Cvijović, Djurdje

Klinowski, Jacek

English

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

Mathematica SlovacaDjurdje Cvijović; Jacek KlinowskiFinite cotangent sums and the Riemann zeta functionMathematica Slovaca, Vol. 50 (2000), No. 2, 149--157Persistent URL: http://dml.cz/dmlcz/131809Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 2000Institute 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.czMathematica Slovaca © 2 0 0 0 . . . _. __, /^rt-.rtx M ^ ->!/-. ni-- Mathematical Insti tute Math . SlOVaCa, 5 0 ( 2 0 0 0 ) , NO. 2 , 1 4 9 - 1 5 7 Slovák Academy of Sciences FINITE COTANGENT SUMS AND THE RIEMANN ZETA FUNCTION D J U R D J E CVIJOVIC — JACEK KLINOWSKI (Communicated by Stanislav Jakubec ) ABSTRACT. We give a closed-form evaluation of the finite cotangent sums given by £«• [ffi±tfc]. £«- ( p=0 L q J p=l ч 'D where n and q are positive integers {q > 2) and £ is a non-integer real number. The lat ter sum enables us to simplify the Apostol formula concerning the Riemann zeta function at integer values of the argument. We demonstrate tha t , for even integers, this formula involves polynomials with rational arguments. 1. Introduction Consider the Riemann zeta function ((z) defined by [1; p. 19] oo .. 1 oo 1 CW = £ F = T^S(^TF Rez>1 k=l k=l v ' oo 1 = r_i--/E(-1)'k"V R e z > 0 ' z*1-k=i z For Re z < 1, z ^ 1, ((z) is defined as the analytic continuation of the foregoing series. The Riemann zeta function is analytic over the whole complex plane, except at z — 1, where it has a simple pole. A p o s t o l [2] gave the following formula £ cot" (-^r) C(n) - ( J ) lim P— (1) v } V 2 j ?->oo qn v y 1991 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n : Pr imary 11L03, 11M99; Secondary 40D99. K e y w o r d s : Riemann zeta function, finite sum. 149 DJURDJE CVIJOVIC — JACEK KLINOWSKI valid for any integer n > 2. For the case when n is even, he found the asymptotic expansion of the finite sum involved, which readily leads to the Euler relation ,n-ЛM.n 2(2n)! ' C(2n) = (-l)"- 1^eVB 2„ ( n > l ) (2) between the even-indexed Bernoulli numbers B2n and ((2n) Here, we deduce summation formulae for Sn(q;0 = Y,ҷЬn P = i ~X Ҷ£ + P K (0 < £ < 1) (Зa) and ł - i ад = Ecot"(т) (зb) p = l v ч J and examine the Apostol formula (1) for even integers. 2. First results Throughout the text [x] denotes the largest integer not exceeding x, while, as usual, (^) is the binomial coefficient given by n\ f n\ n\ ,—l and j - = — - — ' - — ( n . m G N , l < m < n ) . 0/ \mj m\(n — m)\ Further, the principal values of arccot# are defined for any real 6 by 0 < arccot x < 7T. In this section we establish the existence of the polynomials for which all zeros, regardless of the degree, can be given by a simple formula involving the cotangent function. It appears that these polynomials are unknown, and their detailed study is given elsewhere [3]. THEOREM. Let n be a positive integer, and Cn(x;£) a polynomial of degree precisely n. in a variable x with real parameter £, which we define as Cn(x;0=J2(-iyncn,m(0xn-m (4a) m=0 Where Cn,m(0 is c„iTO(f) = Q [cos(m7r/2) + sin(m7r/2) c o t « ) ] (0 < f < 1) • (4b) 150 FINITE COTANGENT SUMS AND THE RIEMANN ZETA FUNCTION Then for any fixed n, the zeros of Cn(x\ £) are all real, simple and given by r „ , m ( 0 = cot ( - i - t p - - - ) m = 0 . 1 . . . . , n - l . (5) Before proving the Theorem we give as an example the first few polynomials Cx (x\ £) = x — cot(7r£), C2 (~; f) = x2 - 2x c o t « ) - 1, C3(x] f) = x3 - 3x2 cot(7rf) - 3x -F COt(7Tf) , C4(x] f) = x4 - 4x3 cot(7r0 - 6x2 + 4~cot(7rO + 1, C5(x; f) = x5 - 5x4 cot(?rO - 10x3 + 10x2 cot(Trf) + 5x - cot(Trf), C6(x] £) = x6 - 6x5 cot(7r0 - 15x4 + 20x3 c o t ^ ) + I5x2 - 6x cot(?rO - 1. P r o o f . First of all, we shall prove that [n/2] , . V ^ ( - l ) m ( )xn~2m = cscn(arccotx)cos(narccotx) (6a) m=o ^ m / and [ (n - l ) / 2 ] , v Yl ( _ 1 ) m ( U ) ^ n - 2 m _ 1 =csc n (arccotx)s in(narccotx) (6b) m=0 V m + / where arccot x takes its principal value, x is real and n is a positive integer. Starting from the well-known Euler-Bernoulli formulae [4; 4.4.4.15, 4.4.4.16] n/2] COS. m=0 [ ( n - l ) / 2 ] sin? m=0 we obtain [n/2] ["/-] / ч iтг6>= ] Г ( - l ) m ( П )cos n - 2 mØsin 2 mé> ^ ' » , s m = Y ( - l ) m ( ) c o s n _ 2 m - 1 Ø s i n 2 m + 1 0 - Ч v y V 2 m + l j ш—-L í n \ Y2 ( - l ) m ( ) t an 2 m = secn cosn m=0 ^ ' У " ( - l ) m ( П ] t a n 2 m + 1 0 = secnØsinnØ ---1 ; V2m+11 m—II N '  [ ( n - l ) / 2 ] where 0 7̂  (2fc+l)7r/2 (k £ Z ) . In view of these identities, upon setting x — cot 0 (0 < 9 7̂  7r/2 < 7r) in (6a), (6b), it follows at once that the proposed formulae 151 DJURDJE CVIJOVIC — JACEK KLINOWSKI hold when x / 0. Moreover, it can be shown by direct verification that they remain valid for x = 0. Next, since the polynomials Cn(x;£) may be expressed in the form [n/2] / . [(n-l)/2] / „ \ cn(^o=E^r(2my-2m-cot(,o £ (-ir(2m+iy ra=0 x ' ra=0 v 7 2 m - l (7) as an immediate consequence of (6) we have Cn(x;£) = cscn(arccotx)[cos(narccotx) - cot(7r£)sin(narccotx)] . (8) To complete the proof we only need to verify that Cn(Tn m(£);£) = 0 for any n and ra = 0 , 1 , . . . , n — 1, regardless of £. Observe that, in general, if k is an integer and 9 is real, then arccot(cot 9) = 9 - kir kir < 9 < (k + 1)TT . Thus (£-r-rn)7r -*ccot[rn>m(0] = П since 0 < ( ! + m K < Î Г ł m = 0 , 1 , . . . ,n — 1, provided that 0 < £ < 1. Finally, it follows readily from (8), that c B ( r B i m (0 ;0 = (-i)m c8c^« + m ) ' ' [C0S(7Г£) - COt(7г£) SІn(7Г<î;)] = 0 . Since Cn(x\^) is of degree n , it vanishes at the n points r n m ( £ ) for ra = 0 , 1 , . . . , n — 1. In view of the properties of the cotangent functions, these zeros are all real and simple. This completes our proof. • A similar result is obtained as follows. Let Cn(x) denote the polynomial given by the expressions in (6b). Several examples are C{(x c;(x c;(x C*A(x c;(x c;(x c;(x c;(x = 1, = 2x, = Зx 2 - 1, = 4x 3 - 4x, = 5x 4 - 10x2 + 1, = 6x 5 - 20x3 + 6x , = 7x6 - 35x4 + 21x2 - 1, = Sx7 - 56x5 + 56x3 - 8x . The determination of the zeros of Cn(x) is clearly equivalent to solving sin(n arccot x) = 0, which gives rise to the following Proposition. 152 FINITE COTANGENT SUMS AND THE RIEMANN ZETA FUNCTION PROPOSITION 1. Assume that n is a positive integer. Let Cn(x) be the poly-nomial in a real variable x of degree n — 1. defined by (6b). Then for any fixed n>2, the zeros of Cn(x) are all real, simple and given by r;,™ = c o t ( ^ ) , m = l , 2 , . . . , n - l . 3. Finite cotangent sums As a straightforward consequence of the Theorem, we derive a closed-form determinant formula for finite cotangent sum Sn(q;£). We note that the sums Sx, S2 and S4 are well known in a slightly different form [4; Section 4.4.7] and [5; Sections 29.1 and 30.1]. We begin by stating some elementary results from the theory of polynomials needed [6; p. 179]. Let xk (1 < k < n) be the roots of an algebraic equation of the form xn + axxn~x H h an_xx + \-an = xn- axxn-x + • • • + ( - l r - X - i * + ( - ! ) X = ° (10) with coefficients a1,a2,... , a n (n > 1). The elementary symmetric functions a1,a2,...ian of xk are then given by the Viete formulae xi + x2 + * *' + xn = "i > X^X2 + X1X^ + ' ' • + "n—\~n — <~2 5 X^2X^ + XXX2X± + ' ' ' + xn-2Xn-lXn = a3 ' x1x2 + "- + xn = an while sk (k = 1,2, . . . ), the kth power sum of zeros, defined as sk = xl + x2 + - • - + xn is related to ak by the Newton identities as follows "i = si > sxax -2a2 = 5 2 , s2al — S-^a2 — oO"3 = 53 , sk°i - sk-i(T2 + ••• + (-l)k+1kak = sk . 153 DJURDJE CVIJOVIC JACEK KLINOWSKI The Newton identities can clearly be considered as a system of linear equations for s-_, s 2 , . . . , sn, which enables us to rewrite them into a more convenient form. Thus by making use of the Cramer rule [6; p. 198] it is easy to see from the first k of the equations, that power sum sk is given by the following determinant Si. = (k 2a2 3<7o kau 1 0"i c k-2 Ţk-1 0 0 .. . 0 1 0 .. . 0 a l 1 .. . 0 1 <т, of the order k. Now, in view of the Theorem, it is obvious that the finite sum in (3a) is in fact the power sum of the zeros of the polynomial in (4). Thus, we can state the following result. PROPOSITION 2. Assume that n is a positive integer, and let Sn(q;£,) be a finite sum in (3a). Then for any fixed n we have sn(q;0 cяЛ 1 0 0 .. . 0 2c„9 q,2 C<7,1 1 0 .. 0 З C < 7 , 3 C ? , 2 c? , l 1 .. . 0 n- -K.n-1 C, q,n- l nc„ ~q,n q,n — l <7,1 where, for brevity, c t stands for c t(0 defined in (4b), and the determinant is of the order n. In particular Sx(q;0 = qcot(nO, S2(q;0 = q2(cot2(7r0 + l)-q, S3(q;0 = <73(cot3(>r£) + c o t ( O ) - gcotfrO , SA(q; 0 = q4(cotA(7rO + (4/3) cot2(7r£) + 1/3) - </2((4/3) c o t 2 « ) + 4/3) + q , S5(q; 0 = q5(cot5(irO + (5/3) cot3(7rO + (2/3) cotfrO) - q3 ((5/3) cot3(7r0 + (5/3) cot3(7re)) + gcot(vrO , S6(q;0 = 96(cot6(7r0 + 2cot4(7r£) + (17/15) cot2(7rO + 2/15) - q4 (2 cot4(7r0 + (8/3) cot2(7r£) + 2/3) + q2 ((23/15) cot2(?r0 + 23/15) - q. 154 FINITE COTANGENT SUMS AND THE RIEMANN ZETA FUNCTION We note that the expressions for Sn(q\l/2), Sn(q]l/4), S2n(q;l/3) and S2n(q;l/6) are the polynomials in q with rational coefficients. Moreover, £___,(.; 1/2) = 0. However, it is easier to find the sums for higher n by using the differential recurrence relation ~W?;0 = -s-fofl - j^ihjnTzs^{q>® {n -1} which is readily proved by differentiating the definition (3a) with respect to £. 4. The Apostol formula In this section we shall derive the closed-form formula for 5* (q) which allows us to investigate the Apostol formula. The properties of the cotangent function readily lead to o - l X> tW^W í + l ( í ? ) = o P=I \ * / and o - l X > Ҷ | И ^ p=\ x ' The first indication about S2n can be obtained from the expressions found in Section 3 and the relation o - l 52n(?) = Jҷ E C O t " ( ^ 7 ^ ) = lҷ[Sn(*0 ~ «*»«/_)] resulting in 1 , 2 ' - ^ = 3 ' " ; з e * ^ ! 4 4 2__ 26 54(9) = 45? - 9 ? + 9 - 4 5 . o . , . 2 6 2 4 23 2 502 5 « ( * ) = 945^-45^ + 4 5 ^ - * + 9 4 5 ' where our S^ and 5 4 are the same as those listed in literature [4; Section 4.4.7]. 155 DJURDJE CVIJOVIC — JACEK KLINOWSKI On the other hand, a straightforward consequence of Proposition 1 is that the sum in (3b) is the power sum of the zeros of Cn(x). In this way we shall evaluate Sn for any n. Note, however, that the polynomial Cn(x) is not monic, that it is of degree n — 1 and that some of its coefficients are zero (compare with Equation 10). PROPOSITION 3. Let n be a positive integer and let Sn(q) be a finite sum defined by (3b). Then for any fixed n we have S*n(q) 0 2c 9,2 0 1 0 C«,2 0 0 1 0 0 1 (n — l)c„ „ - c v / q,n — l < ПC q,n q,n — 2 Cq,n-1 where c { stands for c {(£), and the determinant is of the order n. It is not difficult to verify that S^n-t-i ls a i w a v s zero, and it follows directly that, in general, S%n is the polynomial of degree 2n in q with rational coeffi-cients. Two more special cases are S*s(q) = 1 16 44 4725 s"<") = d b ' " -4 28354 675ч 176 7102 315? + Q 14175' 2835 q° + 86 8505 718 563 ~ 44834 8505q + 945q q 93555 5. Concluding remarks The Apostol formula implies the existence of a certain function A(q; n), such that A(q;n) q-+oo qn We have demonstrated that, for even integers, we have C(„) = ( ЛП liш -tøi-!l (n>2) v ' 4 2 / ?^o  qn v - ' A(«2;2n) = - 5 * n ( 2 g + l ) ( n > l ) , where S^n is fully determined: it is a polynomial in q with rational coefficients of degree 2n and its evaluation is straightforward. 156 FINITE COTANGENT SUMS AND THE RIEMANN ZETA FUNCTION REFERENCES [1] MAGNUS, W . — O B E R G E T T I N G E R , F.—SONI, R. P . : Formulas and Theorems for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin, 1966. [2] APOSTOL, T. M. : Another elementary proof of Euler's formula for £(2), Amer. Math. Monthly 80 (1973), 425-431. [3] CVIJOVIC, D.—KLINOWSKI, J . : A new polynomial system with easily deducible zeros, (Subm i t ted) . [4] PRUDNIKOV, A. P .—BRYCHKOV, YU. A.—MARICHEV, O. I. : Integrals and Series, Vol. 1, Gordon and Breach Science Publishers, New York, 1986. [5] HANSEN, E. R. : A Table of Series and Products, Prentice-Hall, Englewood Cliffs, 1975. [6] COHN, P . M. : Algebra, Vol. 1, John Wiley, Chichester, 1982. Received January 26, 1998 Department of Chemistry University of Cambridge Lensfield Road Cambridge CB2 1EW UNITED KINGDOM 157 

Finite cotangent sums and the Riemann zeta function

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Finite cotangent sums and the Riemann zeta function

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