Journal of Applied Mathematics and Physics, 2021, 9, 1108-1120 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352
Special Values for the Riemann Zeta Function
John H. Heinbockel
Old Dominion University, Norfolk, Virginia, USA
How to cite this paper: Heinbockel, J.H. Abstract (2021) Special Values for the Riemann Zeta Function. Journal of Applied Mathematics The purpose for this research was to investigate the Riemann zeta function at and Physics, 9, 1108-1120. odd integer values, because there was no simple representation for these re- https://doi.org/10.4236/jamp.2021.95077 sults. The research resulted in the closed form expression
Received: April 9, 2021 n 21n+ (2n) (−−4) π E2n 2ψ ( 34) Accepted: May 28, 2021 ζ (2nn+= 1) ,= 1,2,3, 221nn++( 2 21− 12)( n) ! Published: May 31, 2021
for representing the zeta function at the odd integer values 21n + for n a Copyright © 2021 by author(s) and Scientific Research Publishing Inc. positive integer. The above representation shows the zeta function at odd
This work is licensed under the Creative positive integers can be represented in terms of the Euler numbers E2n and Commons Attribution International ψ (2n) 34 License (CC BY 4.0). the polygamma functions ( ) . This is a new result for this study area. http://creativecommons.org/licenses/by/4.0/ For completeness, this paper presents a review of selected properties of the Open Access Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results pre- sented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form ex- pressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are al- so presented.
Keywords Riemann Zeta Function, Zeta (2n), Zeta (2n + 1), Apéry’s Constant, Catalan Constant
1. Introduction
If you do not know about the Riemann zeta function, then do an internet search to observe the extensive research that has been done investigating various prop- erties of this function. A more detailed introduction to the Riemann zeta func- tion can be found in the references [1] [2]. One way of defining this function is
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to express it as an infinite series having the form ∞ 1 111 ζσ( ) =∑ σσσσ =+++,1σ > (1) n=1 n 123
where σ is a real number greater than 1 in order for the infinite series to con- verge. Observing that for σ = 1, the series becomes the harmonic series which slowly diverges. The zeta function was introduced by Leonhard Euler (1707-1783) who considered ζσ( ) to be a function of a real variable. Another form for representing the zeta function is the integral representation
σ −1 ∞ r Γ=(σζσ) ( ) d,r σ > 1 (2) ∫0 e1r −
where Γ(σ ) is the gamma function. Bernhard Riemann (1826-1866) studied the zeta function and changed the independent real variable σ to the complex variable s=σ + it . This notation is still used in current studies of the zeta function. By doing this, Riemann made ζ (s) a function of a complex variable. Riemann discovered that the zeta func- tion satisfied the functional equation
1−s − −s 2 ss1− π Γ=Γ−ζζ(ss) π 2 (1 ) (3) 22
where Γ(s) is the gamma function. Several proofs of the above result can be found in the Titchmarsh reference [3]. Various forms for the functional equa- tion are derived later in this paper. The equation allowed the zeta function to be defined for values Re(σ ) < 1. The point s = 1 is a singular point. Using prop- erties of the gamma function, the functional equation can be expressed in the al- ternative form s−1 ζζ(s) =2( 2π) sin(πs 2) Γ−( 1−< s) ( 1 s) , Re( s) 1 (4)
derived later in this paper. The above results can be used to extend the definition of the zeta function to the whole of the complex plane. Euler also showed that the zeta function can also be expressed using prime numbers
−1 1 ζ =−> (s) ∏1,s Re( s) 1 (5) p p
where the product runs through all primes p = 2,3,5,7,. The equation (5) is known as the Euler product formula. The Euler-Riemann function ζ (s) is an important function in number theory where it is related to the distribution of prime numbers. It also can be found in such diverse study areas as probability and statistics, physics, Diophan- tine equations, modular forms and in many tables of integrals. The Eu- ler-Riemann zeta function evaluated at special integer values for s occurs quite frequently in tables of integrals and in many areas of science and engineering.
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2. Bernoulli and Euler Numbers
In later sections we need knowledge of the Bernoulli numbers Bn and Euler
numbers En . Representation of these numbers can be obtained from reference [4] (24.2), where one finds the generating functions x∞ xn 11 xxx246 1 1 = =−+ − + + x ∑ Bxn 1 e1− n=0 n! 2 6 2! 30 4! 42 6! (6) 2exn∞ x xx24 x 6 x 8 = =−+ − + + 2x ∑ En 1 5 61 1385 e1+ n=0 n! 2! 4! 6! 8! Note that the first few values are
BB01==−==−=1, 1 2, B 2 1 6, B 4 1 30, B 6 1 42, EE=1, =−==−= 1, E 5, E 61, E 1385, 02 4 6 8
Note that for mn, positive integers with n ≥ 1, B21n+ = 0 and for m ≥ 0 ,
E21m+ = 0 .
3. Calculation of ζ (2n) for n = 1,2,3,
Leonhard Euler discovered values for the zeta function at ζζζ(2,) ( 4,) ( 6,) In general for s an even integer, say sn= 2 , for n = 1, 2, 3, , the zeta function ζ (2n) , evaluated at positive even integers takes on the values given by ππ24ππ 6 8 ζζζ(2,4,6,8) =( ) =( ) = ζ( ) = 6 90 945 9450 nn+12 (7) (−12) ( π) and in generalζ ( 2n) = Bn,= 1,2,3, 22( n) ! 2n
where Bn are the Bernoulli numbers. These results were discovered by Leon- hard Euler (1707-1783) sometime around 1724 and are well known. Observe that ζ (2n) is proportional to π2n . The above results can be derived from the following observations. The func- tion gx( ) = xcot ( x) can be expressed in different forms. For example, eeix+ − ix cos( x) e12ix + 2ix g( x) = x = x2 = ix = ix + sin ( x) eeix−− ix e122ix −− e1ix 2i Now compare the last term of the above equation with the previous equation (6) involving the Bernoulli numbers, to obtain m ∞ (2ix) g( x) = ix + ∑ Bm = m! m 0 (8) mm ∞∞(22ix) ( ix) g( x) =++ ix B01 B(21 ix) +∑∑ Bmm =+ B mm=22mm!!= cos( x) One can examine the zeros of the denominator in gx( ) = x and ex- sin ( x) press gx( ) in the alternative form
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∞ x2 = − gx( ) 12∑ 22 2 n=1 nxπ − ∞ x2 1 = − gx( ) 12∑ 22 2 n=1 n π x 1− π n The last term of the above equation can be expanded in a series to obtain 2 j ∞∞xx2 = − gx( ) 12∑∑22 nj=10n π = nπ
One can interchange the order of summation and write ∞∞1 x22j+ = − gx( ) 12∑∑22jj++ 22 jn=01 = n π (9) ∞∞1 x2 j = − gx( ) 12∑∑22jj jn=11 = n π
Now by comparing the coefficients of powers xn in the equations (9) and (8) one obtains the well known result nn+12 (−12) ( π) ζ (2n) = Bn,= 1, 2, 3, 22( n) ! 2n
as previously given in equation (7). A similar derivation can be found in the ref- erence [5].
4. The Zeta Function ζ (2kk+= 1) , 1,2,3,
Note that the reference [2] points out that there is no known formula for the zeta function evaluated at odd positive integers greater than or equal to three. This paper will provide such a formula. It will be demonstrated that for odd positive integers s, say sn=21 + , for n = 1, 2, 3, that (2) π3 E ψ (34) ζ (3) =−−=2 1.20205 , 28 56 (4) π5 E ψ (34) ζ (5) = 4 −=1.03692 1488 11904 (6) π7 E ψ (34) ζ (7) =−−=6 1.00834 , 182880 5852160 (8) π9 E ψ (34) ζ (9) =8 − = 1.00200 41207040 5274501120 where the ellipsis denotes the decimal representations are unending. In general, it will be demonstrated n (−−4) π21n+ E 2ψ (2n) ( 34) ζ (2nn+= 1) 2n ,= 1, 2, 3, (10) 221nn++( 2 21− 12)( n) !
(n) where En are the Euler numbers and ψ ( z) are the polygamma functions.
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Observe the ζ (21n + ) is related to π21n+ . Apéry’s constant is −4π3 E − 2ψ (2) ( 34) ζ (3) = 2 = 1.20205690316 112 5. Polygamma Functions
The digamma function ψ (s) is defined d Γ′(s) ln Γ=(ss) =ψ ( ) (11) dssΓ( )
where Γ(s) is the gamma function.
∞ −− Γ=(s) xsx1e d, x Re( s) > 0 (12) ∫0 The gamma function satisfies the functional equation Γ+=Γ( x1) xx( ) so one can write Γ+=Γ( x1) xx( ) Γ+=+Γ+( x2) ( xx 11) ( ) (13) Γ++=+Γ+( xn1) ( xn) ( xn)
and consequently Γ( xn ++1) Γ=( x) (14) xx( ++12)( x)( x + n)
Take logarithms on both sides of equation (14) and then differentiate to show 11 1 1 ψψ( x) =−− − − − +( xn ++1) x x++12 x xn + Differentiate again and show 11 1 1 ψψ′′= + + + + + ++ ( x) 22 2 2( xn1) x ( x++12) ( x) ( xn +) (15) From the reference [2] or reference [4] (5.15), one can show that in the limit as n increases without bound the derivative term ψ ′(n) behaves like 1/n and approaches zero. By repeated differentiation of equation (15) one can obtain the (n) polygamma functions ψ ( x) defined by
n+1 d n+1 11 ψ (n) = Γ=− + + ( x) n+1ln( xn) ( 1) ! nn++11 dxx( x +1) (16) ∞ n+1 1 ψ (n) = − ( xn) ( 1!) ∑ n+1 =0 ( x + )
for n = 1, 2, 3, .
6. Additional Functions
Related to the study of the zeta function are the Dirichlet1 eta, lambda and beta
1Peter Gustav Lejeune Dirichlet (1805-1859).
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series defined ∞ 1111 k −1 η =−+−+= −−s > (s) ssss ∑( 1) ks , 0, 1234 k =1 ∞ 1111 −s λ =+++ += + > (s) ssss ∑(2ks 1,) 1 1357 k =0 ∞ 1111 ks− β =−+− += − + > (s) ssss ∑( 12) ( ks 1,) 0 1357 k =0
The first two Dirichlet series are related to the zeta function by the identities ζλη(ss) ( ) ( s) = = and ζη(ss) +=( ) 2 λ( s) (17) 2ss 212−− s 2 Make note of the fact that knowing the equations (7) and (10) one can con- struct closed form expressions for the Dirichlet eta and lambda functions eva- luated at odd and even integers greater than one.
7. Preliminary Observations
Define the function 22 2 ∞ 1 = + + += rs( ) ss s 2∑ s (18) 3 7 11 k =0 (34+ k )
where s is a positive integer greater than 1. One can then verify that βλ(s) += rs( ) ( s) =−(12−s ) ζ(s) (19)
We examine the special cases
−+(21k ) βλ(21k++) rk( 21 +=) ( 21 k +=−) ( 12) ζ( 21k +) (20)
from which ζ (21k + ) can be obtained and
−(2k ) βλ(2k) += rk( 2) ( 2 k) =−( 12) ζ( 2k) (21)
from which an expression for β (2k ) , k an integer, can be obtained. From these two equations one can develop closed form expressions for ζ (21k + ) and β (2k ) .
8. Calculation of rk(21+ ) and rk(2 )
Observe that by using equation (16) with nk= 2 , and again with nk=21 − , one can obtain the series representations (2k ) 212∞ − ψ (34) += = rk(21) 21k+∑ 21 kk ++ 21 44=0 3 (2!k ) + 4 (22) (21k − ) 21∞ 2ψ ( 34) = = rk(2 ) 2k∑ 22 kk 4 =0 3 4( 2k − 1!) + 4
These results will be used shortly.
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9. Calculation of β (21k + ) and β (2k )
We begin by examining the trigonometric function fx( ) =sec( x) + tan ( x) cos( x) which can be expressed in many different forms. One form is fx( ) = 1− sin ( x) where one can examine the zeros of the denominator and write ∞ ab fx( ) = ∑ nn+ xx n=0 11−+ ππ (14++nn) ( 34) 22
where abnn, are constants which can be determined from the limits 4 lim( x−+( 1 4 m)π 2) fx( ) = am = xm→+(14 )π 2 (14+ m)π −4 lim( x−+( 3 4 m)π 2) fx( ) = bm = xm→+(3 4 )π 2 (34+ m)π
This produces the expression 41∞ 1 1 1 fx( ) = ∑ − π n=0 14++nnxx34 11−+ (14++nn)ππ22(34 )
which can now be expanded into the series
4 22xx22 2nn x = + + ++ + fx( ) 1 2 n ππππ 4 22xx22 2nn x + + + ++ + + 1 2 2 nn 5π 5π 5π5 π n (23) 4 22xx22 (−12) nnx − − + ++ + 1 2 2 nn 3π 3π 3π3 π n 4 22xx22 (−12) nnx −1 − + ++ + − ππ2 2 nn 7 7 7π 7 π
Another form for fx( ) is ∞ xnn xx2 f( x) =∑ Knn = K01 + Kx + K 2 ++ K + (24) k =0 nn! 2! !
where the coefficients Kn are known as the Euler zigzag numbers. Still another form for fx( ) is
∞∞ nnEB−1 − fx( ) =−∑∑( 1) 22nnx2n +−( 1) 222nn( 2 − 1) x21n (25) nn=01(2!nn) = (2!)
where
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∞ n E sec( xx) =∑( − 1) 2n 2n n=0 (2!n)
and ∞ n−1 B − tan( xx) =−−∑( 1) 22nn( 2 2 1) 2n 21n n=1 (2!n)
with En and Bn denoting the Euler and Bernoulli numbers. Comparing like powers of x from equations (23) and (24) one can establish the relation
nn++11 nn ++ 11 K 2n+2 11 11 n = +− + +− + + n+1 1 (26) n!π 35 79
where the right-hand side of the equation is recognized as the β series or λ series, depending upon the value of n. Replace n by 2n in equation (26) to obtain π21n+ β (21nK+=) (27) 222n+ ( 2!n) 2n
Comparing like powers of x using the equations (24) and (25) one can show
n−1 22nn n (−−1) 22( 1) K=(−=1) EK and − B (28) 2n2 nn21 2n 2n which expresses the zigzag numbers in terms of the Euler and Bernoulli num- bers. Therefore, the equation (27) can be expressed in the alternative form n (−1) π21n+ β (2n+= 1) En,= 1, 2, 3, (29) 222n+ ( 2!n) 2n
a result also found in references [5] [6]. In equation (26) let nm=21 − and show
mm−122m K π2m (−−1) ( 2 12)( π) λ (2mB) = 21m− = (2mm− 1) !221mm++221( 2) ! 2m
and consequently the equation (21) can be written
mm−122m (−−1) ( 2 12)( π) 2ψ (21m− ) ( 34) β (2mB) = − (30) 221mm+ ( 2mm) !2m 42( 2− 1!)
giving a closed form expression for β (2m) where m = 1, 2, 3,. Note Cata- lan’s constant is given by 2 3( 2π) B 2ψ (1) ( 34) β (2) =−=2 0.915965559 16 16
10. Calculation of ζ (21k + )
Use the results from equations (29), (20) and (22) one can demonstrate the equation βλ(21k++) rk( 21 +=) ( 21 k +=−) ( 12−+(21k ) ) ζ( 21k +)
can be expressed in the form
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k 21k + (2k ) (−1) π −2 ψ (34) −+ + =−+(21k ) ζ 22kk++Ek2k 21 (12) ( 2 1) 2( 2!k ) 4 (2!k )
for k = 1, 2, 3,. Solving for ζ (21k + ) one obtains the closed form expression given by equation (10) for the zeta function evaluated at odd positive integers greater than or equal to three.
11. Riemann Zeta Functional Equation
Several derivations of the Riemann zeta functional equation can be found in the ref- erence [3]. One derivation is as follows. Using the definition of the gamma function
∞ α −− Γ=(αα) rr1er d,> 0 (31) ∫0 s make the substitutions α = and r= π nx2 to obtain after simplification 2 −s 2 π s ∞ 2 Γ=xxs21−−edπ nx s ∫0 n 2 A summation of both sides of this equation over the index n and interchang- ing summation and integration one can show the above equation reduces to ∞∞ s 1 ∞ 2 π−s2Γ=xxs21−−edπnx (32) ∑∑s ∫0 2 nn=11n = ∞ ∞ 1 2 = ζ −πnx Here ∑ s (s) is the Riemann zeta function and ∑ e is related to n=1 n n=1 the Jacobi theta function ∞∞ 22 θ ( x) =∑∑e−−ππnx = 12 + enx (33) nn=−∞ =1 ∞ 2 Define φ ( x) = ∑ e−πnx and express equation (33) in the form n=1 θφ( xx) =12 + ( )
11 The Jacobi theta function satisfies the property θθ( x) = which can x x be written in terms of the function φ as 1 1 1 1 11 12+φ( xx) = 12 + φ or φφ( ) = +− (34) xxxxx2 2 The equation (32) can now be expressed s ∞∞1 π−s2Γ=ζφ(s) x ss21 −−( xx)ddd = x21 φ( xx) + x s21 − φ( xx) ∫0 ∫∫ 01 2 The first integral on the right-hand side can be written in a different form as follows.
11ss21−−211 1 11 ∫∫xφθ( xx)dd= x +−x 00xxx 2 2
11s21−11 1ss232−− 1 21 =xφ dd x +− x xx ∫∫00x 22 x
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which simplifies to
11ss2−− 1 2 32 11 ∫∫xφφ( xx)dd= x x+ 00x ss( −1)
1 This integral is further simplified by making the substitution x = to obtain u
1 ∞ 1 xss21−φφ( xx)dd= u−−212 ( uu) + ∫∫01 ss( −1)
This last integral allows one to express the equation (32) in the form
∞ −−ss2 s 21 −+(s 12) 1 π Γ=+ ζφ(s) ∫ ( x x) ( xx)d + (35) 211 ss( − )
Observe that the right-hand side of equation (35) remains unchanged when s is replaced by 1− s . This implies
1−s − 2 1− ss −s 2 ππΓ ζζ(1 −=ss) Γ( ) (36) 22 which is the Riemann zeta functional equation. Multiplication of equation on s +1 both sides by Γ and using the Euler reflection formula 2 π Γ( xx) Γ−(1 ) = sin (πx)
and the Legendre duplication formula
1 12− x Γ( xx) Γ+ =22π Γ( x) 2 the functional equation can be expressed in the alternative form
1−s − π π 2 ζζ(12−=s) π−−ss21 π Γ(ss) ( ) s +1 sin π 2 which simplifies to
1−−ss πs ζζ(1−=s) 2π cosΓ(s) ( s), Res > 0 (37) 2
Replacing s by 1− s the Riemann zeta functional equation can also be ex- pressed in the form
ss−1 πs ζζ(s) =2π sinΓ−( 1s) ( 1 − s) , Res < 1 (38) 2
12. Zeta Function for 0 and Negative Integers
The Riemann zeta functional equation is used to demonstrate ζ (−=2nn) 0, = 1,2,3, (39)
since sin(nπ) = 0 for all values of the integer n. These values for the zeta func-
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tion are known as the trivial zeros. The nontrivial zeros lie in the complex plane. Also the Riemann zeta functional equation gives ζζ(−+=212n) −+21nnπ − 2Γ( 2nn) ( 2)
Using the results from equation (7), this simplifies to −B ζ (−+=21n ) 2n (40) 2n
where Bn are the Bernoulli numbers.
Using the fact that Bn = 0 for odd integers greater than one the equations (39) and (40) can be combined into the form
n B ζ (−=−n) ( 1) n+1 (41) n +1 for n a positive integer or zero. This last equation also gives the integer values 11 ζζ(0) =− ,1( −=−) (42) 2 12 Recall the value ζ (1) does not exist as the series is the harmonic series which diverges for σ = 1. These values added to the values presented earlier will give the value of the zeta function at integer values, different from 1, along the real line. For additional representations involving the zeta function in various forms and evaluated at other values the reader is referred to the references [2] [3] [6] [7] [8].
13. Zeros of the Zeta Function
The Euler product formula is used to demonstrate ζ (s) ≠ 0 whenever Re( s) > 1 . The Dirichlet eta function η (s) is used to study the zeros of the zeta function for Re( s) > 0 , s ≠ 1 , since it is related to the zeta function n+1 ∞ (−1) ηζ= =−1−s >≠ (s) ∑ s (1 2) (s) , Re( s) 0, s 1 (43) n=1 n The eta function is a converging alternating series for Re( s) >≠0, s 1 and is sometimes referred to as the alternating zeta function. The equation (43) shows η (s) = 0 whenever ζ (s) = 0 . The factor ( 12− 1−s ) is zero at the points in2π s =1 + , for all nonzero integer values for n. These are additional zeros of ln 2 the eta function. Writing η(s) = ησ( += it) u( σ,, t) + iv( σ t) where for 01<<σ one can show nn−−11 ∞∞(−−11) ( ) u(σσ, t) = ∑∑σσcos(tn ln) , v( , t) = − sin(tn ln ) (44) nn=11nn= ∂u ∂∂ vv ∂ u and verify that =, = − so the Cauchy-Riemann equations are ∂σσ ∂∂tt ∂
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satisfied. This show η (s) is an holomorphic function which satisfies ηη(ss) = ( ) . This implies that if η (s) = 0 for some value of s, then its conju- gate s satisfies η (s ) = 0 . This demonstrates that the zeros of the zeta function are symmetric about the σ -axis. The equation η (s) = 0 is satisfied if both the real part u and imaginary part v of η are zero simultaneously. The condition u = 0 and v = 0 simultaneously is illustrated in Figure 1 by plotting 22 ut[σσ,,] + vt[ ] vs t in the special case where σ = 12. The special case σ = 12 was selected for Figure 1 because of the Riemann hypothesis which is a conjecture that the nontrivial zeros of the zeta function have a real part equal to
one-half. The values tt12,, are the values of t where u = 0 and v = 0 si- multaneously for σ = 12. Here σ = 12 is called the critical line and the re- gion 01<<σ is called the critical strip. To see the first one hundred imaginary parts of the complex zeros one can visit the web site https://wow.Imfdb.org/zeros/zeta/. A huge number of these complex zeros have been calculated and all lie on the critical line where σ = 12. Currently there is no proof that all of the nontrivial zeros of the zeta function must lie on the criti- cal line.
22 Figure 1. Plot of ut[σσ,,] + vt[ ] vs t, for 0≤ 14. Conclusion A closed form expression for the Riemann zeta function evaluated at odd posi- tive integers greater than three has been presented having the form n (−−4) π21n+ E 2ψ (2n) ( 34) ζ (2nn+= 1) 2n ,= 1,2,3, 221nn++( 2 21− 12)( n) ! (2n) where E2n are the Euler numbers and ψ (34) are the polygamma func- tions. It has been demonstrated that knowing closed form expressions for ζ (2n) and ζ (21n + ) for n = 1,2,3, one can construct closed form ex- pressions for the Dirichlet eta, lambda and beta series at the even and odd integ- ers different from unity. Closed form representations of the Apéry’s constant −4π3 E − 2ψ (2) ( 34) ζ (3) = 2 = 1.20205690316 and the Catalan’s constant 112 DOI: 10.4236/jamp.2021.95077 1119 Journal of Applied Mathematics and Physics J. H. Heinbockel 2 3( 2π) B 2ψ (1) ( 34) β (2) =−=2 0.915965559 are obtained. 16 16 15. The Riemann Hypothesis The Riemann hypothesis is a conjecture that all nontrivial zeros of the zeta function have a real part equal to one-half. If this is true, all nontrivial zeros are complex 1 numbers of the form + it , called the critical line. Whether this is true or not is 2 still an open question. The Clay Mathematics Institute in Petersborough, New Hampshire is offering a one million dollar prize to anyone who can prove this conjecture and show how to calculate all the zeros of the zeta function. For additional information and conditions to be met in order to win the prize, the reader can consult refer- ence [2] under the search name Riemann zeta function prize. Conflicts of Interest The author declares no conflicts of interest regarding the publication of this pa- per. References [1] Whittaker, E.T. and Watson, G.N. (1962) A Course of Modern Analysis. 4th Ed., Cambridge at the University Press, Cambridge. [2] The Riemann Zeta Function. http://www.wikipedia.com/ [3] Titchmarch, E.C. (1986) The Theory of the Riemann-Zeta Function. 2nd Ed., Re- vised by D.R. Heath-Brown, The Clarendon Press, Oxford. [4] Digital Library of Mathematical Functions. https://dlmf.nist.gov/5.15 https://dlmf.nist.gov/24.8 https://dlmf.nist.gov/24.2 [5] Borwein, J.M. and Borwein, P.B. (1987) Pi and the AGM. Wiley, Toronto, 383-385. [6] Abramowitz, M. and Stegun, I.A. (1988) Handbook of Mathematical Functions. Dover Publications, New York, p. 807. [7] Edwards, H.M. (1974) Riemann’s Zeta Function. Academic Press, New York and London. [8] Dwilewicz, R.J. and Mináŏ, J. (2009) Values of the Riemann Zeta Function at Integ- ers. Materials Matemàtics, No. 6, 26 p. DOI: 10.4236/jamp.2021.95077 1120 Journal of Applied Mathematics and Physics