Generalized Rational Zeta Series with Two Parameters

Background Main results Extensions

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 ∈ Q and ζ(n, m) is the Hurwitz zeta function, dened by

∞ 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:

∞ ∞ 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 k constant. k=

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:

∞ ∞ 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 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 ∈ Q and ζ(n, m) is the Hurwitz zeta function, dened by

∞ X 1 ζ(n, m) = . (m + k)n k=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Examples:

∞ ∞ 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 k constant. k=

∞ 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 ∈ Q and ζ(n, m) is the Hurwitz zeta function, dened by

∞ 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:

∞ ∞ 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 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,

∞ k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = , k ∈ 0, n2k 2(2k)! N n=1 where ∞ z X Bn = zn, |z| < 2π. ez − 1 n! n=0

Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function

Denition.

∞ X 1 ζ(s) = , <(s) > 1, (1) ns n=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters More generally,

∞ k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = , k ∈ 0, n2k 2(2k)! N n=1 where ∞ z X Bn = zn, |z| < 2π. ez − 1 n! n=0

Background Rational Zeta Series Main results Zeta and Beta function Extensions Clausen function

Denition.

∞ 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.

∞ X 1 ζ(s) = , <(s) > 1, (1) ns n=1

In 1734, Leonhard Euler showed ζ(2) = π2/6. More generally,

∞ k+1 2k X 1 (−1) B2k (2π) ζ(2k) = = , k ∈ 0, n2k 2(2k)! N n=1 where ∞ z X Bn = zn, |z| < 2π. ez − 1 n! n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Denition.

∞ 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:

∞ n 2n ∞ X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− , |x| < 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:

∞ n 2n ∞ X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− , |x| < π. (2n)! π2n n=0 n=0 (2) Denition.

∞ 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:

∞ n 2n ∞ X (−1) 2 B2n 2 1 X ζ(2n) 2 1 cot(x) = x n− = −2 x n− , |x| < π. (2n)! π2n n=0 n=0 (2) Denition.

∞ 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

∞ Cl2(θ) X ζ(2n) θ 2n = 1 − log |θ| + , |θ| < 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

∞ 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

∞ 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

∞ Cl2(θ) X ζ(2n) θ 2n = 1 − log |θ| + , |θ| < 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:

∞ ∞ 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:

∞ ∞ 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 , |θ| < 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 , |θ| < 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 , |θ| < 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) − Cl2(t) dt, 0 Z z Z z Cl4(z) = Cl3(x) dx = zζ(3) − (z − t) Cl2(t) dt, 0 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

Using these integrals,

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)!

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters p k+3 Z πz 1 b 2 c p p X p!(− ) cot Cl 1 2 x (x) dx = (πz) k k+ ( πz) 0 (p − k)!(2πz) k=0 p p!(−1) 2 + δ p p ζ(p + 1), p ∈ , |z| < 1, (13) b 2 c, 2 2p N

where δj,k is the Kronecker delta function. Proof. Denoting the left hand side as f (z) and the right hand side as g(z), one can show f (0) = g(0) and f 0(z) = g 0(z), so thus f (z) = g(z).

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Theorem 1.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Proof. Denoting the left hand side as f (z) and the right hand side as g(z), one can show f (0) = g(0) and f 0(z) = g 0(z), so thus f (z) = g(z).

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Theorem 1.

p k+3 Z πz 1 b 2 c p p X p!(− ) cot Cl 1 2 x (x) dx = (πz) k k+ ( πz) 0 (p − k)!(2πz) k=0 p p!(−1) 2 + δ p p ζ(p + 1), p ∈ , |z| < 1, (13) b 2 c, 2 2p N where δj,k is the Kronecker delta function.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Theorem 1.

p k+3 Z πz 1 b 2 c p p X p!(− ) cot Cl 1 2 x (x) dx = (πz) k k+ ( πz) 0 (p − k)!(2πz) k=0 p p!(−1) 2 + δ p p ζ(p + 1), p ∈ , |z| < 1, (13) b 2 c, 2 2p N where δj,k is the Kronecker delta function. Proof. Denoting the left hand side as f (z) and the right hand side as g(z), one can show f (0) = g(0) and f 0(z) = g 0(z), so thus f (z) = g(z).

Theorem 1.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters and with (13),

∞ 2 p k+3 X ζ(2n)z n X p!(−1)b 2 c − 2 = Cl 1(2πz) 2n + p (p − k)!(2πz)k k+ n=0 k=0 p p!(−1) 2 + δ p p ζ(p + 1). (14) b 2 c, 2 (2πz)p

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Using (2) we can also say

Z πz Z πz ∞ 2 p cot 2 X ζ( n) 2n−1+p x (x) dx = − 2n x dx 0 0 π n=0 ∞ 2 X ζ(2n)z n = −2(πz)p , 2n + p n=0

Using (2) we can also say

Z πz Z πz ∞ 2 p cot 2 X ζ( n) 2n−1+p x (x) dx = − 2n x dx 0 0 π n=0 ∞ 2 X ζ(2n)z n = −2(πz)p , 2n + p n=0 and with (13),

∞ 2 p k+3 X ζ(2n)z n X p!(−1)b 2 c − 2 = Cl 1(2πz) 2n + p (p − k)!(2πz)k k+ n=0 k=0 p p!(−1) 2 + δ p p ζ(p + 1). (14) b 2 c, 2 (2πz)p

For z = 1/2 and z = 1/4,

p ∞ b 2 c X ζ(2n) X p!(−1)k (4k − 1)ζ(2k + 1) − 2 = log 2 + (2n + p)4n (p − 2k)!(2π)2k n=0 k=1 p p!(−1) 2 ζ(p + 1) + δ p p , (15) b 2 c, 2 πp

p ∞ b 2 c X ζ(2n) 1 1 X p!(−1)k (4k − 1)ζ(2k + 1) −2 = log 2+ (2n + p)16n 2 2 (p − 2k)!(2π)2k n=0 k=1 p+1 b 2 c p π X p!(−4)k β(2k) p!(−1) 2 2pζ(p + 1) − + δ p p . (16) 2 (p + 1 − 2k)!π2k b 2 c, 2 πp k=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters For p = 1,

∞ X ζ(2n) 1 = − log 2, (2n + 1)4n 2 n=0 and ∞ X ζ(2n) 1 2G 2 = − log 2 − . (2n + 1)16n 2 π n=0 Subtracting amd rearranging, we have a nice formula for G:

∞ π X ζ(2n) 2 G = 1 − 2 (2n + 1)4n 4n n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Remark.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Subtracting amd rearranging, we have a nice formula for G:

∞ π X ζ(2n) 2 G = 1 − 2 (2n + 1)4n 4n n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Remark. For p = 1,

∞ X ζ(2n) 1 = − log 2, (2n + 1)4n 2 n=0 and ∞ X ζ(2n) 1 2G 2 = − log 2 − . (2n + 1)16n 2 π n=0

Remark. For p = 1,

∞ X ζ(2n) 1 = − log 2, (2n + 1)4n 2 n=0 and ∞ X ζ(2n) 1 2G 2 = − log 2 − . (2n + 1)16n 2 π n=0 Subtracting amd rearranging, we have a nice formula for G:

∞ π X ζ(2n) 2 G = 1 − 2 (2n + 1)4n 4n n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 1 9ζ(3) = − log 2 + , (2n + 3)4n 2 4π2 n=0

∞ X ζ(2n) 9ζ(3) 93ζ(5) = − log 2 + − , (n + 2)4n π2 2π4 n=0 ∞ X ζ(2n) 1 35ζ(3) 4G = − log 2 + − , (n + 1)16n 2 4π2 π n=0 ∞ X ζ(2n) 1 9ζ(3) 3G 24β(4) = − log 2 + − + . (2n + 3)16n 4 8π2 π π3 n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 9ζ(3) 93ζ(5) = − log 2 + − , (n + 2)4n π2 2π4 n=0 ∞ X ζ(2n) 1 35ζ(3) 4G = − log 2 + − , (n + 1)16n 2 4π2 π n=0 ∞ X ζ(2n) 1 9ζ(3) 3G 24β(4) = − log 2 + − + . (2n + 3)16n 4 8π2 π π3 n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 1 9ζ(3) = − log 2 + , (2n + 3)4n 2 4π2 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 1 35ζ(3) 4G = − log 2 + − , (n + 1)16n 2 4π2 π n=0 ∞ X ζ(2n) 1 9ζ(3) 3G 24β(4) = − log 2 + − + . (2n + 3)16n 4 8π2 π π3 n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 1 9ζ(3) = − log 2 + , (2n + 3)4n 2 4π2 n=0

∞ X ζ(2n) 9ζ(3) 93ζ(5) = − log 2 + − , (n + 2)4n π2 2π4 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 1 9ζ(3) 3G 24β(4) = − log 2 + − + . (2n + 3)16n 4 8π2 π π3 n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 1 9ζ(3) = − log 2 + , (2n + 3)4n 2 4π2 n=0

∞ X ζ(2n) 9ζ(3) 93ζ(5) = − log 2 + − , (n + 2)4n π2 2π4 n=0 ∞ X ζ(2n) 1 35ζ(3) 4G = − log 2 + − , (n + 1)16n 2 4π2 π n=0

Other sums

∞ X ζ(2n) 1 9ζ(3) = − log 2 + , (2n + 3)4n 2 4π2 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Proof. Same as the proof for Theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Theorem 2.

Z 2πz p x Clm(x) dx = 0 m+p+1 1 m k X (2πz)p+m+ −k p!(−1)b 2 c(−1)b 2 c − Cl (2πz) (p + m + 1 − k)! k k=m+1 b m c p+m + δ p+m p+m (−1) 2 p!(−1) 2 ζ(p + m + 1), p ∈ 0, m ∈ . b 2 c, 2 N N (17)

Theorem 2.

Proof. Same as the proof for Theorem 1.

Theorem 2.

Proof. Same as the proof for Theorem 1.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters m−1 b 2 c m−1 2 m−1 X (−1)b 2 c+k (2πz)m+p− k (−1)b 2 c = ζ(2k + 1) + ∗ (m − 1 − 2k)!(m + p − 2k) (m − 3)! k=1 Z 2πz Z x ∞ 2 2n+1 p m−3 log X ζ( n)t x (x −t) t −t t + 2n dt dx 0 0 n(2n + 1)(2π) n=1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Using (5) and (12),

Z 2πz Z 2πz p b m−1 c p x Clm(x) dx = (−1) 2 x ∗ 0 0 m−1 b 2 c 2 1 3 X (−1)k xm− k− Z x (x − t)m− ζ(2k + 1) + Cl2(t) dt dx (m − 2k − 1)! 0 (m − 3)! k=1

Using (5) and (12),

Z 2πz Z 2πz p b m−1 c p x Clm(x) dx = (−1) 2 x ∗ 0 0 m−1 b 2 c 2 1 3 X (−1)k xm− k− Z x (x − t)m− ζ(2k + 1) + Cl2(t) dt dx (m − 2k − 1)! 0 (m − 3)! k=1

m−1 b 2 c m−1 2 m−1 X (−1)b 2 c+k (2πz)m+p− k (−1)b 2 c = ζ(2k + 1) + ∗ (m − 1 − 2k)!(m + p − 2k) (m − 3)! k=1 Z 2πz Z x ∞ 2 2n+1 p m−3 log X ζ( n)t x (x −t) t −t t + 2n dt dx 0 0 n(2n + 1)(2π) n=1

Evaluating the rst integral,

m−1 2 b 2 c m−1 2 Z πz 1 b 2 c+k 2 m+p− k p X (− ) ( πz) x Clm(x) dx = ζ(2k+1) 0 (m − 1 − 2k)!(m + p − 2k) k=1 1 b m− c 2πz m−1 (−1) 2 Z x (H 1 − log x) + xp m− (m − 3)! 0 (m − 1)(m − 2) ∞ 2 1 X 2ζ(2n)Γ(m − 2)xm+ n− + dx, 2n(2n + 1)(2n + 2) ... (2n + m − 1)(2π)2n n=1

Hn is the n-th harmonic number, H0 = 0.

And the second integral...

2 m−1 Z πz 2 p+m 1 b 2 c log 2 p ( πz) (− ) Hm−1 − ( πz) x Clm(x) dx = 0 (m − 1)! (p + m) m−1 b 2 c 1 X (m − 1)!(−1)k ζ(2k + 1) + + (p + m)2 (m − 1 − 2k)!(p + m − 2k)(2πz)2k k=1 ∞ 2 X (m − 1)!ζ(2n)z n + . (18) n(2n + 1) ... (2n + m − 1)(2n + p + m) n=1

Using (17) and rearranging, we nd

∞ 2 X ζ(2n)z n = (−1)mp!∗ n(2n + 1) ... (2n + m − 1)(2n + m + p) n=1 m+p b k+1 c p+m X (−1) 2 Clk+1(2πz) (−1) 2 − δ p+m p+m ζ(p + m + 1) (p + m − k)!(2πz)k b 2 c, 2 (2πz)p+m k=m b m−1 c 2 k log(2πz) − Hm−1 X (−1) ζ(2k + 1) + − (m − 1)!(p + m) (m − 1 − 2k)!(m + p − 2k)(2πz)2k k=1 1 − . (19) (m − 1)!(p + m)2

For z = 1/2,

∞ X ζ(2n) 1 = (−1)m+ p!∗ n(2n + 1) ... (2n + m − 1)(2n + m + p)4n n=1 p+m b 2 c p+m 1 k 4k 1 2 1 1 2 X (− ) ( − )ζ( k + ) (− ) 1 2 +δ p+m p+m ζ(p+m+ ) (p + m − 2k)!(2π) k b 2 c, 2 πp+m m+1 k=b 2 c b m−1 c 2 k log π − Hm−1 X (−1) ζ(2k + 1) + − (m − 1)!(p + m) (m − 1 − 2k)!(p + m − 2k)π2k k=1 1 − , (20) (m − 1)!(p + m)2

∞ X ζ(2n) (−1)mp! = ∗ n(2n + 1) ... (2n + m − 1)(2n + m + p)16n 2 n=1 p+m+1 p+m b 2 c b 2 c X π(−4)k β(2k) X (−1)k (4k − 1)ζ(2k + 1) − (p + m + 1 − 2k)!π2k (p + m − 2k)!(2π)2k m+2 m+1 k=b 2 c k=b 2 c p+m p+m+1 (−1) 2 2 log(π/2) − Hm−1 − δ p+m p+m ζ(p + m + 1) + b 2 c, 2 πp+m (m − 1)!(p + m) m−1 b 2 c X (−4)k ζ(2k + 1) 1 − − . (m − 1 − 2k)!(p + m − 2k)π2k (m − 1)!(p + m)2 k=1 (21)

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Some nice sums (m=1,p=0):

∞ X ζ(2n) = log π − 1, n(2n + 1)4n n=1 and ∞ X ζ(2n) 2G π = − 1 + log , n(2n + 1)16n π 2 n=1 both of which are famous series ([13], [11]).

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Remark 1.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) = log π − 1, n(2n + 1)4n n=1 and ∞ X ζ(2n) 2G π = − 1 + log , n(2n + 1)16n π 2 n=1 both of which are famous series ([13], [11]).

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Remark 1. Some nice sums (m=1,p=0):

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters and ∞ X ζ(2n) 2G π = − 1 + log , n(2n + 1)16n π 2 n=1 both of which are famous series ([13], [11]).

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Remark 1. Some nice sums (m=1,p=0):

∞ X ζ(2n) = log π − 1, n(2n + 1)4n n=1

Remark 1. Some nice sums (m=1,p=0):

∞ X ζ(2n) = log π − 1, n(2n + 1)4n n=1 and ∞ X ζ(2n) 2G π = − 1 + log , n(2n + 1)16n π 2 n=1 both of which are famous series ([13], [11]).

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters The series

∞ X ζ(2n) n(2n + 1)(2n + 3)(n + 2)16n n=1 converges very closely to 0.0023 (0.002299999499895073...). Using (21) a few times for dierent m and p, one can show that

4π4 19 π 96 ζ(5) = − log + β(4) 1581 12 2 π3 ∞ X ζ(2n) + 6 , n(2n + 1)(2n + 3)(n + 2)16n n=1

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Remark 2.

Remark 2. The series

∞ X ζ(2n) n(2n + 1)(2n + 3)(n + 2)16n n=1 converges very closely to 0.0023 (0.002299999499895073...). Using (21) a few times for dierent m and p, one can show that

4π4 19 π 96 ζ(5) = − log + β(4) 1581 12 2 π3 ∞ X ζ(2n) + 6 , n(2n + 1)(2n + 3)(n + 2)16n n=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters 4π4 19 π 96 ζ(5) ≈ −log + β(4)+6∗0.0023 = 1.03692775588... 1581 12 2 π3

which has a small error of 7.395 ∗ 10−10.

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And so

4π4 19 π 96 ζ(5) ≈ −log + β(4)+6∗0.0023 = 1.03692775588... 1581 12 2 π3 which has a small error of 7.395 ∗ 10−10.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 2ζ(3) 31ζ(5) 1 7 = + + log π − n(2n + 1)(n + 1)(n + 2)4n π2 4π4 2 8 n=1

∞ X ζ(2n) ζ(3) ζ(5) 1 137 = − + log π − n(2n + 1) ... (2n + 5)4n 6π2 π4 120 7200 n=1 ∞ X ζ(2n) 35ζ(3) π 3 = + log − n(2n + 1)(n + 1)16n 4π2 2 2 n=1 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 1 π 25 = − + log − n(2n + 1) ... (2n + 4)16n π2 32π4 24 2 288 n=1

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Other sums

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) ζ(3) ζ(5) 1 137 = − + log π − n(2n + 1) ... (2n + 5)4n 6π2 π4 120 7200 n=1 ∞ X ζ(2n) 35ζ(3) π 3 = + log − n(2n + 1)(n + 1)16n 4π2 2 2 n=1 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 1 π 25 = − + log − n(2n + 1) ... (2n + 4)16n π2 32π4 24 2 288 n=1

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Other sums

∞ X ζ(2n) 2ζ(3) 31ζ(5) 1 7 = + + log π − n(2n + 1)(n + 1)(n + 2)4n π2 4π4 2 8 n=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 35ζ(3) π 3 = + log − n(2n + 1)(n + 1)16n 4π2 2 2 n=1 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 1 π 25 = − + log − n(2n + 1) ... (2n + 4)16n π2 32π4 24 2 288 n=1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 2ζ(3) 31ζ(5) 1 7 = + + log π − n(2n + 1)(n + 1)(n + 2)4n π2 4π4 2 8 n=1

∞ X ζ(2n) ζ(3) ζ(5) 1 137 = − + log π − n(2n + 1) ... (2n + 5)4n 6π2 π4 120 7200 n=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 2ζ(3) 527ζ(5) 1 π 25 = − + log − n(2n + 1) ... (2n + 4)16n π2 32π4 24 2 288 n=1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 2ζ(3) 31ζ(5) 1 7 = + + log π − n(2n + 1)(n + 1)(n + 2)4n π2 4π4 2 8 n=1

∞ X ζ(2n) ζ(3) ζ(5) 1 137 = − + log π − n(2n + 1) ... (2n + 5)4n 6π2 π4 120 7200 n=1 ∞ X ζ(2n) 35ζ(3) π 3 = + log − n(2n + 1)(n + 1)16n 4π2 2 2 n=1

Other sums

∞ X ζ(2n) 2ζ(3) 31ζ(5) 1 7 = + + log π − n(2n + 1)(n + 1)(n + 2)4n π2 4π4 2 8 n=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Can we get a similar family of generalized zeta series, but using

cot(x) instead of Clm(x)? Answer: yes! We will investigate Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0 and now we can use theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Answer: yes! We will investigate

Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0 and now we can use theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Can we get a similar family of generalized zeta series, but using cot(x) instead of Clm(x)?

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters We will investigate

Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0 and now we can use theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Can we get a similar family of generalized zeta series, but using cot(x) instead of Clm(x)? Answer: yes!

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0 and now we can use theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Can we get a similar family of generalized zeta series, but using cot(x) instead of Clm(x)? Answer: yes! We will investigate Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters and now we can use theorem 1.

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Can we get a similar family of generalized zeta series, but using cot(x) instead of Clm(x)? Answer: yes! We will investigate Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0

Can we get a similar family of generalized zeta series, but using cot(x) instead of Clm(x)? Answer: yes! We will investigate Z πz Z x p m x (x − t) t cot(t) dt dx, m ∈ N0, p ∈ N0. 0 0

Using the binomial theorem,

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m Z πz Z x X m 1 (−1)j xp+m−j tj+ cot(t) dt dx j 0 0 j=0 and now we can use theorem 1.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m j+1 k+3 1 j 1 1 b 2 c Z πz X X m (− ) (j + )!(− ) p+m+1−k Cl 1 2 k x k+ ( x) dx j (j + 1 − k)!2 0 j=0 k=0 m j+1 1 j 1 1 2 Z πz X (− ) (j + )!(− ) m p+m−j 1 1 2 + δb j+ c, j+ j+1 ζ(j + ) x dx 2 2 2 j 0 j=0 m j+1 k+3 2 1 j 1 1 b 2 c Z πz X X m (− ) (j + )!(− ) p+m+1−k Cl 1 = p+m+2 u k+ (u) du j (j + 1 − k)!2 0 j=0 k=0 m+1 b 2 c 2 2 X (2k)(−1)k m!(πz)p+m+ − k − ζ(2k + 1) 22k (m + 1 − 2k)!(p + m + 2 − 2k) k=1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Z πz Z x xp(x − t)mt cot(t) dt dx = 0 0 m j+1 k+3 1 j 1 1 b 2 c Z πz X X m (− ) (j + )!(− ) p+m+1−k Cl 1 2 k x k+ ( x) dx j (j + 1 − k)!2 0 j=0 k=0 m j+1 1 j 1 1 2 Z πz X (− ) (j + )!(− ) m p+m−j 1 1 2 + δb j+ c, j+ j+1 ζ(j + ) x dx 2 2 2 j 0 j=0 m j+1 k+3 2 1 j 1 1 b 2 c Z πz X X m (− ) (j + )!(− ) p+m+1−k Cl 1 = p+m+2 u k+ (u) du j (j + 1 − k)!2 0 j=0 k=0 m+1 b 2 c 2 2 X (2k)(−1)k m!(πz)p+m+ − k − ζ(2k + 1) 22k (m + 1 − 2k)!(p + m + 2 − 2k) k=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters m Z πz Z x 1 b 2 c p m cot (− ) m! x (x − t) t (t) dt dx = − p+m+2 ∗ 0 0 2 2 m+1 2 Z πz 1 1 b 2 c Z πz p+1 (m + )!(− ) p Cl 1 Cl 2 u m+ (u) du+ p+m+2 u m+ (u) du 0 2 0 m+1 b 2 c 1 2 2 X (2k)(−1)k+ m!(πz)p+m+ − k ζ(2k + 1) + . 22k (m + 1 − 2k)!(p + m + 2 − 2k) k=1

Now we can integrate the rst integral by parts and use theorem 2...

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Evaluating the sums on j and k,

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Now we can integrate the rst integral by parts and use theorem 2...

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Evaluating the sums on j and k,

m Z πz Z x 1 b 2 c p m cot (− ) m! x (x − t) t (t) dt dx = − p+m+2 ∗ 0 0 2 2 m+1 2 Z πz 1 1 b 2 c Z πz p+1 (m + )!(− ) p Cl 1 Cl 2 u m+ (u) du+ p+m+2 u m+ (u) du 0 2 0 m+1 b 2 c 1 2 2 X (2k)(−1)k+ m!(πz)p+m+ − k ζ(2k + 1) + . 22k (m + 1 − 2k)!(p + m + 2 − 2k) k=1

Evaluating the sums on j and k,

Now we can integrate the rst integral by parts and use theorem 2...

Rearranging and simplifying, we will arrive at

Z πz Z x xp(x − t)mt cot(t) dt dx = (p + m + 2)(−1)mp!m!∗ 0 0 3 p+m+ b k c p+m+2 X (−1) 2 (πz) Clk (2πz) 2πz + δ p+m p+m ∗ (p + m + 3 − k)!(2πz)k b 2 c, 2 k=m+3 1 p+m b m+ c p+1 (−1) 2 ζ(p + m + 3) (−1) 2 m!(πz) Cl 2(2πz) − m+ 2m+p+2 2m+1

m+1 b 2 c 1 2 X (2k)(−1)k+ m!(πz)p+m+ ζ(2k + 1) + . (22) (2πz)2k (m + 1 − 2k)!(p + m + 2 − 2k) k=1

Another way to evaluate this double integral is to use the power series for cot(x). Doing so, we see

Z πz Z x xp(x − t)mt cot(t) dt dx 0 0 Z πz ∞ 2 Z x 2 p X ζ( n) m 2n = − x 2n (x − t) t dt dx 0 π 0 n=0 ∞ 2 2 1 Z πz 2 X ζ( n)m!Γ( n + ) 2n+m+p+1 = − 2n x dx π Γ(m + 2n + 2) 0 n=0 ∞ 2 2 X ζ(2n)m!(πz)m+p+ z n = −2 . (2n + 1) ... (2n + m + 1)(2n + m + p + 2) n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ 2 X ζ(2n)z n = (p + m + 2)∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2) n=0 p+m+3 k 1 b 2 c Cl 2 m+1 X πz(− ) k ( πz) (−1) p! + δ p+m p+m ∗ (p + m + 3 − k)!(2πz)k b 2 c, 2 k=m+3 p+m b m+1 c (−1) 2 ζ(p + m + 3) (−1) 2 Cl 2(2πz) + m+ 2(2πz)m+p+2 2(2πz)m+1 m+1 b 2 c X k(−1)k ζ(2k + 1) + . (23) (2πz)2k (m + 1 − 2k)!(m + p + 2 − 2k) k=1

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Putting this and (22) together, we nd

∞ 2 X ζ(2n)z n = (p + m + 2)∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2) n=0 p+m+3 k 1 b 2 c Cl 2 m+1 X πz(− ) k ( πz) (−1) p! + δ p+m p+m ∗ (p + m + 3 − k)!(2πz)k b 2 c, 2 k=m+3 p+m b m+1 c (−1) 2 ζ(p + m + 3) (−1) 2 Cl 2(2πz) + m+ 2(2πz)m+p+2 2(2πz)m+1 m+1 b 2 c X k(−1)k ζ(2k + 1) + . (23) (2πz)2k (m + 1 − 2k)!(m + p + 2 − 2k) k=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) p!(m + p + 2) = ∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2)4n 2 n=0 2 b p+m+ c 2 1 k 4k 1 2 1 1 m X (− ) ( − )ζ( k + ) (− ) 2 − δ p+m p+m ∗ (p + m + 2 − 2k)!(2π) k b 2 c, 2 m+3 k=b 2 c p+m ! m+1 1 (−1) 2 ζ(p + m + 3) (−1) 2 (2m+ − 1)ζ(m + 2) 2 − δ m+1 m+1 1 πm+p+ b 2 c, 2 2(2π)m+

m+1 b 2 c X k(−1)k ζ(2k + 1) + , (24) π2k (m + 1 − 2k)!(m + p + 2 − 2k) k=1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

For z = 1/2,

∞ X ζ(2n) p!(m + p + 2) = ∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2)4n 2 n=0 2 b p+m+ c 2 1 k 4k 1 2 1 1 m X (− ) ( − )ζ( k + ) (− ) 2 − δ p+m p+m ∗ (p + m + 2 − 2k)!(2π) k b 2 c, 2 m+3 k=b 2 c p+m ! m+1 1 (−1) 2 ζ(p + m + 3) (−1) 2 (2m+ − 1)ζ(m + 2) 2 − δ m+1 m+1 1 πm+p+ b 2 c, 2 2(2π)m+

m+1 b 2 c X k(−1)k ζ(2k + 1) + , (24) π2k (m + 1 − 2k)!(m + p + 2 − 2k) k=1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) p!(m + p + 2) = ∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2)16n 4 n=0 p+m+2 p+m+3 b 2 c b 2 c X (−1)m+k (4k − 1)ζ(2k + 1) X π(−1)m(−4)k β(2k) − (p + m + 2 − 2k)!(2π)2k (p + m + 3 − 2k)!π2k m+3 m+4 k=b 2 c k=b 2 c p+m 3 ! m+1 (−1) 2 (−1)m2p+m+ ζ(p + m + 3) (−1)b 2 c −δ p+m p+m 2 +δb m c, m 1 ∗ b 2 c, 2 πp+m+ 2 2 πm+

b m+1 c 2 4 k 2 1 2m 2 X k(− ) ζ( k + ) β(m+ )+ 2 −δ m+1 m+1 ∗ π k (m + 1 − 2k)!(m + p + 2 − 2k) b 2 c, 2 k=1 m+1 1 (−1) 2 (2m+ − 1)ζ(m + 2) . (25) 4(2π)m+1

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x) And for z = 1/4,

∞ X ζ(2n) p!(m + p + 2) = ∗ (2n + 1) ... (2n + m + 1)(2n + m + p + 2)16n 4 n=0 p+m+2 p+m+3 b 2 c b 2 c X (−1)m+k (4k − 1)ζ(2k + 1) X π(−1)m(−4)k β(2k) − (p + m + 2 − 2k)!(2π)2k (p + m + 3 − 2k)!π2k m+3 m+4 k=b 2 c k=b 2 c p+m 3 ! m+1 (−1) 2 (−1)m2p+m+ ζ(p + m + 3) (−1)b 2 c −δ p+m p+m 2 +δb m c, m 1 ∗ b 2 c, 2 πp+m+ 2 2 πm+

b m+1 c 2 4 k 2 1 2m 2 X k(− ) ζ( k + ) β(m+ )+ 2 −δ m+1 m+1 ∗ π k (m + 1 − 2k)!(m + p + 2 − 2k) b 2 c, 2 k=1 m+1 1 (−1) 2 (2m+ − 1)ζ(m + 2) . (25) 4(2π)m+1

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters For m = 0 and p = 0, we have

∞ X ζ(2n) 7ζ(3) = − , (2n + 1)(n + 1)4n 2π2 n=0 and ∞ X ζ(2n) 2G 35ζ(3) = − , (2n + 1)(n + 1)16n π 4π2 n=0 the rst of which was rediscovered by Ewell [8].

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Remark.

Remark. For m = 0 and p = 0, we have

∞ X ζ(2n) 7ζ(3) = − , (2n + 1)(n + 1)4n 2π2 n=0 and ∞ X ζ(2n) 2G 35ζ(3) = − , (2n + 1)(n + 1)16n π 4π2 n=0 the rst of which was rediscovered by Ewell [8].

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 5ζ(3) = − , (2n + 1)(2n + 2)(2n + 3)4n 8π2 n=0

∞ X ζ(2n) ζ(3) 49ζ(5) = − + , (2n + 1) ... (2n + 5)4n 6π2 32π4 n=0 ∞ X ζ(2n) 9ζ(3) G 12β(4) = − + − , (2n + 1)(2n + 3)16n 16π2 π π3 n=0 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 4β(4) = − + − . (2n + 1) ... (2n + 4)16n π2 16π4 π3 n=0

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Other sums

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) ζ(3) 49ζ(5) = − + , (2n + 1) ... (2n + 5)4n 6π2 32π4 n=0 ∞ X ζ(2n) 9ζ(3) G 12β(4) = − + − , (2n + 1)(2n + 3)16n 16π2 π π3 n=0 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 4β(4) = − + − . (2n + 1) ... (2n + 4)16n π2 16π4 π3 n=0

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Other sums

∞ X ζ(2n) 5ζ(3) = − , (2n + 1)(2n + 2)(2n + 3)4n 8π2 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 9ζ(3) G 12β(4) = − + − , (2n + 1)(2n + 3)16n 16π2 π π3 n=0 ∞ X ζ(2n) 2ζ(3) 527ζ(5) 4β(4) = − + − . (2n + 1) ... (2n + 4)16n π2 16π4 π3 n=0

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Other sums

∞ X ζ(2n) 5ζ(3) = − , (2n + 1)(2n + 2)(2n + 3)4n 8π2 n=0

∞ X ζ(2n) ζ(3) 49ζ(5) = − + , (2n + 1) ... (2n + 5)4n 6π2 32π4 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters ∞ X ζ(2n) 2ζ(3) 527ζ(5) 4β(4) = − + − . (2n + 1) ... (2n + 4)16n π2 16π4 π3 n=0

Background Zeta series using cot(x) Main results General zeta series using Clm(x) Extensions General zeta series using cot(x)

Other sums

∞ X ζ(2n) 5ζ(3) = − , (2n + 1)(2n + 2)(2n + 3)4n 8π2 n=0

∞ X ζ(2n) ζ(3) 49ζ(5) = − + , (2n + 1) ... (2n + 5)4n 6π2 32π4 n=0 ∞ X ζ(2n) 9ζ(3) G 12β(4) = − + − , (2n + 1)(2n + 3)16n 16π2 π π3 n=0

Other sums

∞ X ζ(2n) 5ζ(3) = − , (2n + 1)(2n + 2)(2n + 3)4n 8π2 n=0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Using the digamma function and negapolygammas, we can have ζ(2n + 1) on the numerator of these zeta series instead of ζ(2n).

d ψ(0)(z) = ψ(z) = log Γ(z), ψ(−1)(z) = log Γ(z), dz 1 Z z (−n) n−2 ψ (z) = (z − t) log Γ(t) dt, n ∈ N, n ≥ 2. (n − 2)! 0 The Taylor series for log Γ(z) is

∞ X (−1)k ζ(k) log Γ(z) = − log z − γz + zk , |z| < 1. k k=2 We can split the above series into odd k and even k and use the formulas for ζ(2n) to establish rational zeta series for ζ(2n + 1).

Background Main results Extensions

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters d ψ(0)(z) = ψ(z) = log Γ(z), ψ(−1)(z) = log Γ(z), dz 1 Z z (−n) n−2 ψ (z) = (z − t) log Γ(t) dt, n ∈ N, n ≥ 2. (n − 2)! 0 The Taylor series for log Γ(z) is

Background Main results Extensions

Using the digamma function and negapolygammas, we can have ζ(2n + 1) on the numerator of these zeta series instead of ζ(2n).

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters The Taylor series for log Γ(z) is

Background Main results Extensions

Using the digamma function and negapolygammas, we can have ζ(2n + 1) on the numerator of these zeta series instead of ζ(2n).

d ψ(0)(z) = ψ(z) = log Γ(z), ψ(−1)(z) = log Γ(z), dz 1 Z z (−n) n−2 ψ (z) = (z − t) log Γ(t) dt, n ∈ N, n ≥ 2. (n − 2)! 0

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Main results Extensions

Using the digamma function and negapolygammas, we can have ζ(2n + 1) on the numerator of these zeta series instead of ζ(2n).

d ψ(0)(z) = ψ(z) = log Γ(z), ψ(−1)(z) = log Γ(z), dz 1 Z z (−n) n−2 ψ (z) = (z − t) log Γ(t) dt, n ∈ N, n ≥ 2. (n − 2)! 0 The Taylor series for log Γ(z) is

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Main results Extensions ReferencesI

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, New York, NY:Dover, 1972.

T. M. Apostol, Another elementary proof of Euler's formula for ζ(2n), Amer. Math. Monthly, 80 (1973), 425431.

J. Choi, Some integral representations of the Clausen function Cl2(x) and the Catalan constant, East Asian Math. J. 32 (2016), 4346.

J. Choi, H. M. Srivastava, The Clausen function Cl2(x) and its related integrals, Thai J. Math. 12 (2014), 251264.

D. Cvijovic, Closed-form evaluation of some families of cotangent and cosecant integrals, Integral Transf. Spec. Func. 19 (2008), 147155.

D. Cvijovic, J. Klinowski, New rapidly convergent series representations for ζ(2n + 1), Proc. Amer. Math. Soc. 125 (1997), 12631271.

P. J. de Doeder, On the Clausen integral Cl2(θ) and a related integral, J. Comp. Appl. Math. 11 (1984), 325330.

J. A. Ewell, A new series representation for ζ(3), Amer. Math. Monthly 97 (1990), 219220.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Main results Extensions ReferencesII

C. C. Grosjean, Formulae concerning the computation of the Clausen integral Cl(θ), J. Comp. Appl. Math. 11 (1984), 331342.

F.M.S. Lima, An Euler-type formula for β(2n) and closed-form expressions for a class of zeta series, Integral Transf. Spec. Funct. (2011), DOI:10.1080/10652469.2011.622274

C. Lupu, D. Orr, Approximations for Apery's constant ζ(3) and rational series representations involving ζ(2n), (2016) preprint at http://arxiv.org/abs/1605.09541. H. M. Srivastava, M. L. Glasser, V. S. Adamchik, Some denite integrals associated with the Riemann zeta function, Z. Anal. Anwendungen 19 (2000), 831846.

D. Tyler, P. R. Cherno, An old sum reappears-Elementary problem 3103, Amer. Math. Monthly 92 (1985), 507.

J. Wu, X. Zhang, D. Liu, An ecient calculation of the Clausen functions, BIT Numer. Math. 50 (2010), 193206.

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters Background Main results Extensions

Thank You!

Derek Orr, University of Pittsburgh Generalized Rational Zeta Series with Two Parameters