DOCSLIB.ORG

New Inequalities Involving the Dirichlet Beta and Euler's Gamma Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
New Inequalities Involving the Dirichlet Beta and Euler's Gamma Functions Sohag J. Math. 5, No. 1, 9-13 (2018) 9 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/050102 New Inequalities Involving the Dirichlet Beta and Euler's Gamma Functions Kwara Nantomah∗ and Mohammed Muniru Iddrisu Department of Mathematics, University for Development Studies, Navrongo Campus, P. O. Box 24, Navrongo, UE/R, Ghana. Received: 17 Mar. 2017, Revised: 18 Nov. 2017, Accepted: 23 Nov. 2017 Published online: 1 Jan. 2018 Abstract: We present some new inequalities involving the Dirichlet Beta and Euler’s Gamma functions. The concept of monotonicity of Dirichlet Beta function is also discussed. The generalized forms of the H¨older’s and Minkowski’s inequalities among other techniques are employed. Keywords: Dirichlet beta function, Gamma function, Inequality 2010 Mathematics Subject Classification: 33B15, 11M06, 33E20. 1 Introduction where G = 0.915965594177... is the Catalan constant [5]. The Dirichlet beta function (also known as the Catalan For positive integer values of n, the function β(x) may be beta function or the Dirichlet’s L-function) is defined for evaluated explicitly by x > 0by[4, p. 56] n π2n+1 β (−1) E2n ∞ x−1 (2n + 1)= n 1 β 1 t 4 + (2n)! (x)= t −t dt (1) Γ (x) 0 e + e ∞ Z where En are the Euler numbers generated by (−1)n ∑ ∞ = x 1 2 xn n=0 (2n + 1) . = x −x = ∑ En coshx e + e n=0 n! where Γ (x) is the classical Euler’s Gamma function ψ(m) defined as ∞ Also, in terms of the polygamma function (x), the Γ (x)= tx−1e−t dt function β(x) may be written as [6] 0 Z 1 1 3 and satisfying the basic relation β(n)= ψ(n−1) − ψ(n−1) . 22n(n − 1)! 4 4 Γ (x + 1)= xΓ (x). (2) The main objective of this paper is to establish some Let K(x) be defined as inequalities involving the Dirichlet Beta and Euler’s Gamma functions. We begin by recalling the following ∞ tx−1 lemmas which shall be required in order to establish our K(x)= β(x)Γ (x)= dt, x > 0. (3) et + e−t results. Z0 The Dirichlet beta function, which is closely related to the Riemann zeta function, has important applications in 2 Preliminaries Analytic Number Theory as well as other branches of mathematics. See for instance [2], [3] and the related Lemma 1(Generalized Holder’s¨ Inequality). Let β π β references therein. In particular, (1)= 4 and (2)= G, f1, f2,..., fn be functions such that the integrals exist. ∗ Corresponding author e-mail: [email protected] c 2018 NSP Natural Sciences Publishing Cor. 10 K. Nantomah, M. M. Iddrisu: New Inequalities involving the dirichlet... Then the inequality Proof.Let K(x) be defined as in (3). Then by utilizing Lemma 1, we obtain 1 α b n n b i αi ∏ fi(t) dt ≤ ∏ | fi(t)| dt (4) ∑n xi a a n ∞ ( i=1 α )−1 Z i=1 i=1 Z xi t i K ∑ = dt α t −t n 1 i=1 i ! 0 e + e holds for αi > 1 such that ∑ = 1. Z i=1 αi n xi−1 ∞ ∑ α t i=1 i Proof.See page 790-791 of [1]. = dt ∑n 1 0 t −t i=1 α Lemma 2(Generalized Minkowski’s Inequality). Let Z (e + e ) i 1 f , f ,..., f be functions such that the integrals exist. ∞ α 1 2 n n txi−1 i Then the inequality = ∏ t −t dt 0 i=1 e + e 1 Z u u 1 1 b n n b u n ∞ x −1 α u t i i ∑ fi(t) dt ≤ ∑ | fi(t)| dt (5) ≤ ∏ t −t dt a i=1 ! i=1 a 0 e + e Z Z i=1 Z n 1 holds for u ≥ 1. α = ∏(K(xi)) i Proof.See page 790-791 of [1]. i=1 Lemma 3([9]). Let f and h be continuous rapidly which gives the required result (7). decaying positive functions on [0,∞). Further, let F and H be defined as Remark.If n = 2, α1 = a, α2 = b, x1 = x and x2 = y, then, ∞ ∞ we have F(x)= f (t)tx−1 dt and H(x)= h(t)tx−1 dt. 0 0 x y 1 1 Z Z K + ≤ (K(x)) a (K(y)) b f (t) F(x) a b If h(t) is increasing, then so is H(x) . which implies that K(x) is logarithmically convex. Also, Lemma 4([7]). Let f and g be two nonnegative functions since every logarithmically convex function is convex, it of a real variable and m, n be real numbers such that the follows that K(x) is convex. integrals in (6) exist. Then b b Corollary 1.The inequality g(t)( f (t))m dt · g(t)( f (t))n dt Za Za 2 ′ 2 ′′ b m+n β (x) β (x) ≥ g(t)( f (t)) 2 dt (6) − ≤ ψ′(x) (8) β(x) β(x) Za Lemma 5([8]). Let f : (0,∞) → (0,∞) be a differentiable, ψ Γ ′(x) logarithmically convex function. Then the function holds for x > 0, where (x) = Γ (x) is the Digamma function. ( f (x))α g(x)= f (αx) Proof.Since K(x)= β(x)Γ (x) is logarithmically convex, is decreasing if α ≥ 1, and increasing if 0 < α ≤ 1. then (lnK(x))′′ ≥ 0 which results to (8). 3 Main Results Theorem 2.Let xi > 0, i = 1,2,...,n and u ≥ 1. Then the inequality We present the main findings of the paper in this section. 1 n u n Theorem 1.For i = 1,2,...,n, let αi > 1 such that 1 n 1 ∑ β(xi)Γ (xi) ≤ ∑ (β(xi)Γ (xi)) u (9) ∑ α = 1. Then the inequality i=1 i i=1 ! i=1 n x 1 Γ ∑ i α i 1 α ∏n β i = i i=1 ( (xi)) holds. 1 ≤ (7) α n xi ∏n Γ i β ∑ i=1 ( (xi)) i=1 αi Proof.Let K(x) bedefinedasin (3). Then by using the fact is valid for x > 0. ∑n u ∑n u i that i=1 ai ≤ ( i=1 ai) , for ai ≥ 0, u ≥ 1 in conjunction c 2018 NSP Natural Sciences Publishing Cor. Sohag J. Math. 5, No. 1, 9-13 (2018) / www.naturalspublishing.com/Journals.asp 11 with Lemma 2, we obtain Corollary 2.Let xi > 0 for i = 1,2,3,...,n. Then the inequality 1 1 u u n n n ∞ txi−1 n n ∑ K(xi) = ∑ dt ∏β(xi) ≤ β ∑ xi (11) et + e−t i=1 ! i=1 Z0 ! i=1 " i=1 !# 1 u ∞ n txi−1 is valid. = ∑ t −t dt 0 e + e Z i=1 ! Proof.Let xi > 0 for i = 1,2,3,...,n. Then since β(x) is u 1 increasing, we have ∞ n xi−1 u t u = ∑ dt t −t 1 n 0 i=1 "(e + e ) u # ! Z 0 < β(x1) ≤ β ∑ xi , 1 x −1 u i=1 ! ∞ n i u t u ≤ ∑ dt n t −t 1 β β Z0 "i=1 (e + e ) u # ! 0 < (x2) ≤ ∑ xi , i=1 ! u 1 n ∞ xi−1 u t u . ≤ ∑ dt . t t 1 i=1 0 (e + e− ) u Z " # ! n n 1 0 < β(x ) ≤ β ∑ x . = ∑ (K(x )) u n i i i=1 ! i=1 which yields the result (9). Taking products yields n β n n Theorem 3.The function (x) is monotone increasing on β β (0,∞). That is, for 0 < x ≤ y, we have ∏ (xi) ≤ ∑ xi i=1 " i=1 !# β(x) ≤ β(y). (10) as required. Proof.Let F, H, f and h be defined as Remark.In particular, if n = 2, x1 = x and x2 = y in (11), ∞ tx−1 ∞ then we obtain F(x)= dt, H(x)= tx−1e−t dt = Γ (x), et + e−t 2 Z0 Z0 β(x)β(y) ≤ [β(x + y)] . 1 f (t)= and h(t)= e−t . Theorem 4.The inequality et + e−t f t F x β β Then, ( ) = 1 is increasing and by Lemma 3, ( ) is (x + 2) (x + 1) h(t) 1+e−2t H(x) (x + 1)β ≥ x β (12) increasing as well. Thus, for 0 < x ≤ y, we have (x + 1) (x) F(x) F(y) holds for x > 0. ≤ ⇐⇒ F(x)H(y) ≤ F(y)H(x) H(x) H(y) 1 Proof.Let x > 0, g(t)= t −t , f (t)= t, m = x − 1, n = ∞ e +e which implies x + 1, a = 0 and b = . Then by Lemma 4, we have ∞ x−1 ∞ x+1 ∞ x 2 ∞ x−1 ∞ t t t t y−1 −t dt · dt ≥ dt dt · t e dt t −t t −t t −t et + e−t 0 e + e 0 e + e 0 e + e Z0 Z0 Z Z Z ∞ ty−1 ∞ ≤ dt · tx−1e−t dt which implies et + e−t Z0 Z0 β(x)Γ (x)·β(x+2)Γ (x+2) ≥ [β(x + 1)Γ (x + 1)]2 . (13) which further implies By using the functional equation (2), the relation (13) β(x)Γ (x)Γ (y) ≤ β(y)Γ (y)Γ (x). becomes Thus x 1 β x β x 2 x β x 1 2 β(x) ≤ β(y) ( + ) ( ) ( + ) ≥ ( ( + )) as required. which gives the required result. c 2018 NSP Natural Sciences Publishing Cor. 12 K. Nantomah, M. M. Iddrisu: New Inequalities involving the dirichlet..
Load more

Recommended publications
  • The Riemann and Hurwitz Zeta Functions, Apery's Constant and New
    The Riemann and Hurwitz zeta functions, Apery’s constant and new rational series representations involving ζ(2k) Cezar Lupu1 1Department of Mathematics University of Pittsburgh Pittsburgh, PA, USA Algebra, Combinatorics and Geometry Graduate Student Research Seminar, February 2, 2017, Pittsburgh, PA A quick overview of the Riemann zeta function. The Riemann zeta function is defined by 1 X 1 ζ(s) = ; Re s > 1: ns n=1 Originally, Riemann zeta function was defined for real arguments. Also, Euler found another formula which relates the Riemann zeta function with prime numbrs, namely Y 1 ζ(s) = ; 1 p 1 − ps where p runs through all primes p = 2; 3; 5;:::. A quick overview of the Riemann zeta function. Moreover, Riemann proved that the following ζ(s) satisfies the following integral representation formula: 1 Z 1 us−1 ζ(s) = u du; Re s > 1; Γ(s) 0 e − 1 Z 1 where Γ(s) = ts−1e−t dt, Re s > 0 is the Euler gamma 0 function. Also, another important fact is that one can extend ζ(s) from Re s > 1 to Re s > 0. By an easy computation one has 1 X 1 (1 − 21−s )ζ(s) = (−1)n−1 ; ns n=1 and therefore we have A quick overview of the Riemann function. 1 1 X 1 ζ(s) = (−1)n−1 ; Re s > 0; s 6= 1: 1 − 21−s ns n=1 It is well-known that ζ is analytic and it has an analytic continuation at s = 1. At s = 1 it has a simple pole with residue 1.
    [Show full text]
  • On a Double Inequality for the Dirichlet Beta Function ∗
    Tamsui Oxford Journal of Mathematical Sciences 24(3) (2008) 269-276 Aletheia University On a Double Inequality for the Dirichlet Beta Function ∗ P. Ceroneyz School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne 8001, Victoria, Australia. Received February 22, 2007, Accepted March 26, 2007. Abstract Let β (x) be the Dirichlet beta function. Then for all x > 0 c < 3x+1 [β (x + 1) − β (x)] < d with the best possible constant factors π 1 c = 3 − 0:85619449 and d = 2: 4 2 t The above result, and some variants, are used to approximate β at even integers in terms of known β at odd integers. Keywords and Phrases: Dirichlet Beta function, Inequalities, Approxima- tion, Dirichlet L−function. ∗2000 Mathematics Subject Classification. 26D99, 11M06, 26D07. yE-mail: [email protected] zhttp://www.staff.vu.edu.au/RGMIA/cerone/ 270 P. Cerone 1. Introduction The Dirichlet beta function or Dirichlet L−function is given by [5] 1 n X (−1) β (x) = ; x > 0; (1.1) (2n + 1)x n=0 where β (2) = G; Catalan's constant. The beta function may be evaluated explicitly at positive odd integer values of x; namely, E π 2n+1 β (2n + 1) = (−1)n 2n ; (1.2) 2 (2n)! 2 where En are the Euler numbers. The Dirichlet beta function may be analytically continued over the whole complex plane by the functional equation 2 z πz β (1 − z) = sin Γ(z) β (z) : π 2 The function β (z) is defined everywhere in the complex plane and has no P1 1 singularities, unlike the Riemann zeta function, ζ (s) = n=1 ns ; which has a simple pole at s = 1: The Dirichlet beta function and the zeta function have important applica- tions in a number of branches of mathematics, and in particular in Analytic number theory.
    [Show full text]
  • A Series Representation for Riemann's Zeta Function and Some
    Journal of Classical Analysis Volume 17, Number 2 (2021), 129–167 doi:10.7153/jca-2021-17-09 A SERIES REPRESENTATION FOR RIEMANN’S ZETA FUNCTION AND SOME INTERESTING IDENTITIES THAT FOLLOW MICHAEL MILGRAM Abstract. Using Cauchy’s Integral Theorem as a basis, what may be a new series representation for Dirichlet’s function η(s), and hence Riemann’s function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are un- earthed. The incomplete functions ζ ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths be- tween the endpoints. 1. Introduction In a recent paper [1], I have developed an integral equation for Riemann’s func- tion ξ (s), based on LeClair’s series [2] involving the Generalized Exponential Integral. Glasser, in a subsequent paper [3], has shown that this integral equation is equivalent to the standard statement of Cauchy’s Integral Theorem applied to ξ (s), from which he closed the circle [3, Eq. (13)] by re-deriving LeClair’s original series representation. This raises the question of what will be found if Cauchy’s Integral Theorem is applied to other functions related to Riemann’s function ζ(s), notably the so-called Dirichlet function η(s) (sometimes called the alternating zeta function), working backwards to a series representation? That question is the motivation for this work.
    [Show full text]
  • Special Functions Related to Dedekind Type DC-Sums and Their
    SPECIAL FUNCTIONS RELATED TO DEDEKIND TYPE DC-SUMS AND THEIR APPLICATIONS Yilmaz SIMSEK Akdeniz University, Faculty of Arts and Science, Department of Mathematics 07058 Antalya, Turkey [email protected] Abstract In this paper we construct trigonometric functions of the sum Tp(h, k), which is called Dedekind type DC-(Dahee and Changhee) sums. We establish analytic properties of this sum. We find trigonometric represen- tations of this sum. We prove reciprocity theorem of this sums. Furthermore, we obtain relations between the Clausen functions, Polylogarithm function, Hurwitz zeta function, generalized Lambert series (G-series), Hardy-Berndt sums and the sum Tp(h, k). We also give some applications related to these sums and functions. 2000 Mathematics Subject Classification. 11F20, 11B68, 11M35, 11M41, 11S80, 33C10, 33E20. Key Words and Phrases. Dedekind sums, Hardy-Berndt sums, Bernoulli Functions, Euler Functions, Euler numbers and polynomials, Riemann zeta function, Hurwitz zeta function, Lerch zeta function, Dirichlet series for the polylogarithm function, Dirichlet’s eta function, Legendre chi function, trigonometric functions and generalized Lambert series. arXiv:0902.0380v1 [math.CV] 2 Feb 2009 1 Introduction, Definitions and Notations In this section, we give some definitions, notations and results related to the Dedekind sums. Firstly we start with the definition of the classical Dedekind sums. Let x [x] 1 , if x / Z ((x)) = − G − 2 ∈ 0, if x Z, ∈ 1 [x]G being the largest integer x. Let h and k be coprime integers with k > 0, the classical Dedekind sum≤ s(h, k) is defined as follows k−1 a ha s (h, k)= . k k a=1 X The reciprocity law of the classical Dedekind sums is given by 1 1 h k 1 s(h, k)+ s(k, h)= + + + , −4 12 k h hk where (h, k) = 1 and h, k N := 1, 2, 3, ..
    [Show full text]
  • An Experimental Conjecture Involving Closed-Form Evaluation of Series
    M athematical I nequalities & A pplications Volume 16, Number 4 (2013), 971–979 doi:10.7153/mia-16-76 AN EXPERIMENTAL CONJECTURE INVOLVING CLOSED–FORM EVALUATION OF SERIES ASSOCIATED WITH THE ZETA FUNCTIONS JUNESANG CHOI AND HARI M. SR I VA S TAVA (Communicated by Yeol Je Cho) Abstract. The subject of closed-form summation of series involving the Zeta functions has been remarkably widely investigated. Recently, in the course of his trying to give a closed-form expression for the Dirichlet beta function β(2n)(n ∈ N),Lima[16] posed a very interesting experimental conjecture for a closed-form evaluation of a certain class of series involving the Riemann Zeta function ζ(s). Here, in the present sequel to Lima’s work, we aim at verifying correctness of Lima’s conjecture and presenting several general analogues of Lima’s conjecture. Our demonstration and derivations are based mainly upon a known formula for series associated with the Zeta functions. Relevant connections of some specialized results of the main identities presented here with those obtained in earlier works are also pointed out. 1. Introduction and preliminaries A rather classical (over two centuries old) theorem of Christian Goldbach (1690– 1764), which was stated in a letter dated 1729 from Goldbach to Daniel Bernoulli (1700–1782), was revived in 1986 by Shallit and Zikan [20] as the following problem: − ∑ (ω − 1) 1 = 1, (1.1) ω∈S where S denotes the set of all nontrivial integer kth powers, that is, S := nk : n,k ∈ N \{1} . (1.2) In terms of the Riemann Zeta function ζ(s) defined by ⎧ ∞ ∞ ⎪ 1 1 1 ⎪ ∑ = ∑ ℜ(s) > 1 ⎨ s − −s ( − )s n=1 n 1 2 n=1 2n 1 ζ(s) := (1.3) ⎪ 1 ∞ (−1)n−1 ⎩⎪ ∑ ℜ(s) > 0; s = 1 , − 1−s s 1 2 n=1 n Mathematics subject classification (2010): Primary 11M35, 11M36; Secondary 11M06, 33B15.
    [Show full text]
  • An Alternative Form of the Functional Equation for Riemann's Zeta Function
    Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia, 56 (2008-2009), 95-111. AN ALTERNATIVE FORM OF THE FUNCTIONAL EQUATION FOR RIEMANN′S ZETA FUNCTION ANDREA OSSICINI Abstract. In this paper we present a simple method for deriving an alternative form of the functional equation for Riemann’s Zeta function. The connections between some functional equations obtained implicitly by Leonhard Euler in his work ”Remarques sur un beau rapport entre les series des puissances tant directes que reciproques” in Memoires de l’Academie des Sciences de Berlin 17, (1768), permit to define a special function, named A (s), which is fully symmetric and is similar to Riemann’s ξ function1. To be complete we find several integral representations of the A (s) function and as a direct consequence of the second integral representation we obtain also an analytic continuation of the same function using an identity of Ramanujan. 2000 Mathematics Subject Classification: Primary 11M35; Secondary 11B68, 11M06. Keywords: Riemann Zeta, Dirichlet Beta, Riemann Hypothesis, series representations. 1. INTRODUCTION arXiv:0709.4173v25 [math.HO] 8 Feb 2010 Formulae (1) and (2) below can be found in the chapter devoted to Euler’s Gamma function in [4]. These are namely two functional equations for the Eulerian Zeta and for the alternat- ing Zeta, connected with the odd numbers, best known as Dirichlet’s Beta function and Catalan’s Beta function, see [4, pag. 35, formulae (24) and (29)]. Both of them were discovered, over 100 years before G.F.B. Riemann and O. Schl¨omilch [8, notes on chapter II], by L.
    [Show full text]
  • Families of Log Legendre Chi Function Integrals
    Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. x (xxxx), xxx{xxx. doi:10.2298/AADMxxxxxxxx FAMILIES OF LOG LEGENDRE CHI FUNCTION INTEGRALS Anthony Sofo In this paper we investigate the representation of integrals involving the prod- uct of the Legendre Chi function, polylogarithm function and log function. We will show that in many cases these integrals take an explicit form involv- ing the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed. 1. INTRODUCTION PRELIMINARIES AND NOTATION In this paper we investigate the representations of integrals of the type 1 Z a b m (1) x χp (x) Liq(δx ) ln (x) dx; 0 in terms of special functions such as zeta functions, Dirichlet eta functions, polylog- arithmic functions, beta functions and others. We note that χp (x) is the Legendre- b Chi function (LCF), Liq(δx ) is the polylogarithmic function, a 2 R, a ≥ −2, b 2 R+, p 2 N, q 2 N, δ 2 [−1; 1] n f0g and m 2 N for the set of complex num- bers C, natural numbers N, the set of real numbers R and the set of positive real numbers, R+: The following notation and results will be useful in the subsequent λ sections of this paper. A generalized binomial coefficient µ (λ, µ 2 C) is defined, in terms of the familiar (Euler's) gamma function, by λ Γ(λ + 1) := (λ, µ 2 ); µ Γ(µ + 1) Γ (λ − µ + 1) C 2020 Mathematics Subject Classification.
    [Show full text]
  • A Series Representation for Riemann's Zeta Function and Some
    Journal of Classical Analysis Volume 17, Number 2 (2021), 129–167 doi:10.7153/jca-2021-17-09 A SERIES REPRESENTATION FOR RIEMANN’S ZETA FUNCTION AND SOME INTERESTING IDENTITIES THAT FOLLOW MICHAEL MILGRAM Abstract. Using Cauchy’s Integral Theorem as a basis, what may be a new series representation for Dirichlet’s function η(s), and hence Riemann’s function ζ(s), is obtained in terms of the Exponential Integral function Es(iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are un- earthed. The incomplete functions ζ ±(s) and η±(s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths be- tween the endpoints. Mathematics subject classification (2010): 11M06, 11M26, 11M35, 11M99, 26A09, 30B40, 30E20, 33C20, 33B20, 33B99. Keywords and phrases: Riemann zeta function, Dirichlet eta function, alternating zeta function, incom- plete zeta function, generalized exponential integral, Bernoulli numbers, Euler numbers, harmonic numbers, infinite series, evaluation of integrals. REFERENCES [1] MICHAEL MILGRAM, An integral equation for Riemann’s Zeta function and its approximate solution, Abstract and Applied Analysis, /2020/1832982(1832982), May 2020, https://doi.org/10.1155/2020/1832982. [2] ANDRE LECLAIR, An electrostatic depiction of the validity of the Riemann Hypothesis and a for- mula for the N-th zero at large N, Int.
    [Show full text]
  • Approximations for Apery's Constant Ζ(3) and Rational
    APPROXIMATIONS FOR APERY'S CONSTANT ζ(3) AND RATIONAL SERIES REPRESENTATIONS INVOLVING ζ(2n) CEZAR LUPU, DEREK ORR Abstract. In this note, using an idea from [4] we derive some new series rep- resentations involving ζ(2n) and Euler numbers. Using a well-known series rep- resentation for the Clausen function, we also provide some new representations of Apery's constant ζ(3). In particular cases, we recover some well-known series representations of π. 1. Introduction and Preliminaries In 1734, Leonard Euler produced a sensation when he proved the following for- mula: 1 X 1 π2 = : n2 6 n=1 Later, in 1740, Euler generalized the above formula for even positive integers: B 22n−1π2n (1) ζ(2n) = (−1)n+1 2n ; (2n)! 1 X 1 where ζ(s) = is the celebrated Riemann zeta function. The coefficients B ns n n=1 are the so-called Bernoulli numbers and they are defined in the following way: 1 z X Bn = zn; jzj < 2π: ez − 1 n! n=0 An elementary proof of (1) can be found in [5]. In [4], de Amo, Carrillo and Sanchez produced another proof of Euler's formula (1) using the Taylor series ex- pansion of the tangent function and Fubini's theorem. In this paper, using similar ideas but for other functions, we provide a new proof for the Clausen acceleration formula that will serve as an application in displaying some fast representations for Apery's constant ζ(3). Moreover, the Taylor series Key words and phrases. Riemann zeta function, Clausen integral, rational zeta series represen- tations, Apery's constant.
    [Show full text]
  • New Inequalities Involving the Dirichlet Beta and Euler's Gamma Functions
    Sohag J. Math. 5, No. 1, 9-13 (2018) 9 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/050102 New Inequalities Involving the Dirichlet Beta and Euler's Gamma Functions Kwara Nantomah∗ and Mohammed Muniru Iddrisu Department of Mathematics, University for Development Studies, Navrongo Campus, P. O. Box 24, Navrongo, UE/R, Ghana. Received: 17 Mar. 2017, Revised: 18 Nov. 2017, Accepted: 23 Nov. 2017 Published online: 1 Jan. 2018 Abstract: We present some new inequalities involving the Dirichlet Beta and Euler’s Gamma functions. The concept of monotonicity of Dirichlet Beta function is also discussed. The generalized forms of the H¨older’s and Minkowski’s inequalities among other techniques are employed. Keywords: Dirichlet beta function, Gamma function, Inequality 2010 Mathematics Subject Classification: 33B15, 11M06, 33E20. 1 Introduction where G = 0.915965594177... is the Catalan constant [5]. The Dirichlet beta function (also known as the Catalan For positive integer values of n, the function β(x) may be beta function or the Dirichlet’s L-function) is defined for evaluated explicitly by x > 0by[4, p. 56] n π2n+1 β (−1) E2n ∞ x−1 (2n + 1)= n 1 β 1 t 4 + (2n)! (x)= t −t dt (1) Γ (x) 0 e + e ∞ Z where En are the Euler numbers generated by (−1)n ∑ ∞ = x 1 2 xn n=0 (2n + 1) . = x −x = ∑ En coshx e + e n=0 n! where Γ (x) is the classical Euler’s Gamma function ψ(m) defined as ∞ Also, in terms of the polygamma function (x), the Γ (x)= tx−1e−t dt function β(x) may be written as [6] 0 Z 1 1 3 and satisfying the basic relation β(n)= ψ(n−1) − ψ(n−1) .
    [Show full text]
  • Bounds for Zeta and Related Functions
    Journal of Inequalities in Pure and Applied Mathematics BOUNDS FOR ZETA AND RELATED FUNCTIONS volume 6, issue 5, article 134, P. CERONE 2005. School of Computer Science and Mathematics Received 13 April, 2005; Victoria University accepted 24 August, 2005. PO Box 14428, MCMC 8001 Communicated by: F. Qi VIC, Australia. EMail: [email protected] URL: http://rgmia.vu.edu.au/cerone/ Abstract Contents JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 114-05 Abstract Sharp bounds are obtained for expressions involving Zeta and related functions at a distance of one apart. Since Euler discovered in 1736 a closed form ex- pression for the Zeta function at the even integers, a comparable expression for the odd integers has not been forthcoming. The current article derives sharp bounds for the Zeta, Lambda and Eta functions at a distance of one apart. The methods developed allow an accurate approximation of the function values at Bounds for Zeta and Related the odd integers in terms of the neighbouring known function at even integer Functions values. The Dirichlet Beta function which has explicit representation at the odd P. Cerone integer values is also investigated in the current work. Cebyševˇ functional bounds are utilised to obtain tight upper bounds for the Zeta function at the odd integers. Title Page Contents 2000 Mathematics Subject Classification: Primary: 26D15, 11Mxx, 33Exx; Sec- ondary: 11M06, 33E20, 65M15. JJ II Key words: Euler Zeta function, Dirichlet beta, eta and lambda functions, Sharp bounds, Cebyševˇ functional. J I This paper is based on the talk given by the author within the “International Go Back Conference of Mathematical Inequalities and their Applications, I”, December 06- 08, 2004, Victoria University, Melbourne, Australia [http://rgmia.vu.edu.au/ Close conference ] Quit Page 2 of 42 J.
    [Show full text]
  • Special Values and Integral Representations for the Hurwitz-Type Euler Zeta Functions
    J. Korean Math. Soc. 55 (2018), No. 1, pp. 185{210 https://doi.org/10.4134/JKMS.j170110 pISSN: 0304-9914 / eISSN: 2234-3008 SPECIAL VALUES AND INTEGRAL REPRESENTATIONS FOR THE HURWITZ-TYPE EULER ZETA FUNCTIONS Su Hu, Daeyeoul Kim, and Min-Soo Kim Abstract. The Hurwitz-type Euler zeta function is defined as a defor- mation of the Hurwitz zeta function: 1 X (−1)n ζE (s; x) = : (n + x)s n=0 In this paper, by using the method of Fourier expansions, we shall eval- uate several integrals with integrands involving Hurwitz-type Euler zeta functions ζE (s; x). Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given. 1. Introduction The Hurwitz zeta function is defined by 1 X 1 (1.1) ζ(s; x) = (n + x)s n=0 for s 2 C and x 6= 0; −1; −2;:::; and the series converges for Re(s) > 1; so that ζ(s; x) is an analytic function of s in this region. Setting x = 1 in (1.1), it reduces to the Riemann zeta function 1 X 1 (1.2) ζ(s) = : ns n=1 1 s From the well-known identity ζ(s; 2 ) = (2 − 1)ζ(s); we see that the only real 1 zeroes of ζ(s; x) with x = 2 are s = 0; −2; −4;:::: Its special values at nonpositive integers are Bernoulli polynomials, that is, 1 (1.3) ζ(1 − m; x) = − B (x); m 2 m m N Received February 11, 2017; Revised May 8, 2017; Accepted September 5, 2017.
    [Show full text]