A Generalisation of an Expansion for the Riemann Zeta Function Involving Incomplete Gamma Functions

Transcription

1 Applied Mathematical Science, Vol. 3, 009, no. 60, A Generaliation of an Expanion for the Riemann Zeta Function Involving Incomplete Gamma Function R. B. Pari Diviion of Complex Sytem Univerity of Abertay Dundee, Dundee DD HG, UK R.Pari@abertay.ac.uk Abtract We derive an expanion for the Riemann zeta function ζ() involving incomplete gamma function with their econd argument proportional to n p, where n i the ummation index and p i a poitive integer. The poibility i examined of reducing the number of term below the value N t (t/π) / in the finite main um appearing in aymptotic approximation for ζ() on the critical line = + it a t. It i hown that the expanion correponding to quadratic dependence on n (p = ) i the bet poible repreentation of thi type for ζ(). Mathematic Subject Claification: M06, 33B0, 34E05, 4A60 Keyword: Zeta function; Incomplete gamma function; Aymptotic. Introduction A repreentation of the Riemann zeta function ζ() valid for all value of ( ) i given by [6] ζ() = n Q(, πn η)+χ() n Q(, πn /η)+ξ() (.) where η i a parameter atifying arg η π, Q(a, z) =Γ(a, z)/γ(a) i the normalied incomplete gamma function defined by Q(a, z) = u a e u du ( arg z <π) (.) Γ(a) z

2 974 R. B. Pari and χ() =π Γ( ) Γ, Ξ() = (πη) Γ η. (.3) From the tandard aymptotic behaviour for fixed a [, p. 63] Γ(a, z) z a e z (z in arg z < 3 π), (.4) 4 the late term (n ) in the um in (.) behave like n exp( πn γ), where γ = η for the firt um and γ =/η for the econd um. Both um therefore converge abolutely provided arg η π. The particular cae of thi expanion with η = wa effectively embodied in Riemann 859 paper [, p. 99], though he did not explicitly identify the incomplete gamma function. The reult (.) i the expanion given in [3] for the Dirichlet L- function pecialied to ζ(). To preerve the ymmetry of (.) we take η = and et η = e iφ, with φ π. In numerical computation on the upper half of the critical line = + it (t 0), the term Ξ() ha the controlling behaviour exp{( π φ)t} for large t. When φ< π, thi repreent a numerically large 4 factor which would require the evaluation of the term of (.) to exponentially mall accuracy. Thu, we are effectively forced to et φ = π to avoid uch numerically undeirable term when computing ζ() high up on the critical line. It i cutomary to define the real function Z(t) =e iθ(t) ζ( + it), where Θ(t) = arg {π it Γ( + it)}. From the reult χ( + it) = exp{ iθ(t)}, it 4 then follow from (.) that Z(t) = Re eiθ(t) n Q(, πn i) π 4 e 4 πi Γ, (.5) where in thi expreion = + it, t 0. An aymptotic formula for Z(t) at + baed on (.5) ha been given in [6] by exploiting the uniform aymptotic of the incomplete gamma function. The main characteritic of (.5) i the moothing property of the incomplete gamma function on the original Dirichlet erie ζ() = n in Re() >. Thi can be een from the integral defining Q(a, z) in (.) which, for large a, ha a addle point at u = a. Thi addle point coalece with the lower endpoint of integration when a z, o that the behaviour of Q(a, z) can then be expected to change uddenly. For the incomplete gamma function appearing in (.5), thi occur when πn i; that i, when n attain the There are compenating exponentially large term preent in the incomplete gamma function in (.) when φ< π.

3 Generaliation of an expanion for the Riemann zeta function 975 critical value N t =[ (t/π)], where the quare bracket denote the integer part. More pecifically, the uniform aymptotic expanion of the incomplete gamma function [8] how that for n N t on the critical line [6] Q(, πn i) { ( π t erfc n t ) } e 4 πi, t +, π where erfc denote the complementary error function. Thu for large t, the function Q(, πn i) when n < N t and decay algebraically to zero when n > N t. Looely peaking therefore, the term in the um in (.5) effectively witch off when n N t. To illutrate thi moothing behaviour we how in Fig. a plot of Q(, πn i) a a function of n when = + it with t = 00.. Q n Figure : The behaviour of the normalied incomplete gamma function Q(, πn i) when = + 00i a a function of n (regarded a a continuou variable for clarity). The aymptotic formula that reult from (.5) for Z(t) on the critical line conit of the finite-main um contribution over N t term N t n co{θ t log n} (.6) together with a equence of aymptotic correction term, and o i of the computationally more powerful Riemann-Siegel type [6]. In thi paper we generalie the expanion (.) to examine the poibility of reducing the number of term in the finite-main um by allowing the dependence on the index n in the incomplete gamma function to be proportional to n p, where p i a poitive integer, intead of being quadratic. Thi generaliation will involve the introduction of a generalied incomplete gamma function Q p (a, z) which we dicu in. Our analyi will demontrate that (.) i the bet poible repreentation of thi type for ζ().

4 976 R. B. Pari. A generalied incomplete gamma function Let p be a poitive integer. We define the generalied incomplete gamma function Γ p (a, z), for arg z <π,by Γ p (a, z) = z τ a F p (τ) dτ (p =,, 3,...), (.) where F p (τ) = ( ) k τ k/p k + k!γ(k + )Γ p k=0 = p πi c+ i c i Γ(p) Γ( p)γ ( τ < ) p τ d (c >0); (.) ee [5, p. 56]. When p =, it i een that F (τ) =e τ and Γ (a, z) reduce to the tandard incomplete gamma function Γ(a, z). Henceforth we exclude the cae p = from our deliberation. The function F p (τ) i related to the generalied hypergeometric function (or the Wright function) Ψ m (z) = k=0 z k k!γ(k + )Γ k + m ( z < ; m>). The aymptotic of thi function for large z can be obtained from the aymptotic theory of hypergeometric function; ee, for example, [5,.3]. We define the parameter κ = ( m ), h =(/m)/m, ϑ = m, X = κ(hz)/κ (.3) and note that κ>0 for m>. The expanion of Ψ m (z) conit of an exponentially large expanion together with an algebraic expanion in the ector arg z πκ given by Ψ m (z) E(z)+H(ze πi ) ( z in arg z πκ), where the upper or lower ign i choen according a arg z>0or arg z<0, repectively. The expanion E(z) and H(z) denote the formal aymptotic um E(z) :=X ϑ e X A j X j, A 0 = κ ( mκ) ϑ, (.4) j=0 The choice of the ign in the algebraic expanion i a conequence of arg z = 0 being a Stoke line for Ψ m (z).

5 Generaliation of an expanion for the Riemann zeta function 977 with the A j (j ) being computable coefficient that we do not dicu here, and H(z) := m ( ) k Γ( mk + ) k=0 k! Γ( z k. mk) In the ector arg( z) < π( κ), the expanion E(z) become exponentially mall and the dominant expanion of Ψ m (z) in thi ector i then H(ze πi ). If m =p then H(z) 0. In thi cae the expanion of Ψ m (z) i exponentially mall in the ector arg( z) < π( κ) and we find [0] E(z) ( arg z <π) Ψ m (z) (.5) E(z)+E(ze πi ) ( arg z π), where the upper or lower ign i choen according a arg z>0 or arg z< 0, repectively. From (.5) and the fact that F p (τ) =Ψ p ( τ /p ), it then follow that F p (τ) i exponentially mall for large poitive τ with the leading behaviour F p (τ) A 0 T ϑ e T co(π/κ) co{t in(π/κ)+πϑ /κ} (.6) a τ + for p, where T = κ(τ/p) /κp and the parameter κ, ϑ and A 0 are given by (.3) and (.4) with m =p. Since <κ<, thi reult how that the integrand in (.) decay exponentially for large τ and hence that, provided z 0,Γ p (a, z) i defined without retriction on the parameter a. The aymptotic expanion of Γ p (a, z) for large z ha been dicued in [4] where it wa hown to conit of p exponential expanion, all of which are exponentially mall (of different degree of ubdominance) in the ector arg z < π. From Eq. (.0) of thi reference, the dominant behaviour3 i given by z a Γ p (a, z) B 0 Z ϑ e Z co(π/κ) co{z in(π/κ)+πϑ/κ}, (.7) for z in arg z π and p, where Z = κ(z/p) /κp, ϑ = 3p p, B 0 = κ (κp) ϑ. A z on the ray arg z = ± π, the dominant behaviour of Γ p(a, z) i therefore algebraic of O(z a+ϑ/κp ). The normalied incomplete gamma function i defined by Q p (a, z) =Γ p (a, z)/λ p (a), (.8) 3 The dominant expanion in [4, Eq. (.0)] correpond to r = N λ and r = N, where N =[ p] and λ =0(p odd), (p even).

6 978 R. B. Pari where the normaliing factor Λ p (a) i pecified by the requirement that Q p (a, 0) = (when Re(a) > 0). Thi yield Λ p (a) = 0 τ a F p (τ) dτ, Re(a) > 0, o that Λ p (a) can be conidered a the Mellin tranform of F p (τ). It then follow upon inverion that F p (τ) = c+ i Λ p ()τ d (c >0). πi c i Comparion with (.) then how that Λ p (a) = which correctly reduce to Γ(a) when p =. pγ(ap) Γ( ap)γ p a, (.9) We have the identity 3. A generalied expanion for ζ() P (a, z)+q(a, z) =, where P (a, z) = γ(a, z)/γ(a) i the complementary normalied incomplete gamma function defined by P (a, z) = z u a e u du (Re(a) > 0). Γ(a) 0 We chooe the parameter a = /m and the variable z = π m/ n m η, where m> i (for the moment) arbitrary and η i a complex parameter atifying φ = arg η < π. Then, from the Dirichlet erie repreentation for ζ() valid in Re() > we find ζ() = n = = n P } n {P m, (π n) m η + Q m, (π n) m η ( m, (π n) m η ) + n Q m, (π n) m η. (3.) From (.4), the late term (n ) in the econd um in (3.) involving Q poe the behaviour n m exp{ (π n) m η} o that thi um converge abolutely for all when φ π.

7 Generaliation of an expanion for the Riemann zeta function 979 For Re() >, the firt um in (3.) can be written a S = π Γ( m ) n P m, (π n) m η = π (π n n) m η u (/m) e u du 0 η = x (/m) ψ m (x) dx, ψ m (x) = exp{ (π n) m x} (3.) 0 upon reveral of the order of ummation and integration. In [5, 8.] it i hown that, for arbitrary m> and Re(x) > 0, the function ψ m (x) atifie the relation { } ψ m (x)+= x /m F m (π m/ n m /x)+π Γ( m m ), (3.3) where F m (u) denote the generalied hypergeometric function F m (u) = k=0 ( ) k u k/m k!γ(k + )Γ k + m ( u < ; m>). When m =,F (u) =e u and (3.3) reduce to the well-known Poion ummation formula [9, p. 4] ψ (x)+=x {ψ (/x)+} relating the behaviour of ψ (x) to that of ψ (/x) when Re(x) > 0. It then follow from (3.) and (3.3) that, provided φ < π and Re() >, { S = η /m π )η /m Γ( m m } + τ ( )/m F m (π m/ n m τ) dτ. m /η Now we et m = p, where p i a poitive integer, o that from (.6) F p (π p n p τ) i exponentially mall a τ +, and define the quantity Ξ p () = π η /p Γ( p ) { π )η /p Γ( p p } Then we obtain upon reveral of the order of ummation and integration (jutified by abolute convergence) and with the change of variable u =(πn ) p τ S = π Γ( p )Ξ p()+ π p = π Γ( p )Ξ p()+ π p n u ( )/p F p (u) du (πn ) p /η n Γ p p, (πn ) p /η,.

8 980 R. B. Pari where Γ p (a, z) i the generalied incomplete gamma function defined in (.). It then follow that n P p, (πn ) p η =Ξ p ()+ π pγ( p ) n Γ p p, (πn ) p /η =Ξ p ()+χ() n Q p p, (πn ) p /η, (3.4) where Q p (a, z) =Γ p (a, z)/λ p (a) i the normalied incomplete gamma function and, from (.9), we have ued the fact that π pγ( )Λ p = χ(). p p Then, from (3.) and (3.4), we finally obtain the deired generalied expanion given by ζ() = n Q p, (πn ) p η + χ() n Q p p, (πn ) p /η +Ξ p () (3.5) valid for Re() >, φ < π and integer p. From the aymptotic behaviour of Γ p (a, z) in (.7) we have ( ) Γ p p, (πn ) p /η = O n (/κ) exp{an /κ co ω} (3.6) a n and p, where A = κ(πh η /p ) /κ, ω = πp φ κp = π + π φ κp and the parameter κ and h are defined in (.3) with m =p. The upper or lower ign in ω i choen according a φ > 0orφ < 0, repectively. Since π<ω< 3π when p and φ < π, the econd um in (3.5) i een to be abolutely convergent for all. When φ = ± π, we have ω = π and the exponential in (3.6) i ocillatory; the late term in the econd um in (3.5) then behave like n /κ a n and o thi latter um therefore converge abolutely for all when φ π. Since the firt um in (3.5) ha been hown to converge abolutely for all when φ π, it follow by analytic continuation that the expanion (3.5) hold for all ( ) and arg η π. When p = it i een that (3.5) reduce to (.).

9 Generaliation of an expanion for the Riemann zeta function Dicuion Initially we put η = in (3.5) and let η = e iφ. The late term in the firt um in (3.5) then behave ultimately like n p exp{ (πn ) p co φ}. A explained in, we are forced to et φ = arg η = π on the critical line = + it (t 0); thee term then poe the algebraic decay n p. But more importantly the cut-off in Q(/p, (πn ) p i) now occur when /p π p n p i; that i, when n = n where t /p n (pπp ) /p. (4.) π Thu the firt um in (3.5) witche off more rapidly when p with the term in the tail of thi um decaying more rapidly. The decay of the generalied incomplete gamma function in (3.5) i imilarly no longer exponential, but algebraic given by n /κ, when φ = π. In Fig. we how a plot of Q p (( )/p, (πn ) p )i) when p = and = + it with t = 50, which clearly how that thi function alo exhibit a cut-off tructure. We note that a convenient way of numerically computing Q p (a, z). Q p n Figure : The behaviour of the generalied incomplete gamma function Q p (a, z), where a =( )/p, z = i(πn ) p, for p = and = +50i a a function of n (regarded a a continuou variable for clarity). The dahed curve denote the aymptotic approximation in (.7) and (.8). when Re(a) > 0 follow from the reult Q p (a, z) = z τ a F p (τ) dτ Λ p (a) 0 = pza ( ) k z k/p Γ((k + )/p) Λ p (a) k!γ(k + ), (4.) k + ap k=0 obtained by ubtitution of the erie expanion for F p (τ) in (.) followed by term-by-term integration. The late term in the econd um in (3.5) therefore

10 98 R. B. Pari behave like n /κ, o that when p, the convergence of thi um i weakened: the late term behave like n when p = and approach the behaviour n 3/ for large p. Of greater importance, however, i the location of the tranition point of the function Q p (a, z) for large value of a and z, which determine the cutoff point in the econd um in (3.5). Thi correpond to the addle point of the integrand in (.). From the firt equation in (.5) combined with F p (u) =Ψ p ( u /p ), the leading aymptotic behaviour of F p (u) i decribed by the ingle term F p (u) Cu ϑ/κp exp{κ(u/p) /κp e πi/κ } a u in the ector arg(e πi u /p ) <π, where C i a contant that we do not need to pecify here. For large a, the location of the addle point u i then controlled by the term u a exp{κ(u/p) /κp e πi/κ } in the integrand and o i given by u = p(ap) κp e πi(p ). When a =( )/p, thi correpond approximately to u ip(t/) κp for large poitive t on the critical line. Hence the behaviour of Q p (( )/p, (πn ) p i) can be expected to change from ocillating about the value unity to decaying to zero when u z = (πn ) p i; that i, when n ha the value t κ/ n (pπp ) /p. (4.3) π Thu the cut-off in the econd um in (3.5) occur for an n value caling like t κ/. Since <κ<, thi indicate that the number of term from thi um that will make a contribution to the finite main um (.6) will increae with increaing p. We oberve that the introduction of a more rapid growth in the argument of the incomplete gamma function ha reulted in a certain aymmetry in the expanion (3.5): the firt um effectively witche off after n N t term wherea the econd um witche off after n N t term. Thi imbalance can be retored if we take η. Thu, if we et η = i/k, where K denote a poitive contant, it i eaily een that the correponding cut-off value in (4.) and (4.3) now become and t /p n (pπ p ) /p K /p π t κ/ n (pπ p ) /p K /p. π

11 Generaliation of an expanion for the Riemann zeta function 983 By appropriate choice of K it i then poible to increae n and decreae n. The maximum amount by which we can meaningfully decreae n occur when n n. Thi i readily een to arie when K ha the particular caling with t given by K = p(t/) p, whereupon n n (t/π)/ N t. Thu, it doe not appear poible to imultaneouly reduce both n and n below the value N t, thereby howing that the repreentation (.) i optimal in thi ene. Finally, we remark that a repreentation for ζ( + it) involving the the original Dirichlet erie n moothed by a imple Gauian exponential exp{ (n/n) } ha been given in [7]. Thi wa hown to reult in a computationally le powerful Gram-type expanion ince the index N had to be choen to atify N > t/(π) for the correction term to poe an aymptotic character. Reference [] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Function, Dover, New York, 965. [] H. M. Edward, Riemann Zeta Function, Academic Pre, New York, 974. [3] A. F. Lavrik, An approximate functional equation for the Dirichlet L- function, Tran. Mocow Math. Soc. 8 (968), 0 5. [4] R. B. Pari, The aymptotic expanion of a generalied incomplete gamma function, J. Comput. Appl. Math. 5 (003), [5] R. B. Pari and D. Kaminki, Aymptotic and Mellin-Barne Integral, Cambridge Univerity Pre, Cambridge, 00. [6] R. B. Pari and S. Cang, An aymptotic repreentation for ζ( + it), Method Appl. Anal. 4 (997), [7] R. B. Pari and S. Cang, An exponentially-moothed Gram-type formula for the Riemann zeta function, Method Appl. Anal. 4 (997), [8] N. M. Temme, The aymptotic expanion of the incomplete gamma function, SIAM J. Math. Anal. 0 (979), [9] E. T. Whittaker and G. N. Waton, Modern Analyi, Cambridge Univerity Pre, Cambridge, 95.

12 984 R. B. Pari [0] E. M. Wright, The aymptotic expanion of the generalied hypergeometric function, Proc. Lond. Math. Soc. () 46 (940), Received: April, 009