View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 190 (2006) 339–353 www.elsevier.com/locate/cam Uniform asymptotic approximations for incomplete Riemann Zeta functions T.M. Dunster∗ Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA Received 14 August 2004; received in revised form 5 November 2004 Dedicated to Roderick Wong on the occasion of his 60th birthday Abstract An incomplete Riemann Zeta function Z1(,x)is examined, along with a complementary incomplete Riemann ={ − 1− }−1 x −1 t + −1 Zeta function Z2( ,x). These functions are defined by Z1( ,x) (1 2 ) ( ) 0 t (e 1) dt and Z2(,x) = () − Z1(,x), where () is the classical Riemann Zeta function. Z1(,x) has the property that limx→∞ Z1(,x)= () for Re > 0 and = 1. The asymptotic behaviour of Z1(,x) and Z2(,x) is studied for the case Re = > 0 fixed and Im = →∞, and using Liouville–Green (WKBJ) analysis, asymptotic approximations are obtained, complete with explicit error bounds, which are uniformly valid for 0x<∞. © 2005 Elsevier B.V. All rights reserved. MSC: Primary 33E20; Secondary 11M06, 34E20 1. Introduction It is well known that the Riemann Zeta function has the integral representation 1 ∞ t−1 = ( ) 1− t dt, (1.1) (1 − 2 )() 0 e + 1 ∗ Tel.: +1 619 594 5968; fax: +1 619 594 2029. E-mail address: [email protected]. 0377-0427/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2004.11.051 340 T.M. Dunster / Journal of Computational and Applied Mathematics 190 (2006) 339–353 which converges and defines () for all Re > 0 (except at the pole = 1). With this definition in mind, we introduce an incomplete Riemann Zeta function 1 x t−1 = = Z1( ,x) 1− t dt(Re > 0, 1), (1.2) (1 − 2 )() 0 e + 1 along with a complementary incomplete Riemann Zeta function 1 ∞ t−1 = = Z2( ,x) 1− t dt( 1). (1.3) (1 − 2 )() x e + 1 In the integrands of (1.2) and (1.3) the principal branch of t is taken. Here and throughout we assume that x is real with 0x<∞, and from now on that is not equal to 1, and satisfies = + i, > 0, 0 < ∞. (1.4) From (1.1)–(1.3) it is evident that for 0x<∞ () = Z1(,x)+ Z2(,x). (1.5) Therefore Z1(,x) and Z2(,x) are linearly independent (as functions of x) if and only if () = 0. The definitions (1.2) and (1.3) are analogous to those of the incomplete and complementary incomplete Gamma functions x − − (,x)= t 1e t dt(Re()>0), (1.6) 0 ∞ − − (,x)= t 1e t dt, (1.7) x which of course satisfy () = (,x)+ (,x). Kölbig [6,7] undertook a numerical study of the trajectory of the zeros of the incomplete Riemann Zeta function (1.2) in the complex plane, with x regarded as a real parameter. However, he remarked that the expansions he employed suffered from numerical instabilities as increased. For instance, in a footnote in [7] Kölbig mentions that it would be interesting to study the zero trajectories near a zero-free Gram interval, the first of these being near = 282, but alluded that he was unable to do so as this range was “far beyond the range of the present calculations.” The purpose of this paper is to study the asymptotics of Z1(,x)and Z2(,x); we shall obtain leading term approximations as →∞, complete with explicit error bounds. In a subsequent paper we will investigate the case where the argument x is complex, as well as extensions to asymptotic expansions. Our approach here will be via a differential equation: specifically, a straightforward differentiation shows that y = Z1(,x)and y = Z2(,x)(along with a trivial solution y = 1) satisfy the second order linear differential equation d2y − 1 − x 1 dy = + . (1.8) dx2 x ex + 1 dx Then, removing the first derivative in a standard manner yields d2w 2 − 1 (1 − )ex ex(ex + 2) = + + w, (1.9) dx2 4x2 2x(ex + 1) 4(ex + 1)2 T.M. Dunster / Journal of Computational and Applied Mathematics 190 (2006) 339–353 341 (1−)/2 x 1/2 with solutions w=x (e +1) y, where y=Z1(,x), y=Z2(,x), as well as y=1. The differential (1.9) will be the main focus of our attention. We next record the following behaviour of the functions at the endpoints of the x interval: x Z (,x)= {1 + O(x)} (x → 0), (1.10) 1 2(1 − 21−)() Z2(,x)= () + O(x )(x→ 0), (1.11) −1 −x Z1(,x)= () + O(x e )(x→∞), (1.12) −1 −x x e − Z (,x)= {1 + O(x 1)} (x →∞). (1.13) 2 (1 − 21−)() Thus, provided () = 0, Z1(,x)is recessive and Z2(,x)is dominant at x = 0, with the roles reversed at x =∞. Consider then the following singular eigenvalue problem: for what values of (with Re > 0) does there exist a solution w = (,x), say, of (1.9) that is recessive at both x = 0 and x =∞? From (1.10)–(1.13) it is clear that when () = 0 no such eigensolution exists. However, when () = 0we have from (1.3) that Z1(,x)=−Z2(,x), and consequently (1−)/2 x 1/2 (1−)/2 x 1/2 (,x)= x (e + 1) Z1(,x)=−x (e + 1) Z2(,x) (1.14) is the unique (to within a multiplicative constant) eigensolution of (1.9) that is recessive at both x = 0 and ∞: the corresponding eigenvalues of this singular boundary value problem are of course the non-trivial zeros of (). The Riemann Hypothesis is therefore equivalent to the above singular eigenvalue value = 1 problem having eigenvalues which all lie on the critical line Re 2 . The importance of the complete Riemann Zeta function () and its zeros is well documented: amongst the many applications in diverse areas, we mention its role in prime number theory, quantum wave physics, and dynamical chaos; see [3]. Whilst there has been an intensive study on the asymptotics of the Riemann Zeta function (for recent results, see [1,2,10,11]), with the exception of [6] and [7] there appear to be few results in the literature (asymptotic or otherwise) for the incomplete Riemann Zeta functions in the above form. In addition to being regarded as eigensolutions of the singular eigenvalue value problem described above, the incomplete Riemann Zeta functions are closely related to several important special functions of mathematical physics and chemistry, specifically Debye [8], Bose-Einstein and Fermi-Dirac functions [4]. We refer the reader to [7] for further details. The plan of this paper is as follows. In Section 2 we derive a Dirichlet-type asymptotic expansion for Z2(,x). This is achieved by expanding the integral (1.3) as a convergent series involving (,x), which in turn is approximated by an established uniform asymptotic approximation. The resulting expansion for Z2(,x) is valid for unbounded x, but is not valid for small x due to the lack of uniformity of the re-expansions in terms of the incomplete Gamma function. In Section 3 we construct Liouville–Green (WKBJ) asymptotic solutions to (1.9), complete with explicit error bounds. In Section 4 we match these Liouville–Green solutions with the exact solutions Z1(,x)and Z2(,x), and by means of a differentiation convert these into a particularly simple form. The resulting approximation for Z1(,x) is uniformly valid for 0x () and the approximation for Z2(,x) is uniformly valid for ()x<∞, where 0 < ()<∞ is defined in terms of the Lambert W function. The connection formula (1.5) extends the approximations to all non-negative x. Finally, in Section 5 we give some numerical calculations for 342 T.M. Dunster / Journal of Computational and Applied Mathematics 190 (2006) 339–353 the relative errors of our new uniform approximations, and we also demonstrate how the stated intervals of validity can be extended for both Z1(,x)and Z2(,x). 2. Dirichlet-type expansions For x > 0 we can expand the denominator of the integrand of (1.2) as the following geometric series x ∞ 1 − − Z (,x)= t 1 e kt dt, (2.1) 1 (1 − 21−)() 0 k=1 and by reversing the integration and summation, and then referring to (1.6), we arrive at the following Dirichlet-type series ∞ 1 + (,kx) Z (,x)= (−1)k 1 (x > 0). (2.2) 1 (1 − 21−) ()k k=1 Likewise, from (1.3) and (1.7), ∞ 1 + (,kx) Z (,x)= (−1)k 1 (x > 0). (2.3) 2 (1 − 21−) ()k k=1 Consider (2.3). From Temme [12] we have the asymptotic approximation √ (,z) 1 e− (,z) 2 1 ∼ erfc( 1/2(,z))+ √ − ( →∞), (2.4) () 2 2 z − 1/2(,z) where (,z)= z − − ln(z/). (2.5) The asymptotic expansion (2.4) is valid for Im = →∞uniformly for 0 <z<∞. We now set z = kx in (2.4) and then insert into (2.3), which immediately gives √ ∞ − (,kx) 1 + 1 e 2 1 Z (,x)∼ (−1)k 1 erfc( 1/2(,kx))+ √ − , 2 2(1 − 21−) k kx− 1/2(,kx) k=1 (2.6) as →∞, uniformly for 0 < x<∞. We can simplify (2.6) by using the following well-known asymptotic behaviour of the complementary error function e− 3 3 erfc( 1/2) ∼ √ →∞, | arg( )| − < . (2.7) 2 0 2 With (2.7) in mind, and recalling that = + i, we have from (2.5) = − 1 + { }+ + 1 + −1 →∞ ( ,kx) kx 2 i ln /(ekx) ln( /(kx)) 2 i O( )( ), (2.8) T.M.