Background Main results Extensions Generalized Rational Zeta Series with Two Parameters Derek Orr, University of Pittsburgh Algebra Research Seminar - University of Pittsburgh February 9, 2017 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters P That is, they are of the form qnζ(n; m) where qn 2 Q and ζ(n; m) is the Hurwitz zeta function, dened by 1 X 1 ζ(n; m) = : (m + k)n k=0 It can be shown that any real number can be written as a rational zeta series. Examples: 1 1 X X ζ(n) − 1 [ζ(n) − 1] = 1; = 1 − γ n n=2 n=2 n X 1 where γ = lim − log(n) is the Euler-Mascheroni n!1 1 k constant. k= Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function A rational zeta series is a series consisting of rational numbers and either the Hurwitz zeta function or the Riemann zeta function. Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters It can be shown that any real number can be written as a rational zeta series. Examples: 1 1 X X ζ(n) − 1 [ζ(n) − 1] = 1; = 1 − γ n n=2 n=2 n X 1 where γ = lim − log(n) is the Euler-Mascheroni n!1 1 k constant. k= Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function A rational zeta series is a series consisting of rational numbers and either the Hurwitz zeta function or the Riemann zeta function. P That is, they are of the form qnζ(n; m) where qn 2 Q and ζ(n; m) is the Hurwitz zeta function, dened by 1 X 1 ζ(n; m) = : (m + k)n k=0 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Examples: 1 1 X X ζ(n) − 1 [ζ(n) − 1] = 1; = 1 − γ n n=2 n=2 n X 1 where γ = lim − log(n) is the Euler-Mascheroni n!1 1 k constant. k= Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function A rational zeta series is a series consisting of rational numbers and either the Hurwitz zeta function or the Riemann zeta function. P That is, they are of the form qnζ(n; m) where qn 2 Q and ζ(n; m) is the Hurwitz zeta function, dened by 1 X 1 ζ(n; m) = : (m + k)n k=0 It can be shown that any real number can be written as a rational zeta series. Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function A rational zeta series is a series consisting of rational numbers and either the Hurwitz zeta function or the Riemann zeta function. P That is, they are of the form qnζ(n; m) where qn 2 Q and ζ(n; m) is the Hurwitz zeta function, dened by 1 X 1 ζ(n; m) = : (m + k)n k=0 It can be shown that any real number can be written as a rational zeta series. Examples: 1 1 X X ζ(n) − 1 [ζ(n) − 1] = 1; = 1 − γ n n=2 n=2 n X 1 where γ = lim − log(n) is the Euler-Mascheroni n!1 1 k constant. k= Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters In 1734, Leonhard Euler showed ζ(2) = π2=6: More generally, 1 k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = ; k 2 0; n2k 2(2k)! N n=1 where 1 z X Bn = zn; jzj < 2π: ez − 1 n! n=0 Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Denition. 1 X 1 ζ(s) = ; <(s) > 1; (1) ns n=1 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters More generally, 1 k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = ; k 2 0; n2k 2(2k)! N n=1 where 1 z X Bn = zn; jzj < 2π: ez − 1 n! n=0 Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Denition. 1 X 1 ζ(s) = ; <(s) > 1; (1) ns n=1 In 1734, Leonhard Euler showed ζ(2) = π2=6: Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Denition. 1 X 1 ζ(s) = ; <(s) > 1; (1) ns n=1 In 1734, Leonhard Euler showed ζ(2) = π2=6: More generally, 1 k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = ; k 2 0; n2k 2(2k)! N n=1 where 1 z X Bn = zn; jzj < 2π: ez − 1 n! n=0 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Denition. 1 X (−1)n β(s) = ; <(s) > 0: (3) (2n + 1)s n=0 Note that β(2) = G is known as Catalan's constant. Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function One of many appearances: 1 n 2n 1 X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− ; jxj < π: (2n)! π2n n=0 n=0 (2) Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Note that β(2) = G is known as Catalan's constant. Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function One of many appearances: 1 n 2n 1 X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− ; jxj < π: (2n)! π2n n=0 n=0 (2) Denition. 1 X (−1)n β(s) = ; <(s) > 0: (3) (2n + 1)s n=0 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function One of many appearances: 1 n 2n 1 X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− ; jxj < π: (2n)! π2n n=0 n=0 (2) Denition. 1 X (−1)n β(s) = ; <(s) > 0: (3) (2n + 1)s n=0 Note that β(2) = G is known as Catalan's constant. Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Its Taylor series is given by 1 Cl2(θ) X ζ(2n) θ 2n = 1 − log jθj + ; jθj < 2π: (5) θ n(2n + 1) 2π n=1 Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function The Clausen function (see [3], [4], [7], [9], [11], [12], [14]) or Clausen's integral is dened as 1 sin Z θ Cl X (kθ) log 2 sin φ (4) 2(θ) := 2 = − dφ. k 0 2 k=1 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function The Clausen function (see [3], [4], [7], [9], [11], [12], [14]) or Clausen's integral is dened as 1 sin Z θ Cl X (kθ) log 2 sin φ (4) 2(θ) := 2 = − dφ. k 0 2 k=1 Its Taylor series is given by 1 Cl2(θ) X ζ(2n) θ 2n = 1 − log jθj + ; jθj < 2π: (5) θ n(2n + 1) 2π n=1 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Using (1) and (3), we have (4m − 1)ζ(2m + 1) Cl2 (π) = 0; Cl2 1(π) = − ; (7) m m+ 4m and (4m − 1)ζ(2m + 1) Cl2 (π=2) = β(2m); Cl2 1(π=2) = − : (8) m m+ 24m+1 Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Higher order Clausen-type functions: 1 1 X sin(kθ) X cos(kθ) Cl2 (θ) := ; Cl2 1(θ) := : (6) m k2m m+ k2m+1 k=1 k=1 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Higher order Clausen-type functions: 1 1 X sin(kθ) X cos(kθ) Cl2 (θ) := ; Cl2 1(θ) := : (6) m k2m m+ k2m+1 k=1 k=1 Using (1) and (3), we have (4m − 1)ζ(2m + 1) Cl2 (π) = 0; Cl2 1(π) = − ; (7) m m+ 4m and (4m − 1)ζ(2m + 1) Cl2 (π=2) = β(2m); Cl2 1(π=2) = − : (8) m m+ 24m+1 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Z θ Z θ Cl2m(x) dx = ζ(2m+1)−Cl2m+1(θ); Cl2m−1(x) dx = Cl2m(θ): 0 0 (10) Note Cl log 2 sin θ 2 (11) 1(θ) = − 2 ; jθj < π: Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function We can also see d d Cl2 (θ) = Cl2 1(θ); Cl2 1(θ) = − Cl2 (θ); (9) dθ m m− dθ m+ m Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Note Cl log 2 sin θ 2 (11) 1(θ) = − 2 ; jθj < π: Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function We can also see d d Cl2 (θ) = Cl2 1(θ); Cl2 1(θ) = − Cl2 (θ); (9) dθ m m− dθ m+ m Z θ Z θ Cl2m(x) dx = ζ(2m+1)−Cl2m+1(θ); Cl2m−1(x) dx = Cl2m(θ): 0 0 (10) Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function We can also see d d Cl2 (θ) = Cl2 1(θ); Cl2 1(θ) = − Cl2 (θ); (9) dθ m m− dθ m+ m Z θ Z θ Cl2m(x) dx = ζ(2m+1)−Cl2m+1(θ); Cl2m−1(x) dx = Cl2m(θ): 0 0 (10) Note Cl log 2 sin θ 2 (11) 1(θ) = − 2 ; jθj < π: Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Z z Cl3(z) = ζ(3) − Cl2(t) dt; 0 Z z Z z Cl4(z) = Cl3(x) dx = zζ(3) − (z − t) Cl2(t) dt; 0 0 By induction, for m ≥ 3, b m−1 c 2 k m−2k−1 m−1 X (−1) z Cl (z) = (−1)b 2 c ζ(2k + 1) m (m − 2k − 1)! k=1 ! Z z (z − t)m−3 + Cl2(t) dt : (12) 0 (m − 3)! Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Using these integrals, Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Z z Z z Cl4(z) = Cl3(x) dx = zζ(3) − (z − t) Cl2(t) dt; 0 0 By induction, for m ≥ 3, b m−1 c 2 k m−2k−1 m−1 X (−1) z Cl (z) = (−1)b 2 c ζ(2k + 1) m (m − 2k − 1)! k=1 ! Z z (z − t)m−3 + Cl2(t) dt : (12) 0 (m − 3)! Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Using these integrals, Z z Cl3(z) = ζ(3) − Cl2(t) dt; 0 Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters By induction, for m ≥ 3, b m−1 c 2 k m−2k−1 m−1 X (−1) z Cl (z) = (−1)b 2 c ζ(2k + 1) m (m − 2k − 1)! k=1 ! Z z (z − t)m−3 + Cl2(t) dt : (12) 0 (m − 3)! Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function Using these integrals, Z z Cl3(z) = ζ(3)