 4.n/r 4.0/ C r 4.0/r 4. n / C 0 lc mdn We make use of Equation 24 to derive N.;/.n/ D 8.n/ 32. n 4 / C 8. n / 32. n 52 / C We observe that lc mdn lc mdn r 4.l/r 4.m/:.8.l/ 32. l 4 //.8.m/ 32. m 4 //:.8.l/ 32. l 4 //.8.m/ 32. m 4 // D 64.l/.m/ 256. l 4 /.m/ 256.l/. m 4 / C 024. l 4 /. m 4 /: The evaluation of W.;/.n/ D lc mdn.l/.m/ is shown by E. Royer [0, Thrm.3]. We map l to 4l to infer W.4;/.n/ D. l 4 /.m/ D lc mdn 4 lc mdn.l/.m/: The evaluation of W.4;/.n/ is given in Equation 2. We next map m to 4m to conclude W.;44/.n/ D.l/. m 4 / D.l/.m/: lc mdn lc44 mdn The evaluation of W.;44/.n/ is provided by Equation 20. We simultaneously map l to 4l and m to 4m to deduce. l 4 /. m 4 / D.l/.m/ D W.;/. n 4 /: lc mdn lc md n 4 Again, E. Royer [0, Thrm.3] has proved the evaluation of W.;/.n/. We then put these evaluations together to obtain the stated result for N.;/.n/. Download Date 2/8/8 5:2 M

12 Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and Tables Table 3. Exponents of -functions being basis elements of S // Table 4. Exponents of -functions being basis elements of S // Acknowledgement: I am indebtedly thankful to the anonynuous referee for fruitful comments and suggestions on a draft of this paper. References [] M. Besge. Extrait d une lettre de M Besge à M Liouville. J Math ure Appl, 885, 7, 256. Download Date 2/8/8 5:2 M

13 458 E. Ntienjem [2] James Whitbread Lee Glaisher. On the square of the series in which the coefficients are the sums of the divisors of the exponents. Messenger Math, 862, 4, [3] S. Ramanujan. On certain arithmetical functions. T Cambridge hil Soc, 96, 22, [4] J. G. Huard, Z. M. Ou, B. K. Spearman, and Kenneth S. Williams. Elementary evaluation of certain convolution sums involving divisor functions. Number Theory Millenium, 2002, 7, A K eters, Natick, MA. [5] Mathieu Lemire and Kenneth S. Williams. Evaluation of two convolution sums involving the sum of divisors function. Bull Aust Math Soc, 2006, 73, [6] S. Cooper and. C. Toh. Quintic and septic Eisenstein series. Ramanujan J, 2009, 9, [7] Şaban Alaca and Kenneth S. Williams. Evaluation of the convolution sums.l/.m/ and.l/.m/. J lc6mdn 2lC3mDn Number Theory, 2007, 24(2), [8] Kenneth S. Williams. The convolution sum.m/.n 8m/. ac J Math, 2006, 228, m< n 8 [9] Kenneth S. Williams. The convolution sum m< n 9.m/.n 9m/. Int J Number Theory, 2005, (2), [0] Emmanuel Royer. Evaluating convolution sums of divisor function by quasimodular forms. Int J Number Theory, 2007, 3(2), [] Ayşe Alaca, Şaban Alaca, and Kenneth S. Williams. Evaluation of the convolution sums.l/.m/ and lc2mdn.l/.m/. Adv Theor Appl Math, 2006, (), lC4mDn [2] Ayşe Alaca, Şaban Alaca, and Kenneth S. Williams. Evaluation of the convolution sums.l/.m/ and lc8mdn.l/.m/. Int Math Forum, 2007, 2(2), lC9mDn [3] Ayşe Alaca, Şaban Alaca, and Kenneth S. Williams. Evaluation of the convolution sums.l/.m/ and lc24mdn.l/.m/. Math J Okayama Univ, 2007, 49, 93. 3lC8mDn [4] Ayşe Alaca, Şaban Alaca, and Kenneth S. Williams. The convolution sum.m/.n 6m/. Canad Math Bull, 2008, 5(), 3 4. [5] B. Ramakrishnan and B. Sahu. Evaluation of the convolution sums.l/.m/ and.l/.m/. Int J Number lc5mdn 3lC5mDn Theory, 203, 9(3), [6] Shaun Cooper and Dongxi Ye. Evaluation of the convolution sums.l/.m/;.l/.m/ and lc20mdn 4lC5mDn.l/.m/. Int J Number Theory, 204, 0(6), lC5mDn [7] H. H. Chan and Shaun Cooper. owers of theta functions. ac J Math, 2008, 235, 4. [8] E.. W. ia,. L. Tian, and O.. M. Yao. Evaluation of the convolution sum.l/.m/. Int J Number Theory, 204, lc25mdn 0(6), [9] Şaban Alaca and Yavuz Kesicio Lglu. Evaluation of the convolution sums.l/.m/ and.l/.m/. Int J lc27mdn lc32mdn Number Theory, 206, 2(), 3. [20] Dongxi Ye. Evaluation of the convolution sums.l/.m/ and.l/.m/. Int J Number Theory, 205, lc36mdn 4lC9mDn (), [2] Ebénézer Ntienjem. Evaluation of the convolution sums.l/.m/; where. ; ˇ/ is in lcˇ mdn f.; 4/;.2; 7/;.; 26/;.2; 3/;.; 28/;.4; 7/;.; 30/;.2; 5/;.3; 0/;.5; 6/g. Master s thesis, School of Mathematics and Statistics, Carleton University, 205. [22] Neal Koblitz. Introduction to Elliptic Curves and Modular Forms, volume 97 of Graduate Texts in Mathematics. Springer Verlag, New York, 2 nd edition, 993. [23] William A. Stein. Modular Forms, A Computational Approach, volume 79. American Mathematical Society, Graduate Studies in Mathematics, [24] Morris Newman. Construction and application of a class of modular functions. roc Lond Math Soc, 957, 7(3), [25] Morris Newman. Construction and application of a class of modular functions II. roc Lond Math Soc, 959, 9(3), m< n 6 [26] Gérard Ligozat. Courbes modulaires de genre. Bull Soc Math France, 975, 43, [27] L. J.. Kilford. Modular forms: A classical and computational introduction. Imperial College ress, London, [28] G. Köhler. Eta roducts and Theta Series Identities, volume 3733 of Springer Monographs in Mathematics. Springer Verlag, Berlin Heidelberg, 20. [29] Toshitsune Miyake. Modular Forms. Springer monographs in Mathematics. Springer Verlag, New York, 989. [30] Kenneth S Williams. Number Theory in the Spirit of Liouville, volume 76 of London Mathematical Society Student Texts. Cambridge University ress, Cambridge, 20. Download Date 2/8/8 5:2 M