GENERALIZED RATIONAL ZETA SERIES FOR ζ(2n) AND ζ(2n + 1) DEREK ORR Abstract. In this paper, we find rational zeta series with ζ(2n) in terms of ζ(2k +1) and β(2k), the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function and use those results to discover a second family of generalized rational zeta series. As a special case of our results from Theorem 3.1, we prove a conjecture given in 2012 by F.M.S. Lima. Later, we use the same analysis but for the digamma function ψ(x) and negapolygammas ψ(−m)(x). With these, we extract the same two families of generalized rational zeta series with ζ(2n+1) on the numerator rather than ζ(2n). Contents 1. Introduction 1 1.1. Organization of the Paper 5 2. Rational ζ(2n) series with cot(x) 5 3. General ζ(2n) series using Clm(x) 8 4. General ζ(2n) series using cot(x) 12 5. Rational ζ(2n + 1) series using ψ(x) 16 6. General ζ(2n + 1) series using ψ(−m)(x) 18 7. General ζ(2n + 1) series using ψ(x) 22 References 26 1. Introduction arXiv:1606.04850v4 [math.NT] 12 Feb 2017 In 1734, Leonard Euler proved an amazing result, now known as the celebrated Euler series: ∞ 1 1 1 1 π2 ζ(2) = =1+ + + + ··· = , n2 4 9 16 6 n=1 X where ζ(s) is the Riemann zeta function, defined as ∞ 1 ζ(s)= , ℜ(s) > 1. ns n=1 X Key words and phrases. Riemann zeta function, Dirichlet beta function, Clausen integral, ne- gapolygammas, rational zeta series, polygamma function. 2010 Mathematics Subject Classification. Primary 40C10, 11M99. Secondary 41A58. 1 2 DEREK ORR Later, Euler gave the formula ∞ 1 (−1)k+1B (2π)2k ζ(2k)= = 2k , k ∈ N , n2k 2(2k)! 0 n=1 X where Bn are the Bernoulli numbers, defined by z ∞ B = n zn, |z| < 2π. ez − 1 n! n=0 X These Bernoulli numbers also arise in certain power series, namely of the cotangent function ∞ (−1)n22nB ∞ ζ(2n) (1) cot(x)= 2n x2n−1 = −2 x2n−1, |x| < π. (2n)! π2n n=0 n=0 X X The odd arguments of ζ(s) are the interesting as they do not have a closed form, though many mathematicians have studied them in detail. In 1979, Roger Apery [3] proved that ζ(3) is irrational using the fast converging series 5 ∞ (−1)n−1 ζ(3) = . 2 3 2n n=1 n n X It is still unknown whether ζ(5) is irrational however, it is known that at least one of ζ(5), ζ(7), ζ(9), and ζ(11) is irrational (see [18]). The rational zeta series in this paper will involve odd arguments of ζ(s) among other things. Another main function needed is the Clausen function (or Clausen’s integral), ∞ sin(kθ) θ φ (2) Cl2(θ) := 2 = − log 2 sin dφ. k 0 2 Xk=1 Z The Clausen function also has a power series representation which will be used later in the paper. It is given as ∞ Cl (θ) ζ(2n) θ 2n (3) 2 =1 − log |θ| + , |θ| < 2π. θ n(2n + 1) 2π n=1 X There are also higher order Clausen-type function defined as ∞ sin(kθ) ∞ cos(kθ) (4) Cl (θ) := , Cl (θ) := . 2m k2m 2m+1 k2m+1 Xk=1 Xk=1 GENERALIZED RATIONAL ZETA SERIES FOR ζ(2n) AND ζ(2n + 1) 3 The Clausen function is widely studied and has many applications in mathematics and mathematical physics ([5], [6], [9], [12], [14], [15], [17]). We will also discuss the Dirichlet beta function ∞ (−1)n β(s)= , ℜ(s) > 0. (2n + 1)s n=0 X When s = 2, ∞ (−1)n β(2) = G = (2n + 1)2 n=0 X is known as Catalan’s constant. Using this and the Riemann zeta function, we find (4m − 1)ζ(2m + 1) (5) Cl (π)=0, Cl (π)= − , 2m 2m+1 4m and (4m − 1)ζ(2m + 1) (6) Cl (π/2) = β(2m), Cl (π/2) = − . 2m 2m+1 24m+1 Also, from (4) we can see d d (7) Cl (θ) = Cl (θ), Cl (θ)= − Cl (θ), dθ 2m 2m−1 dθ 2m+1 2m θ θ (8) Cl2m(x) dx = ζ(2m + 1) − Cl2m+1(θ), Cl2m−1(x) dx = Cl2m(θ). Z0 Z0 Using (4) and (7), we find θ (9) Cl (θ)= − log 2 sin , |θ| < 2π. 1 2 Writing others out, z Cl3(z)= ζ(3) − Cl2(t) dt, Z0 z z x z Cl4(z)= Cl3(x) dx = zζ(3) − Cl2(t) dt dx = zζ(3) − (z − t) Cl2(t) dt, Z0 Z0 Z0 Z0 4 DEREK ORR z 1 1 z Cl (z) = ζ(5) − Cl (x) dx = ζ(5) − z2ζ(3) + (z − t)2 Cl (t) dt, 5 4 2 2 2 Z0 Z0 and by induction, for m ≥ 3, − ⌊ m 1 ⌋ 2 k m−2k−1 m−1 (−1) z (10) Cl (z)=(−1)⌊ 2 ⌋ ζ(2k + 1) m (m − 2k − 1)! k=1 X − m 1 z (−1)⌊ 2 ⌋ + (z − t)m−3 Cl (t) dt. (m − 3)! 2 Z0 In the latter sections of this paper, we will focus on the polygamma function. Its definition is given by dn+1 ψ(n)(z) := log Γ(z), n ∈ N . dzn+1 0 This paper will discuss when n = 0. For n = 0, ψ(0)(z) = ψ(z) is called the digamma function. It has been shown (see [2]) that there is a closed form of z xnψ(x) dx in terms of sums involving Harmonic numbers, Bernoulli numbers 0 andZ Bernoulli polynomials. There are also definitions for negative order polygamma functions, called negapolygammas, given by ψ(−1)(z) = log Γ(z), z ψ(−2)(z)= log Γ(x) dx, Z0 z x z z z ψ(−3)(z)= log Γ(t) dtdx = log Γ(t) dxdt = (z − t) log Γ(t) dt, Z0 Z0 Z0 Zt Z0 and by induction, for n ≥ 2, 1 z (11) ψ(−n)(z)= (z − t)n−2 log Γ(t) dt. (n − 2)! Z0 Note the Taylor series for log Γ(z) is ∞ (−1)kζ(k) (12) log Γ(z)= − log z − γz + zk, |z| < 1, k Xk=2 where γ is the Euler-Mascheroni constant. The last function to introduce is the Hurwitz zeta function, defined by GENERALIZED RATIONAL ZETA SERIES FOR ζ(2n) AND ζ(2n + 1) 5 ∞ 1 ζ(s, a)= , a ∈ R\ − N, ℜ(s) > 1. (k + a)s Xk=0 The negapolygammas are related to the derivative of the Hurwitz zeta function with respect to the first variable (see [2]). πz p 1.1. Organization of the Paper. We begin by investigating 0 x cot(x) dx for |z| < 1, which is studied in [15]. We then compute this same integral using the power series for the cotangent and obtain a rational zeta series rRepresentation with ζ(2n) on the numerator for z = 1/2 and z = 1/4. Afterwords, we work on a generalized rational zeta series with ζ(2n) on the numerator and an arbitrary number of monomials on the denominator using the generalized Clausen function Clm(z). As a special case, we immediately prove a conjecture given in 2012 by F.M.S. Lima. After doing this, we go back to the cotangent function and discover a separate class of generalized ζ(2n) series. Lastly, we perform the same analysis but for the digamma function ψ(x) instead of cot(x) and instead of Clm(x), we use the negapolygammas ψ(−m)(x). Using the ζ(2n) sums from earlier, we extract the same rational zeta series representations with ζ(2n + 1) on the numerator for z =1/2 and z =1/4. Acknowledgements. I would like to thank Cezar Lupu, Tom Hales, and George Sparling for their interest in my paper as well as their advice which led to some improvements of the paper. 2. Rational ζ(2n) series with cot(x) Theorem 2.1. For p ∈ N and |z| < 1, (13) p k+3 p πz ⌊ 2 ⌋ 2 p p p!(−1) p!(−1) x cot(x) dx =(πz) Cl (2πz)+δ p p ζ(p+1), k k+1 ⌊ 2 ⌋, 2 p 0 (p − k)!(2πz) 2 Z Xk=0 where δj,k is the Kronecker delta function. Proof. Let f(z) be the left hand side of the equation and let g(z) be the right hand side. Note that f ′(z)= πp+1zp cot(πz). Using (7) and (9), p k+3 1 p!(−1)⌊ 2 ⌋(p − k)(2π)p−kzp−k−1 (πz)p2π cos(πz) g′(z)= Cl (2πz)+ 2p (p − k)! k+1 2 sin(πz) k=0 X p k+3 1 p!(−1)⌊ 2 ⌋(2πz)p−k + (−1)k+12π Cl (2πz) 2p (p − k)! k Xk=1 n o 6 DEREK ORR p p! Clk(2πz) k+2 k+3 = πp+1zp cot(πz)+(πz)p (−1)⌊ 2 ⌋ +(−1)⌊ 2 ⌋(−1)k . (p − k)!zk+1(2π)k−1 Xk=1 So indeed, g′(z)= πp+1zp cot(πz)= f ′(z). Clearly, f(0) = 0. For g(z), note that all terms in the sum are zero except when k = p. So we have p 2 1 ⌊ p+3 ⌋ p!(−1) g(0) = p!(−1) 2 Clp+1(0) + δ p p ζ(p + 1). 2p ⌊ 2 ⌋, 2 2p From (4), Cl (0) = δ p p ζ(p + 1).