THE RIEMANN ZETA FUNCTION (s; a) := n= (n + a) s (R(s) > ; a 6= ; ; ; :::) ; (.) can indeed be extended meromorphically to the whole complex s-plane e

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1 SOME DEFINITE INTEGRALS ASSOCIATED WITH THE RIEMANN ZETA FUNCTION H.M. Srivastava Department of Mathematics Statistics University of Victoria Victoria, British Columbia V8W P4, Canada M.L. Glasser Department ofphysics Clarkson University Potsdam, New York , U.S.A. Victor S. Adamchik Department of Applied Computer Science Illinois State University Normal, Illinois , U.S.A. Abstract The authors aim at deriving a family of series representations for (n + )(n N) by evaluating certain trigonometric integrals in several different ways. They also show how the results presented in this paper relate to those that were obtained in other works. Finally, some illustrative computational examples, using Mathematica (Version 4.) for Linux, are considered.. Introduction The Riemann Zeta function (s) the Hurwitz (generalized) Zeta function (s; a), which are defined usually by (s) := 8 >< >: P n= n = s s P s n= P n= () n (n ) s (R(s) > ) n s (R(s) > ; s 6= ) Mathematics Subject Classification. Primary M6; Secondary M5, 4A. Key words phrases. Zeta functions, trigonometric integrals, Clausen functions, series representations, trigonometric sums, Mellin transforms.. (.)

2 THE RIEMANN ZETA FUNCTION (s; a) := n= (n + a) s (R(s) > ; a 6= ; ; ; :::) ; (.) can indeed be extended meromorphically to the whole complex s-plane except for a simple pole at s = with residue (see, for details, Titchmarsh []). It is easily seen from (.) (.) that (s; ) = (s) =( s ) s; (s; ) = (s) : (.) In recent years there has been a renewed interest in representing (n +)(n N), N being the set of positive integers, by menas of series which converge more rapidly than the defining series in (.). These developments seem to have stemmed from the use of the familiar series representation (see, e.g., Hjortnaes [4] Gosper []): () = 5 () k (.4) k k k in Apéry's proof [] of the irrationality of (), as well as from Ewell's yet another rediscovery [8] of Euler's formula (see, e.g., Ayoub [, pp ]; see also Ramaswami [9] Srivastava [, p. 7, Equation (.)]): () = 4ß 7 : (.5) (k + )(k + )k The works of (among others) Ewell [9], Dοabrowski [7], Zhang Williams [4], Cvijović Klinowski [6], Srivastava ([] []) may be cited in connection with the aforementioned developments. The main object of this paper is to derive a family of series representations for (n +)(n N) by evaluating certain trigonometric integrals in several different ways to show how the results presented here relate to those that were obtained in other works. We also consider some illustrative computational examples by using Mathematica (Version 4.) for Linux.. A Cosecant Integral Its Consequences We begin by considering the cosecant integral: I s () := Z ß= which, upon integration by parts, readily yields s I s () = cot t s csc tdt (R(s) > ; >) ; (.) (R(s) > ; >) : Z ß= + s t s cot tdt (.)

3 THE RIEMANN ZETA FUNCTION The cotangent integral in (.) can, in fact, be evaluated in several different ways. First of all, since (cf., e.g., Magnus et al. [7, p. 6]) z k z cot z = ß (jzj <ß) ; (.) we find from (.) that I s () = s cot s s (R(s) > ; >) : (s +k ) k (.4) In its special case when =, the integral formula (.4) reduces immediately to the form: Z ß= t s csc tdt= s s (R(s) > ) ; (s +k ) k (.5) which, for s N nfg, appears in the works of (for example) Glasser [, p. 446, Equation ()] Prudnikov et al. [8, p. 88, Entry (.5.4.6)] (see also Gradshteyn Ryzhik [, p. 48, Entry.748()]). Furthermore, if we set = 4 in (.4), we shall readily obtain the special case: Z ß=4 t s csc ß s s tdt= s 4 4 (.6) (s +k )4 k (R(s) > ) ; which, for s N nfg, is recorded (among other places) in Prudnikov et al. [8, p. 88, Entry (.5.4.)] (see also Gradshteyn Ryzhik [, p. 48, Entry.748()]). The integral formula (.4) also simplifies when = = 6, giving us the following special cases: Z ß= t s csc tdt= p s s s (s +k ) k (.7) Z ß=6 (R(s) > ) t s csc tdt= p s s s 6 6 (s +k )6 k (.8) (R(s) > ) ; which do not seem to have been recorded earlier. Next, in terms of the incomplete Gamma function fl(z; ff) defined by fl(z; ff) := Z ff t z e t dt (.9)

4 4 THE RIEMANN ZETA FUNCTION it is easily seen that Thus, by writing (R(z) > ; jarg(ff)j 5 ß ; <<ß) ; Z fi t e μt dt = μ fl(; μfi) (R() > ) : (.) cot t = i + i p := e it (.) making use of (.), it is not difficult to deduce from (.) that s h i fl(s; kßi=) I s () = cot + i is (.) (ki) s (R(s) > ; >) : Setting s =n (n N) noting that (cf., e.g., [, p. 94, Entry 8.5()] fl(m e ff (.) yields the integral formula: I n () = n + + n n cot m j= ff j j (n) (n j ) ß (n j) j ß j A (m N := N [fg) ; (.) n (n N; >) ; (n ) cos(kß=) k j+ cos(kß=) sin(kß=) 5 (.4) k j it being assumed here, throughout this paper, that an empty sum is (as usual) nil. On the other h, if we set s =n +(n N) in (.) apply the reduction formula (.), we shall obtain n+ I n+ () = cot n (n + ) +() n (n +) n (n + ) 4 cos (kß=)) n () j j + (n) k (n j) ß cos(kß=) k j+ + n (n j + ) ß (n N; >): j k sin(kß=) 5 (.5) k j

5 Since (cf., e.g., [, p. 9, Entry 7..5)] with y =)» sin THE RIEMANN ZETA FUNCTION 5 cos(kx) = log k x ( <x<ß); (.6) in terms of the generalized Clausen functions C`n C`n+ defined by (cf. Lewin [6, p. 9, Equations (7.9) (7.)]) C`n (x) := sin(kx) k n C`n+ (x) := it is easily seen from our evaluations (.4), (.4), (.5) that h i (k +n ) = k log sin + + (n ) n (n) =()n (k +n) k n+ ß + (n) n 4 n (n j) (n N; >) (n j ) ß ß j C`j ß n (n +) log h sin (n j + ) ß cos(kx) k n+ (n N); (.7) j ß C`j+ 5 (.8) i j» (n j + )C`j+ ß + ß C`j ß (.9) (n N; >): In its special case when n =, (.8) would readily yield a known result [, p. 56, Entry (54.5.4)] in the form: (k)) h i (k +) = k log sin ß 4ß C` ( >): (.) Moreover, since [6, pp. -4, Equations (4.4) (4.5)] C` ß = G = C` ß ; (.)

6 6 THE RIEMANN ZETA FUNCTION where G denotes Catalan's constant defined by G := by setting = 4 = 4 Z () k ß= (k +) =» t log sin dt ο= : :::; (.) in (.), we obtain its further special cases: (k + )4 = G k ß log (.) 4 k + in terms of the Catalan constant G defined by (.). k = G 4 ß log (.4) 4. Series Representations for (n +) In every situation in which the Clausen functions C`n (x) C`n+ (x) can be expressed in closed forms, each of our formulas (.8) (.9) will readily yield a series representation for (n + )(n N): We begin by considering the following rather simple special case. Case. Let =. It immediately follows from the definitions (.) (.7) that C`n (ß) = C`n+ (ß) = n (n +) (n N): (.) Thus the formulas (.8) (.9) yield the series representations: (n +)=() n (ß) n (n + ) ( n ) n +(n + ) (n +)=() n (ß) n (n) ( n+ ) log + (n j + ) log + n +(n) (k +n + ) k k) (k + n) k (n j) j (ß) j j (ß) j (j +) 5 (n N) (.) (j +) 5 (n N); (.) respectively. For n =, (.) immediately reduces to the following series representation for (): () = ß 9 ψ log + (k + ) k ; (.4)

7 THE RIEMANN ZETA FUNCTION 7 which was proven independently by (among others) Glasser [, p. 446, Equation ()], Zhang Williams [4, p. 585, Equation (.)], Dοabrowski [7, p. 6] (see also Chen Srivastava [5, p. 8, Equation (.5)]). And a special case of (.) when n = yields (cf. Dοabrowski [7, p. ]; see also Chen Srivastava [5, p. 9, Equation (.9)]) ψ () = ß log + 7 (k + ) k : (.5) In view of the known sum: (k + ) = log k ; (.6) which incidentally results also from (.8) in the special case when = n =, Euler's formula (.5) is indeed a simple consequence of (.5). We remark in passing that an integral representation for (n +); which is easily seen to be equivalent to the series representation (.), was given by Dοabrowski [7, p., Equation (6)], who [7, p. 6] mentioned the existence of (but did not fully state) the series representation (.) as well. The series representation (.) is derived also in a forthcoming paper by Borwein et al. (cf. [4, Equation (57)]). The generalized Clausen function C`n+ (x) defined in (.7) is known to be expressible in a closed form in at least three other cases, we have (cf. Lewin [6, p. 98]) C`n+ ß = n n (n +) (n N); (.7) C`n+ ß = n n (n +) (n N); (.8) C`n+ ß = n (n +) (n N): (.9) With a view to evaluating the generalized Clausen function C`n (x), also defined in (.7), when x = ß; ß; ß; we recall the following result recorded (for example) by Hansen [, p., Entry (4.4.)]:» sin(kx + y) = (ß)s k s (s) csc(ßs) x cos y ßs s; ß cos y + ßs s; x ß (R(s) > ; <x<ß) ; which, for (.) x = ß ( >) y =;

8 8 THE RIEMANN ZETA FUNCTION reduces at once to the form: sin(kß=) k s = (ß)s 4(s) csc ßs» s; s; (.) Now, making use of the identity: (s) = q q s (R(s) > ; >) : s; j q (q N nfg) ; (.) which follows readily from the definitions (.) (.), the Rademacher formula (cf., e.g., Magnus et al. [7, p. ]): = ( s)(qß) s 4 ßs q pjß sin cos s; p q + cos ßs q q pjß sin q s; j q s; j 5 (p; q N); (.) q as well as the familiar special case of (.) when p = q =: ßs (s) = (ß) s sin ( s) ( s) (.4) or, equivalently, ßs ( s) = (ß) s cos (s) (s); (.5) we find from (.) the relevant definition in (.7) that C`n ß = 4n n; 4 ß C`n = p ρ»6 n n; + n; 6 ß C`n = p» n n; n (n) (n N); (.6) ff n n (n) (n) (n N); (.7) (n N); (.8) in which each (n) can be replaced by its value in terms of the Bernoulli numbers B n by appealing to the well-known relationship (cf., e.g., [7, p. 9]): (n) =() n+ (ß)n (n) B n (n N) : (.9)

9 THE RIEMANN ZETA FUNCTION 9 Much more general trigonometric sums than the ones involved in (.7) to (.9) (.6) to (.8) can also be evaluated by applying the aforementioned technique or (alternatively) by appealing appropriately to the elementary series identity: f(k) = its straightforward consequence (.). (.7) in the form: cos( kß) q f(qk + j) (q N) (.) Thus, for example, we can derive a generalization of k s = s s (s) (R(s) > ) ; (.) which immediately yields (.7) in the special case when s = n + (n N). Analogous generalizations of the remaining evaluations of the Clausen functions C`n+ (x) C`n (x) can be derived similarly; we choose to skip the details involved. In view of the evaluations (.7) to (.9) (.6) to (.8), we are led easily to the following additional special cases of our general results (.8) (.9). Case. Let =. Then, making use of (.9) (.8) in each of our formulas (.8) (.9), we obtain the series representations: (n +)=() n (ß) n (n + ) ( n ) n +(n + ) n+ (n + ) p (n j + ) (n j + ) log + 4 j (ß) j (j +) (k +n + ) k j; j (j) 5 (n N) (ß) j (.) (n +)=() n (ß) n (n) ( n+ ) n +(n) (n) p n (n j) (n j + ) log + j (ß) j (j +) (k) (k + n) k j; j (j) (ß) j 5 (n N) : (.)

10 THE RIEMANN ZETA FUNCTION In particular, when n =, (.) (.) yield the following (presumably new) series representations for () : () = ß log + 4 (k + ) k + p () = ß log + j; j (j) (.4) (4 j) (ß) j ρ ff# (k + ) + p k ; 4 () : (.5) ß Case. Let =4. Then, by applying the evaluations (.7) (.6) in each of our formulas (.8) (.9), we obtain the series representations: (n +)=() n (ß) n (n + ) ( n ) n +(n + ) n+ (n + ) (n j + ) (n j + ) (n +)=() n (ß) n (n) ( 4n+ + n ) n +(n) (n) n (n j) (n j + ) log + 4 j (ß) j (j +) (k +n + )4 k j; 4 j j (j) 5 (n N) (ß) j (.6) log + j (ß) j (j +) (k + n)4 k j; 4 j j (j) 5 (n N): (ß) j (.7) In their special cases when n =, (.6) (.7) yield the following (presumably new) series representations for (): () = ß 9 +6 log + 4 (4 j) k) (k + )4 k 5 (ß) j (.8) j; 4 j j (j)

11 () = ß 5 THE RIEMANN ZETA FUNCTION log + (k + )4 k + ß ρ ; 4 ff# 6 () : (.9) Case 4. Let =6. Then, in view of the evaluations (.8) (.7), our formulas (.8) (.9) give us the series representations: (n +)=() n (ß) n ( n ) ( n ) n + + p n+ (n j + ) 4 (n + ) ψ j j (ß) j (k +n + )6 k (j +) j; + j; 6 j j (j) (n j + ) 5 (ß) j (.) (n +)=() n (ß) n n + n +6 n n p (n j) n (n N) (n) ψ j j (ß) j (k + n)6 k (j +) j; + j; 6 j j (j) (n j + ) 5 (ß) j (.) (n N) For n =, these last results (.) (.) yield the following (presumably new) series representations for (): () = ß 8 + p (k + )6 k j; + j; 6 j j (j) (4 j) 5 (ß) j (.) ρ ff# () = ß (k + )6 + p k ; + ; 6 () : (.) ß 6

12 THE RIEMANN ZETA FUNCTION 4. An Alternative Derivation of the Series Representation (.) In view of the following known representation of (s) as a Mellin transform [7, p. ]: (s) = s ( s )(s +) Z t s sech tdt (R(s) > ; s 6= ); (4.) we propose to give here an interesting alternative derivation of the very first of the set of eight series representations for (n + )(n N), which we have presented in the preceding section. Indeed, from the work of Glasser [, p. 445, Equation (4) with ff ], it is known also that Z tan Z z n ß= I := z dz = t n csc tdt =() n n I ρz (log u iß) n (u +) du ff (n N): (4.) Now evaluate the cosecant integral in (4.) by means of the known result (.5) with s =n (n N). Thus, setting log u = t, we find from the last member of (4.) that ()n n (Z ) = (k +n )k 4n ß I t + iß n sech tdt (n N): By the binomial expansion, we have t + n iß n n = t iß nk k k n n n nj n = () nj t j i () nj j j + j= (n N) ; j= t j+ nj where, as also elsewhere in this work, an empty sum in interpreted to be nil. Upon replacing n in (4.) by n +, making use of (4.4), we obtain (k +n + ) = n n + j k j + j ß j= Z (4.4) t j+ sech tdt (n N) : (4.5) The infinite integral in (4.5) can be evaluated by means of the known result (4.) except when j =, in which case it is easily seen by integrating by parts that (cf. [, p. 5, Entry.57(4)]) Z We thus find from (4.5) that (k +n + ) k = t sech tdt= t lim (t tanh t log cosh t) = log : (4.6) n n + j (j) j (ß) j (j +) log (n N ) : (4.7)

13 THE RIEMANN ZETA FUNCTION For n =, (4.7) immediately yields the known sum (.6), which (as we observed in the proceding section) is derivable also from (.8) for = n =. More interestingly, just as we indicated in Section, since (k + )(k +) = k + k + ; (4.8) by suitably combining the series representation (.5) with (.6), we can easily deduce Euler's formula (.5). In the case when n N, by separating the term for j = n in (4.7), we readily obtain the desired series representation (.) in its (obviously equivalent) form: (n +)=() n (ß) n (n + ) ( n ) n + n + j log + (j) j (k +n + ) k (ß) j (j +) 5 (n N) : (4.9) 5. A Unification of the Series Evaluations (.8) (.9) With a view to unifying the series evaluations (.8) (.9), we begin by considering the formulas (.) (.4), which readily yield the following representation: Z S p := (k + p) = ß ßt k t p cot dt (5.) (p N; jj > ) ; where, for the purpose of this section, we have only considered a special case when s = p+(p N). Now change the variable of integration in (5.) by letting Since we thus find from (5.) that S p = Z z (log t) p t t = i log z: ß dt = zp+ p + ßi (p +) + i p Z Ω ß (p N) ; (5.) (log z) p dz (5.) z where, for convenience, Finally, by setting (p N; j > ) ; ßi Ω:=exp z = ( Ω)t (jj > ) : (5.4)

14 4 THE RIEMANN ZETA FUNCTION in (5.), we obtain or, equivalently, S p = ßi (p +) i p Z flog ( ( Ω)t)g p dt ß t (k + p) = ßi k (p +) p p i S ;p e ßi= in terms of Nielsen's generalized Polylogarithmic function S n;p (z) defined by (cf., e.g., Kölbig [5, p., Equation (.)]) S n;p (z) := ()n+p (n )p Z ß (log t) n flog ( zt)g p dt (n; p N; z C ) ; which is known to play ar^ole in the computation of higher-order radiative corrections in quantum electrodynamics. For the ordinary Polylogarithmic function Li n (z) defined by Li n (z) := ()n (n ) Z (log t) n log( zt) dt (n N nfg; z C ) t (5.5) (5.6) (5.7) t = S n;(z) (5.8) or, more generally, by Li s (z) := z k k s (5.9) (s C when jzj < ; R(s) > when jzj =); it is known that (cf. [5, p. 4, Equation (5.6)]) S ;p (z) = (p +)+ p () k flog( z)g k Li pk+( z): (5.) k Thus, in view of (5.), we have proved the following unification of our series evaluations (.8) (.9): (k + p) k = ßi (p +) log e ßi= p i ß p (p +)+ p (p N; jj > ) ; p k i k ß k Li k e ßi= (5.)

15 THE RIEMANN ZETA FUNCTION 5 which, in view of (5.), immediately yields the integral formula: Z t p cot(νt)dt = i p + + νi ν log e + p p i p (p +) p i k ν ν ν k ν k Li k+ e νi (5.) (p N; ν C nfg) : There is yet another approach for the derivation of a closed-form expression for the infinite series in (5.). This general (generating-function) approach is based upon a familiar integral representation for the Zeta function; it was fully described illustrated by Adamchik Srivastava [], whose technique is also implemented in Mathematica (Version.). Here we choose to present three examples which demonstrate how Mathematica (Version 4.) for Linux can be used to evaluate the infinite series in (5.) for certain values of the parameters p. h i In[] := Sum Zeta[k]=((k + ) q (k)); fk; ; Infinityg Out[] = Log[] h In[] := Sum Zeta[k]=((k + ) q (k)); fk; ; Infinityg Out[] = Pi Pi Log[] + 7Zeta[] 4Pi In[] := Sum h Zeta[k]=((k + ) q (k)); fk; ; Infinityg Out[] = Pi 6Pi Log[] + 7Zeta[] Pi Since () = ; these examples completely agree with the special cases (.6), (.5), (.4), respectively, which have already been shown here to follow from our general series representations considered in Section. Acknowledgments The present investigation was supported, in part, by the Natural Sciences Engineering Research Council of Canada under Grant OGP75. References i i [] V.S. Adamchik H.M. Srivastava, Some series of the Zeta related functions, Analysis 8(998), -44. [] R. Apéry, Irrationalité de () et (), in Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, 978), pp. -, Astérisque 6, Soc. Math. France, Paris, 979. [] R. Ayoub, Euler the Zeta function, Amer. Math. Monthly 8(974), [4] J.M. Borwein, D.M. Bradley, R.E. Crall, Computational strategies for the Riemann Zeta function, J. Comput. Appl. Math. (to appear). [5] M.-P. Chen H.M. Srivastava, Some families of series representations for the Riemann (), Resultate Math. (998), [6] D. Cvijović J. Klinowski, New rapidly convergent series representations for (n + ); Proc. Amer. Math. Soc. 5(997), 6-7.

16 6 THE RIEMANN ZETA FUNCTION [7] A. Dοabrowski, A note on the values of the Riemann Zeta function at positive odd integers, Nieuw Arch. Wisk. (4) 4(996), [8] J.A. Ewell, A new series representation for (), Amer. Math. Monthly 97(99), 9-. [9] J.A. Ewell, On the Zeta function values (k + ); k = ; ; :::; Rocky Mountain J. Math. 5(995), -. [] M.L. Glasser, Some integrals of the arctangent function, Math. Comput. (968), [] R.W. Gosper, Jr., A calculus of series rearrangements, in Algorithms Complexity: New Directions Recent Results (J. F. Traub, Editor), pp. -5, Academic Press, New York London, 976. [] I.S. Gradshteyn I.M. Ryzhik, Table of Integrals, Series, Products (Corrected Enlarged Edition Prepared by A. Jeffrey), Academic Press, New York, London, Toronto, 98. [] E.R. Hansen, A Table of Series Products, Prentice-Hall, Englewood Cliffs, New Jersey, 975. [4] M.M. Hjortnaes, Overffiring av rekken P =k til et bestemt integral, Proceedings of the Twelfth Scanavian Mathematical Congress (Lund; August -5, 95), pp. -, Scanavian Mathematical Society, Lund, 954. [5] K.S. Kölbig, Nielsen's generalized Polylogarithms, SIAM J. Math. Anal. 7(986), -58. [6] L. Lewin, Polylogarithms Associated Functions, Elsevier (North Holl) Science Publishers, New York, London, Amsterdam, 98. [7] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas Theorems for the Special Functions of Mathematical Physics, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtingung der Anwendungsgebiete, Bd. 5, Springer-Verlag, New York, 966. [8] A.P. Prudnikov, Yu. A. Brychkov, O.I. Marichev, Integrals Series: Elementary Functions, Nauka, Moscow, 98 (in Russian). [9] V. Ramaswami, Notes on Riemann's -function, J. London Math. Soc. 9(94), [] H.M. Srivastava, A unified presentation of certain classes of series of the Riemann Zeta function, Riv. Mat. Univ. Parma (4) 4(988), -. [] H.M. Srivastava, Further series representations for (n + ), Appl. Math. Comput. 97(998), -5. [] H.M. Srivastava, Some rapidly converging series for (n + ), Proc. Amer. Math. Soc. 7(999), [] E.C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University (Clarendon) Press, Oxford London, 95; Second Edition (Revised by D.R. Heath-Brown), 986. [4] N.-Y. Zhang K.S. Williams, Some series representations of (n + ), Rocky Mountain J. Math. (99),