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Special Values for the Riemann Zeta Function

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Special Values for the Riemann Zeta Function 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, 21nn++ 21− Published: May 31, 2021 2( 2 12)( n) ! 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 DOI: 10.4236/jamp.2021.95077 May 31, 2021 1108 Journal of Applied Mathematics and Physics J. H. Heinbockel 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. DOI: 10.4236/jamp.2021.95077 1109 Journal of Applied Mathematics and Physics J. H. Heinbockel 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, EE02=1, =−==−= 1, E 4 5, E 6 61, E 8 1385, 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=0 m! (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 DOI: 10.4236/jamp.2021.95077 1110 Journal of Applied Mathematics and Physics J. H. Heinbockel ∞ 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. DOI: 10.4236/jamp.2021.95077 1111 Journal of Applied Mathematics and Physics J. H. Heinbockel 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. ∞ sx−−1 Γ=(s) xe 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.
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