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A Few Remarks on Values of Hurwitz Zeta Function at Natural and Rational A FEW REMARKS ON VALUES OF HURWITZ ZETA FUNCTION AT NATURAL AND RATIONAL ARGUMENTS PAWELJ. SZABLOWSKI Abstract. We exploit some properties of the Hurwitz zeta function ζ(n,x) 1 ∞ n in order to study sums of the form πn Pj=−∞ 1/(jk + l) and 1 ∞ j n πn Pj=−∞(−1) /(jk + l) for 2 ≤ n,k ∈ N, and integer l ≤ k/2. We show that these sums are algebraic numbers. We also show that 1 < n ∈ N and p ∈ Q ∩ (0, 1) : the numbers (ζ(n,p) + (−1)nζ(n, 1 − p))/πn are algebraic. On the way we find polynomials sm and cm of order respectively 2m + 1 and 2m + 2 such that their n−th coefficients of sine and cosine Fourier transforms are equal to (−1)n/n2m+1 and (−1)n/n2m+2 respectively. 1. Introduction Firstly we find polynomials sm and cm of orders respectively 2m +1 and 2m +2 such that their n th coefficients in respectively sine and cosine Fourier series are of the form ( 1)n−/n2m+1 and ( 1)n/n2m+2. Secondly using these polynomials we study sums of− the form: − ∞ 1 ∞ ( 1)j S(n,k,l)= , Sˆ(n,k,l)= − . (jk + l)n (jk + l)n j=−∞ j=−∞ X X for n, k, l N such that n 2, k 2 generalizing known result of Euler (recently proved differently∈ by Elkies≥ ([7]))≥ stating that the number S(n, 4, 1)/πn is ratio- nal. In particular we show that the numbers S(n,k,j)/πn and Sˆ(n,k,j)/πn for N n, k 2, and 1 j < k are algebraic. ∋The results≥ of the≤ paper are somewhat in the spirit of [4] 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. arXiv:1405.6270v2 [math.NT] 8 Dec 2014 The paper is organized as follows. In the present section we recall basic defini- tions 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 ζ(s, x)= 1/(n + x)s, nX≥0 Date: April 2014. 2000 Mathematics Subject Classification. Primary 11M35, 11M06, Secondary 11M36, 11J72; Key words and phrases. Riemann zeta function, Hurwitz zeta function, Bernoulli numbers, Dirichlet series. 1 2 PAWELJ. SZABLOWSKI defined for all Re(s) > 1and 1 Re(x) > 0 is called Hurwitz zeta function while df ≥ the function ζ(s, 1) = ζ(s) is called the Riemann zeta function. One shows that functions ζ(s, x) and ζ(s) can be extended to meromorphic functions of s defined for all s = 1. Let us6 consider slight generalization of the Hurwitz zeta function namely the function; ζˆ(s, x)= ( 1)n/(n + x)s, − nX≥0 for the same as before s and x. Using multiplication theorem for ζ(s, x) and little algebra we get: (1.1) ζ(s, x) = (ζ(s,x/2)+ ζ(s, (1 + x)/2))/2s, ζˆ(s, x) = (ζ(s,x/2) ζ(s, (1 + x)/2))/2s, − (1.2) 1 l l 1 l l S(n,k,l)= (ζ(n, )+( 1)nζ(n, 1 )), Sˆ(n,k,l)= (ζˆ(n, )+( 1)n+1ζˆ(n, 1 )), kn k − −k kn k − −k (1.3) S(n,k,l)= S(n, 2k,l) + ( 1)nS(n, 2k, k l), − − (1.4) Sˆ(n,k,l)= S(n, 2k,l) + ( 1)n+1S(n, 2k, k l), − − (1.5) S(n,k,k l) = ( 1)nS(n,k,l), Sˆ(n,k,k l) = ( 1)n+1Sˆ(n,k,l). − − − − Hence there is sense to define sums S and Sˆ 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 rational { } numbers. Bk will denote k th Bernoulli number. It is known that B2n+1 = 0 for n 1 and B =1/2. One knows− also that ≥ 1 (2π)2l (1.6) ζ(2l) = ( 1)l+1B − 2l 2(2l)! for all l N. For more∈ properties of the Hurwitz zeta function and the Bernoulli numbers see [5], [3], [9] and [2]. Having this and (1.3) and applying multiplication theorem for ζ(s, x) we can easily generalize result of Euler reminded by Elkies in [7]). Let us recall that since Euler times it is known that. 2l (2 −1) l+1 1 2(2l)! ( 1) B2l if n =2l, S(n, 4, 1) = −l πn (−1) E2l ( 22l+2(2l)! if n =2l +1. Apart from this result one can easily show (manipulating multiplication theorem for Hurwitz zeta function) that for for N: ∈ (2l)! 2(2l)! 1 S(2l, 2, 1) = ζ(2l, )=(22l 1)( 1)l+1B , π2l (2π)2l 2 − − 2l 2(2l)! 2 S(2l, 3, 1) = ( )2l( 1)l+1(32l 1)B , π2l 3 − − 2l 2(2l)! S(2l, 6, 1) = (22l 1)(1 3−2l)( 1)l+1B . π2l − − − 2l ZETA FUNCTIONS 3 Notice also that from (1.1) it follows that S(2l +1, 2, 1) = 0 for l N and from ∈ (1.1) and (1.2) we get: Sˆ(2l, 2, 1)= 0 and Sˆ(2l +1, 2, 1)=2S(2l +1, 4, 1). Results concerning S(2l, 6, 1) were also obtained in [4](16b). 2. Main results To get relationships between particular values of ζ(n, q) for n N and q Q we will use some elementary properties of the Fourier transforms. ∈ ∈ Let us denote 1 π 1 π s(f(.)) = s = f(x) sin(nx)dx, c(f(.)) = c = f(x)cos(nx)dx, Fn Fn π Fn Fn π Z−π Z−π s c s c for n N0. We will abbreviate n(f(.)) and n(f(.)) to n and n if the function f is given.∈ We have: F F F F 1 f(x)= c + c cos(nx)+ s sin(nx) 2F0 Fn Fn nX≥1 nX≥1 for all points of continuity of f and (f(x+)+ f (x−)) /2 if at x f has jump discon- tinuity as a result of well known properties of Fourier series. Let us define two families of polynomials: x2( 1)m+1 m 2m +2 (2.1) c (x)= − x2(m−j)π2j B (22j−1 1), m (2m + 2)! 2j 2j − j=0 X x ( 1)m m 2m +1 (2.2) s (x)= − x2(m−j)π2j B (22j−1 1). m (2m + 1)! 2j 2j − j=0 X for all m N0. We have∈ the following auxiliary lemma. Lemma 1. For n N, m N ∈ ∈ 0 ( 1)n ( 1)n (2.3) s(s (.)) = − , c(c (.)) = − . Fn m n2m+1 Fn m n2m+2 2(22m+1 1) c(c (.)) = π2m+2( 1)mB − =2ζ(2m + 2)(1 2−2m−1) F0 m − 2m+2 (2m + 2)! − 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) = ( 1) s and DS(s, x) = ( 1) s , for particular j≥1 − j j≥1 − j values of s. Namely we have showed that P P (22m+1 1) DC(2m +2, x) = c (x)+ π2m+2( 1)m+1B − , m − 2m+2 (2m + 2)! DS(2m +3, x) = sm+1(x) and for x π, m N . What are the values of these series for s =1, 2, 3,...? | |≤ ∈ 0 6 πl 2m+2 πl 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. ∈ ∈ ≤ As a consequence we get the following theorem that allows calculating values of S(n,k,l) and Sˆ(n,k,l) for N n, k 2 and l k. ∋ ≥ ≤ 4 PAWELJ. SZABLOWSKI Theorem 1. For m N0, i N, i) ∈ ∈ (2l 1)π ( 1)mB π2m+2 1 (2.4) c ( − )= − 2m+2 (22m+1(1 + ) 1) m 2i +1 (2m + 2)! (2i + 1)2m+2 − i j(2l 1)π + ( 1)j cos( − )S(2m +2, 2i +1, j), − 2i +1 j=1 X for l =1,...,i 1. ii) − (2l 1)π π2m+2( 1)mB (22m+1 1) (2.5) c ( − )= − 2m+2 − (1 (2i)−2m−2) m 2i (2m + 2)! − i−1 (2l 1)jπ + ( 1)j cos( − )Sˆ(2m +2, 2i, j), − 2i j=1 X for l =1,...,i. iii) 2lπ π2m+2( 1)mB (22m+1 1) (2.6) c ( )= − 2m+2 − (1 (2i + 1)−2m−2) m 2i +1 (2m + 2)! − i 2ljπ + ( 1)j cos( )Sˆ(2m +2, 2i +1, j), − 2i +1 j=1 X for l =1,...,i. iv) For m, k N: ∈ i (2l 1)π (2l 1)jπ (2.7) s ( − )= ( 1)j sin( − )S(2m +1, 2i +1, j), m 2i +1 − 2i +1 j=1 X for l =1,...i 1. v) − (2l 1)π 1 (2.8) s ( − ) = ( 1)i+l+1 S(2m +1, 4, 1) m 2i − i2m+1 i−1 (2l 1)jπ + ( 1)j sin( − )Sˆ(2m +1, 2i, j), − 2i j=1 X for l =1,...i. vi) 2lπ i 2ljπ (2.9) s ( )= ( 1)j sin( )Sˆ(2m +1, 2i +1, j), m 2i +1 − 2i +1 j=1 X for l =1, .
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