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A Few Remarks on Values of Hurwitz Zeta Function at Natural and Rational Arguments

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A Few Remarks on Values of Hurwitz Zeta Function at Natural and Rational Arguments Mathematica Aeterna, Vol. 5, 2015, no. 2, 383 - 394 A few remarks on values of Hurwitz Zeta function at natural and rational arguments Pawe÷J. Szab÷owski Department of Mathematics and Information Sciences, Warsaw University of Technology ul. Koszykowa 75, 00-662 Warsaw, Poland Abstract We exploit some properties of the Hurwitz zeta function (n, x) in order to 1 n study sums of the form n j1= 1/(jk + l) and 1 j n 1 n j1= ( 1) /(jk + l) for 2 n, k N, and integer l k/2. We show 1 P ≤ 2 ≤ that these sums are algebraic numbers. We also show that 1 < n N and P n n 2 p Q (0, 1) : the numbers ((n, p) + ( 1) (n, 1 p))/π are algebraic. 2 \ On the way we find polynomials sm and cm of order respectively 2m + 1 and 2m + 2 such that their n th coeffi cients of sine and cosine Fourier transforms are equal to ( 1)n/n2m+1and ( 1)n/n2m+2 respectively. Mathematics Subject Classification: Primary 11M35, 11M06, Sec- ondary 11M36, 11J72 Keywords: Riemann zeta function, Hurwitz zeta function, Bernoulli num- bers, Dirichlet series 1 Introduction Firstly we find polynomials sm and cm of orders respectively 2m+1 and 2m+2 such that their n th coeffi cients in respectively sine and cosine Fourier series are of the form (1)n/n2m+1 and ( 1)n/n2m+2. Secondly using these polyno- mials we study sums of the form: 1 1 1 ( 1)j S(n, k, l) = , S^(n, k, l) = . (jk + l)n (jk + l)n j= j= X1 X1 for n, k, l N such that n 2, k 2 generalizing known result of Euler (re- cently proved2 differently by≥ Elkies≥ ([6])) stating that the number S(n, 4, 1)/πn 384 Pawe÷J. Szab÷owski is rational. In particular we show that the numbers S(n, k, j)/πn and S^(n, k, j)/πn for N n, k 2, and 1 j < k are algebraic. The3 results≥ of the paper≤ are somewhat in the spirit of [5] in the sense that the sums we study are equal to certain combinations of values of the Hurwitz zeta function calculated for natural and rational arguments. The paper is organized as follows. In the present section we recall basic definitions and a few elementary properties of the Hurwitz and Riemann zeta functions. We present them for completeness of the paper and also to introduce in this way our notation. In Section 2 are our main results. Longer proof are shifted to Section 3. To fix notation let us recall that the following function (s, x) = 1/(n + x)s, n 0 X≥ defined for all Re(s) > 1 and 1 Re(x) > 0 is called Hurwitz zeta function df ≥ while the function (s, 1) = (s) is called the Riemann zeta function. One shows that functions (s, x) and (s) can be extended to meromorphic func- tions of s defined for all s = 1. Hurwitz zeta function has been studied from different points of view in e.g.6 [7], [12], [4]. Let us consider slight generalization of the Hurwitz zeta function namely the function; ^(s, x) = ( 1)n/(n + x)s, n 0 X≥ for the same as before s and x. Using multiplication theorem for (s, x) and little algebra we get: (s, x) = ((s, x/2)+(s, (1+x)/2))/2s, ^(s, x) = ((s, x/2) (s, (1+x)/2))/2s, (1) 1 l l S(n, k, l) = ((n, ) + ( 1)n(n, 1 )), (2) kn k k 1 l l S^(n, k, l) = (^(n, ) + ( 1)n+1^(n, 1 )), kn k k S(n, k, l) = S(n, 2k, l) + ( 1)nS(n, 2k, k l), (3) S^(n, k, l) = S(n, 2k, l) + ( 1)n+1S(n, 2k, k l), (4) S(n, k, k l) = ( 1)nS(n, k, l), S^(n, k, k l) = ( 1)n+1S^(n, k, l). (5) Hence there is sense to define sums S and S^ for l k/2 only. ≤ In the sequel N and N0 will denote respectively sets of natural and natural plus 0 numbers while Z and Q will denote respectively sets of integer and f g Zeta functions 385 rational numbers. Bk will denote k th Bernoulli number. It is known that B2n+1 = 0 for n 1 and B1 = 1/2. One knows also that ≥ 2l l+1 (2) (2l) = ( 1) B2l (6) 2(2l)! for all l N. For more2 properties of the Hurwitz zeta function and the Bernoulli numbers see [9], [3], [11] and [2]. Having this and (3) and applying multiplication theorem for (s, x) we can easily generalize result of Euler reminded by Elkies in [6]). Let us recall that since Euler times it is known that. (22l 1) l+1 1 2(2l)! ( 1) B2l if n = 2l, S(n, 4, 1) = l n ( 1) E2l ( 22l+2(2l)! if n = 2l + 1. Apart from this result one can easily show (manipulating multiplication theo- rem for Hurwitz zeta function) that for for N: 2 (2l)! 2(2l)! 1 2l l+1 S(2l, 2, 1) = (2l, ) = (2 1)( 1) B2l, 2l (2)2l 2 2(2l)! 2 2l l+1 2l S(2l, 3, 1) = ( ) ( 1) (3 1)B2l, 2l 3 2(2l)! 2l 2l l+1 S(2l, 6, 1) = (2 1)(1 3 )( 1) B2l. 2l Notice also that from (1) it follows that S(2l + 1, 2, 1) = 0 for l N and from (1) and (2) we get: S^(2l, 2, 1) = 0 and S^(2l + 1, 2, 1) = 2S(2l + 1,24, 1). Results concerning S(2l, 6, 1) were also obtained in [5](16b). 2 Main results To get relationships between particular values of (n, q) for n N and q Q we will use some elementary properties of the Fourier transforms.2 2 Let us denote s s 1 c c 1 n(f(.)) = n = f(x) sin(nx)dx, n(f(.)) = n = f(x) cos(nx)dx, F F F F Z Z s c s c for n N0. We will abbreviate n(f(.)) and n(f(.)) to n and n if the function2 f is given. We have: F F F F 1 f(x) = c + c cos(nx) + s sin(nx) 2F0 Fn Fn n 1 n 1 X≥ X≥ 386 Pawe÷J. Szab÷owski + for all points of continuity of f and (f(x ) + f (x)) /2 if at x f has jump discontinuity as a result of well known properties of Fourier series. Let us define two families of polynomials: 2 m+1 m x ( 1) 2m + 2 2(m j) 2j 2j 1 cm(x) = x B2j(2 1), (7) (2m + 2)! 2j j=0 m Xm x ( 1) 2m + 1 2(m j) 2j 2j 1 s (x) = x B (2 1), (8) m (2m + 1)! 2j 2j j=0 X for all m N0. Let us consider also the following polynomials: 2 2m+1 m 2(2) x chm(x) = ( 1) B2m+2( ), (2m + 2)! 2 2m m 1 2(2) x shm(x) = ( 1) B2m+1( ). (2m + 1)! 2 Basing partially on [8] we have the following auxiliary lemma. Lemma 1 For n N, m N0 2 2 n n s ( 1) c ( 1) (sm(.)) = , (cm(.)) = , (9) Fn n2m+1 Fn n2m+2 s 1 c 1 (shm(.)) = , (chm(.)) = , (10) Fn n2m+1 Fn n2m+2 2m+1 c 2m+2 m 2(2 1) 2m 1 (cm(.)) = ( 1) B2m+2 = 2(2m + 2)(1 2 ), F0 (2m + 2)! c (chm(.)) = 0. (11) F0 Proof. Is shifted to Section 3. Remark 1 Notice that we have just summed the two following Dirichlet series: j cos(jx) j sin(jx) DC(s, x) = j 1( 1) js and DS(s, x) = j 1( 1) js , for particular values of s. Namely≥ we have showed that ≥ P P 2m+1 2m+2 m+1 (2 1) DC(2m + 2, x) = cm(x) + ( 1) B2m+2 , (2m + 2)! DS(2m + 3, x) = sm+1(x) and for x , m N0. What are the values of these series for s = 1, 2, 3,...? j j ≤ 2 6 l 2m+2 l 2m+1 Remark 2 Notice that cm( k )/π and sm( k )/π for all k N, m N0 and l k, are rational numbers. 2 2 ≤ Zeta functions 387 As a consequence we get the following theorem that allows calculating values of S(n, k, l) and S^(n, k, l) for N n, k 2 and l k. 3 ≥ ≤ Theorem 2 For m N0, i N, i) 2 2 m 2m+2 (2l 1) ( 1) B2m+2 2m+1 1 cm( ) = (2 (1 + ) 1) (12) 2i + 1 (2m + 2)! (2i + 1)2m+2 i j(2l 1) + ( 1)j cos( )S(2m + 2, 2i + 1, j), 2i + 1 j=1 X for l = 1, . , i 1. ii) 2m+2 m 2m+1 (2l 1) ( 1) B2m+2(2 1) 2m 2 cm( ) = (1 (2i) ) (13) 2i (2m + 2)! i 1 (2l 1)j + ( 1)j cos( )S^(2m + 2, 2i, j), 2i j=1 X for l = 1, . , i. iii) 2m+2 m 2m+1 2l ( 1) B2m+2(2 1) 2m 2 cm( ) = (1 (2i + 1) ) (14) 2i + 1 (2m + 2)! i 2lj + ( 1)j cos( )S^(2m + 2, 2i + 1, j), 2i + 1 j=1 X for l = 1, . , i. iv) For m, k N: 2 (2l 1) i (2l 1)j s ( ) = ( 1)j sin( )S(2m + 1, 2i + 1, j), (15) m 2i + 1 2i + 1 j=1 X for l = 1, .
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