-1-
Uniform asymptotic expansions for incomplete Riemann Zeta functions
T. M. Dunster
Department of Mathematics and Statistics
San Diego State University
San Diego, CA 92182-7720
U.S.A.
Email: dunster@math.sdsu.edu
Tel. (619) 594 5968
Fax. (619) 594 6746
Abbreviated title: Incomplete Riemann Zeta functions
Dedicated to Roderick Wong on occasion of his 60th birthday.
-2-
Abstract
An incomplete Riemann Zeta function Ζ 1 (α , x ) is examined, along with a complementary
incomplete
Riemann
{(
)
Zeta
}
Ζ 1 (α , x ) = 1 − 21−α Γ(α )
function
−1 x
tα − 1
0
∫
(e t + 1)
Ζ 2 (α , x ) .
−1
These
functions
are
defined
by
d t and Ζ 2 (α , x ) = ζ (α ) − Ζ 1 (α , x ) , where ζ (α ) is
the classical Riemann Zeta function. Ζ 1 (α , x ) has the property that lim x → ∞ Ζ 1 (α , x ) = ζ (α ) for
Re α > 0 and α ≠ 1. The asymptotic behaviour of Ζ 1 (α , x ) and Ζ 2 (α , x ) is studied for the case
Re α = σ > 0 fixed and Imα = τ → ∞, and using Liouville-Green (WKBJ) analysis, asymptotic
approximations are obtained, complete with explicit error bounds, which are uniformly valid for
0 ≤ x < ∞.
Mathematics Subject Classification 2000:
Primary:
• 33E20 (Special functions defined by integrals)
Secondary:
•
11M06 (Zeta functions)
•
34E20 (WKB methods)
-3-
1. Introduction. It is well-known that the Riemann Zeta function has the integral representation
ζ (α ) =
(1.1)
1
(
)
1 − 21−α Γ(α )
∞ tα − 1
0 t
∫
e +1
d t,
which converges and defines ζ (α ) for all Re α > 0 (except at the pole α = 1). With this definition
in mind, we introduce an incomplete Riemann Zeta function
(1.2)
Ζ 1 (α , x ) =
x tα − 1
0 t
1
(1 − 21−α )Γ(α ) ∫
e +1
d t ( Re α > 0, α ≠ 1),
along with a complementary incomplete Riemann Zeta function
(1.3)
Ζ 2 (α , x ) =
1
(1 − 21−α )Γ(α ) ∫
∞ tα − 1
x t
e +1
d t ( α ≠ 1).
In the integrands of (1.2) and (1.3) the principal branch of tα is taken. Here and throughout we
assume that x is real with 0 ≤ x < ∞ , and from now on that α is not equal to 1, and satisfies
(1.4)
α = σ + iτ , σ > 0, 0 ≤ τ < ∞ .
From (1.1) - (1.3) it is evident that that for 0 ≤ x < ∞
(1.5)
ζ (α ) = Ζ 1 (α, x ) + Ζ 2 (α, x ) .
Therefore Ζ 1 (α , x ) and Ζ 2 (α , x ) are linearly independent (as functions of x) if and only if
ζ (α ) ≠ 0. The definitions (1.2) and (1.3) are analogous to those of the incomplete and
complementary incomplete Gamma functions
x
(1.6)
γ (α, x ) = ∫ tα − 1e − t d t ( Re(α ) > 0),
(1.7)
Γ(α , x ) = ∫ tα − 1e − t d t ,
0
∞
x
which of course satisfy Γ(α ) = γ (α , x ) + Γ(α , x )
Kolbig [5], [6] undertook a numerical study of the trajectory of the zeros of the incomplete
Riemann Zeta function (1.2) in the complex α plane, with x regarded as a real parameter. However,
he remarked that the expansions he employed suffered from numerical instabilities as τ increased.
-4For instance, in a footnote in [6] Kolbig mentions that it would be interesting to study the zero
trajectories near a zero-free Gram interval, the first of these being near τ = 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 Ζ 1 (α , x ) and Ζ 2 (α , x ) ; we shall obtain
leading term approximations as τ → ∞, 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 = Ζ 1 (α , x ) and y = Ζ 2 (α , x ) (along with a trivial solution y = 1) satisfy
the second order linear differential equation
d 2 y α − 1 − x
1 dy
=
+ x  .
2
x
dx
e + 1 d x

(1.8)
Then, removing the first derivative in a standard manner yields
(1.9)
(
)


x x
d 2 w α 2 − 1 (1 − α )e x e e + 2 
=
+
+
,
2 w
d x2  4x2
2x ex + 1 4 ex + 1 


(
(
) (
)
)
1/ 2
with solutions w = x (1−α ) / 2 e x + 1 y , where y = Ζ 1 (α , x ) , y = Ζ 2 (α , x ) , as well as y = 1. The
differential equation (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:
(1.10)
Ζ 1 (α , x ) =
(
xα
)
2α 1 − 21−α Γ(α )
{1 + O( x )}
( )
(1.11)
Ζ 2 (α , x ) = ζ (α ) + O xα
(1.12)
Ζ 1 (α , x ) = ζ (α ) + O xα − 1e − x
(1.13)
(
Ζ 2 (α , x ) =
xα − 1e − x
(1 − 2
1−α
( x → 0),
( x → 0),
)
( x → ∞),
1+ O x
)Γ(α ) { ( )}
−1
( x → ∞).
Thus, provided ζ (α ) ≠ 0, Ζ 1 (α , x ) is recessive and Ζ 2 (α , x ) is dominant at x = 0, with the roles
reversed at x = ∞. Consider then the following singular eigenvalue problem: for what values of α
(with Re α > 0) does there exist a solution w = φ (α , x ) , say, of (1.9) that is recessive at both x = 0
-5and x = ∞? From (1.10) – (1.13) it is clear that when ζ (α ) ≠ 0 no such eigensolution exists.
However, when ζ (α ) = 0 we have from (1.3) that Ζ 1 (α , x ) = −Ζ 2 (α , x ) , and consequently
(1.14)
(
)
(
)
1/ 2
1/ 2
φ (α, x ) = x (1−α ) / 2 e x + 1 Ζ 1 (α, x ) = − x (1−α ) / 2 e x + 1 Ζ 2 (α, x )
is the unique (to within a multiplicative constant) eigensolution of (1.9) that is recessive at both
x = 0 and x = ∞: the corresponding eigenvalues α of this singular boundary value problem are of
course the non-trivial zeros of ζ (α ) . The Riemann Hypothesis is therefore equivalent to the above
singular eigenvalue value problem having eigenvalues α which all lie on the critical line Re α = 12 .
The importance of the complete Riemann Zeta function ζ (α ) 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 [2] . Whilst there has been an intensive study on
the asymptotics of the Riemann zeta function (for recent results, see [1], [8], [9]), with the exception
of [5] and [6] 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 [7], Bose-Einstein and Fermi-Dirac functions [3]. We refer the
reader to [6] for further details.
The plan of this paper is as follows. In section 2 we derive a Dirichlet-type asymptotic
expansion for Ζ 2 (α , x ) . This is achieved by expanding the integral (1.3) as a convergent series
involving Γ(α , x ) , which in turn is approximated by an established uniform asymptotic
approximation. The resulting expansion for Ζ 2 (α , 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 Ζ 1 (α , x ) and Ζ 2 (α , x ) , and by means of a differentiation convert these into a
particularly simple form. The resulting approximation for Ζ 1 (α , x ) is uniformly valid for
0 ≤ x ≤ Ω(σ ) and the approximation for Ζ 2 (α , x ) is uniformly valid for Ω(σ ) ≤ x < ∞ , where
0 < Ω(σ ) < ∞ 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 the relative errors of our new uniform approximations, and we also demonstrate
how the stated intervals of validity can be extended for both Ζ 1 (α , x ) and Ζ 2 (α , x ) .
-62. Dirichlet-type Expansions. For x ≥ δ > 0 we can expand the denominator of the integrand of
(1.2) as the following geometric series
Ζ 1 (α , x ) =
(2.1)
1
x α −1
t
0
∫
Γ(α )
(1 − 2 )
1− α
∞
∑ e − kt d t ,
k =1
and then by reversing the integration and summation and referring to (1.6), we arrive at the
following Dirichlet-type series
Ζ 1 (α , x ) =
(2.2)
γ (α, kx )
∞
1
∑ (−1)k +1 Γ(α )kα
(1 − 2 ) k =1
1− α
( x ≥ δ > 0).
Likewise, from (1.3) and (1.7),
Γ(α , kx )
∞
1
Ζ 2 (α , x ) =
(2.3)
∑ (−1)k +1 Γ(α )kα
(1 − 21−α ) k =1
( x ≥ δ > 0).
Consider (2.3). From Temme [11] we have the asymptotic approximation
(2.4)

Γ(α , z) 1
e −ω (α ,z )  2α
1
− 1/ 2
~ erfc ω 1/ 2 (α , z) +

 ( α → ∞),
Γ(α )
2
2 π  z − α ω (α , z) 
(
)
where
ω (α, z) = z − α − α ln( z / α ) .
(2.5)
The asymptotic expansion (2.4) is valid for Re α = τ → ∞ uniformly for 0 < z < ∞ . We now
set z = kx in (2.4) and then insert into (2.3), which immediately gives
(2.6) Ζ 2 (α , x ) ~
(
∞
1
2 1 − 21−α
) k∑=1
(−1)
k +1

e −ω (α ,kx )  2α
1 
1
1/ 2
erfc
ω
(
α
,
kx
)
+
−


 ,
π  kx − α ω 1/ 2 (α, kx ) 
kα 
(
)
as τ → ∞, uniformly for 0 < δ ≤ x < ∞ . We can simplify (2.6) by using the following well-known
asymptotic behaviour of the complementary error function
(2.7)
( )
erfc ω
1/ 2
e −ω
~
πω
( ω → ∞, arg(ω ) ≤ 32 π − δ 0 < 32 π ).
With (2.7) in mind, and recalling that α = σ + iτ , we have from (2.5)
-7-
(2.8)
( )
ω (α, kx ) = kx − 12 πτ + iτ ln{τ / (ekx )} + σ ln(τ / ( kx )) + 12 πiσ + O τ −1
( τ → ∞),
uniformly for k ≥ 1 and 0 < δ ≤ x < ∞ . Using (2.8) it is not difficult to show that
− 32 π + δ 0 ≤ arg(ω ) < 0 ( δ 0 > 0 ), for all k ≥ 1 and x satisfying 0 < δ ≤ x < ∞ , provided τ is
sufficiently large. Setting kx = t in (2.8), Fig. 1 depicts a typical trajectory of the curve given
parametrically by ω (α ,t) , for 0 < t0 ≤ t < ∞ , where t0 is arbitrary and α = σ + iτ is fixed (with τ
large). Observe that on this curve ω indeed satisfies − 32 π + δ 0 ≤ arg(ω ) < 0 (for some fixed
δ 0 > 0 ).
t = t0
t = e −1τ + O(1)
(0)
t = 12 πτ + O(1)
t→∞
Fig.1 ω plane.
Since (2.7) applies for all terms in the series (2.6) and we may use it to replace all the erfc terms
in the series. Thus we arrive at
(2.9)
Ζ 2 (α , x ) ~
α
1
2π 1 − 21−α
(
∞
∑ (−1)k +1
) k =1
e −ω (α ,kx )
( τ → ∞, 0 < δ ≤ x < ∞ ).
kα ( kx − α )
From (2.8) we observe that exp{−ω (α , kx )} is exponentially large for kx < 12 πτ , but becomes
exponentially small for kx > 12 πτ (see also Fig. 1): taking into account the term kα in the
denominator of the right-hand side of (2.9), we deduce that terms in the series beyond
k =  12 πτ / x  are certainly negligible. Thus, using (2.8), we finally arrive at
-8-
(2.10) Ζ 2 (α , x ) ~
α
1
2π 1 − 21−α
(
 2 πτ / x 
1
)
∑ (−1)
k =1
k +1
e − kx
 ekx 


α
k ( kx − α )  α 
α
( τ → ∞, 0 < δ ≤ x < ∞ ).
In the next sections we shall take a different approach, by using theory of asymptotics of
differential equations. We shall obtain improved approximations which (with the aid of the
connection formula (1.5)) are uniformly valid for 0 ≤ x < ∞ , and moreover are complete with
explicit error bounds.
3. The Liouville-Green Transformation. Considering again the differential equation (1.9), we
shall derive Liouville-Green approximations directly from this equation. We first observe that the
equation has regular singularities at x = 0 and x = ±(2 k + 1)πi ( k = 0,1, 2,L). It also has an
irregular singularity at x = ∞. Since we are only considering positive values of x, the complex
poles are not of a direct concern to us, although we do note that they are infinite in number and the
sequence of poles tends to the singularity at infinity in the complex x plane.
The appropriate general theory is given by Olver in [10, Chap. 6, Theorem 11.1], which
provides Liouville-Green solutions compete with explicit error bounds. In order for these
approximations to be uniformly valid at both x = 0 and x = ∞ it turns out that we must rescale the
independent variable in the following manner
x = τxˆ .
(3.1)
It will become clear later why this is required. On writing w ( x ) = wˆ ( xˆ ) we then express (1.9) in the
form
(3.2)
{
}
d 2 wˆ
= τ 2 f (α , xˆ ) + g( xˆ ) wˆ ,
d xˆ 2
where
(3.3)
α2
1
(
1 − α )eτxˆ
1
f (α , xˆ ) = + 2 2 +
−
,
2
ˆ
τ
x
4 4τ xˆ
2τxˆ e + 1 4 eτxˆ + 1
(
and
(3.4)
g( xˆ ) = −
1
.
4 xˆ 2
) (
)
-9On referring to (1.4) it is readily verified that for τ large f (α , xˆ ) has no zeros on or near the real x̂
axis, and therefore the differential equation (3.2) is free of turning points on the positive real x̂ axis.
Therefore asymptotic solutions only involve elementary functions. To obtain these solutions, we
apply the well-known Liouville transformation, which involves a new independent variable defined
by
ξ = ∫ f 1/ 2 (α, xˆ ) d xˆ ;
(3.5)
(see [10, Chap. 6, §1]). The square root branch is chosen so that
f 1/ 2 (α , xˆ ) =
(3.6)
1 1− α
 1
+
+ O 2  ( x̂ → ∞),
 xˆ 
ˆ
2 2τx
and by continuity for all other positive x̂: in particular, from (3.3),
f 1/ 2 (α , xˆ ) = −
(3.7)
α 1− α
−
+ O( xˆ ) ( x̂ → 0).
2τxˆ
4α
We note in passing that the new variable ξ is complex, and before proceeding, we record the
limit at the singular endpoints x̂ = 0 and x̂ = ∞. Firstly, assume that the integration constant in
(3.5) is such that ξ = 0 when x̂ = a , where 0 < a < ∞ can be arbitrarily assigned. We can then
rewrite (3.5) in the form
(3.8)
a1
a  1/ 2
α 
∫xˆ t d t − ∫xˆ  f (α, t) + 2τt  d t ,
ξ=
α
2τ
ξ=
α  a
ln  − c (α ) + O( xˆ ) ( x̂ → 0),
2τ  xˆ  1
and consequently we find that
(3.9)
where
(3.10)
a
α 
c1 (α ) = ∫  f 1/ 2 (α , t) +
d t .
0
2τt 
From (3.7) it follows that this integral converges at the lower limit.
Likewise, for the behaviour at x̂ = ∞, we express (3.5) in the form
(3.11)
xˆ  1 1 − α 
xˆ 
1 1− α 
ξ=∫  +
d t + ∫  f 1/ 2 (α , t) − −

d t,
a 2
a
2τt 
2 2τt 


-10and as a result we deduce that
ξ=
(3.12)
1
1 − α  xˆ 
 1
ln  + c 2 (α ) + O  ( x̂ → ∞),
( xˆ − a) +
 a
 xˆ 
2
2τ
where
∞
1 1− α 
c 2 (α ) = ∫  f 1/ 2 (α , t) − −
d t ,
a
2 2τt 

(3.13)
which converges at the upper limit on account of (3.6).
In Fig. 2 we show a typical trajectory of ξ where xˆ ∈ (0, ∞) is regarded as a parameter, and
α = σ + iτ is fixed.
x̂ → 0
xˆ = xˆ 0
x̂ → ∞
Fig. 2 Trajectory of ξ ( x̂ ) in complex plane.
The figure suggests that for each positive σ and τ, there exists a value xˆ 0 ∈ (0, ∞) such that Re ξ
is decreasing for 0 < xˆ < xˆ 0 and increasing for xˆ 0 < xˆ < ∞. To confirm that this is indeed
generally true, we first note from the definition (3.5) and a continuity argument that, for increasing
positive x̂, Re ξ changes from decreasing to increasing (or vice versa) when Re f 1/ 2 (α , xˆ ) has a
simple zero (at xˆ = xˆ 0 say): the point(s) in question are those for which Im f (α , xˆ 0 ) = 0 with
-11Re f (α , xˆ 0 ) < 0 . Now, on setting α = σ + iτ in (3.3) we find that a unique xˆ 0 ∈ (0, ∞) satisfying
the requirements exists for each fixed positive σ and τ. In particular, using Im f (α , xˆ 0 ) = 0 we find
that x̂ 0 is the solution of the following implicit equation
τxˆ 0 = σ (1 + exp{−τxˆ 0 }) ,
(3.14)
and using (3.5), (3.6) and (3.7) we can confirm that Re ξ is decreasing for 0 < xˆ < xˆ 0 and
increasing for xˆ 0 < xˆ < ∞.
From (3.14) we can express x̂ 0 in the form
x̂ 0 = τ −1Ω(σ ) ,
(3.15)
where, in terms of the Lambert W function (see, for example, [12]),
(
)
Ω(σ ) = W σe − σ + σ .
(3.16)
( 12 ) = 0.738835L, and
We remark that Ω(σ ) is increasing for σ > 0, with Ω(0) = 0 , Ω
Ω(1) = 1.27846L.
Having made the Liouville transformation, we now apply [10, Chap. 6, Theorem 11.1]. In doing
so we identify Olver’s z, w ( z) and f ( z) with our x̂, wˆ ( xˆ ) and τ 2 f (α , xˆ ) , respectively. We take
the so-called reference points of Olver’s Theorem as a1 = 0 and a2 = ∞. Consider first the
asymptotic solution of (3.2) which is to be recessive at x̂ = 0 (reference point a1 = 0 ): since
Re ξ = ∞ when x̂ = 0 (see (1.4) and (3.9)) the required solution comes from [10, Chap. 6, Eq.
(11.05)] with j = 2: on writing wˆ (1) (α , xˆ ) and ε (1) (α , x̂ ) for Olver’s w 2 ( z) and ε 2 ( z) , respectively,
we thus obtain the following asymptotic solution of (3.2)
{
}
wˆ (1) (α , xˆ ) = f −1/ 4 (α , xˆ )e − τξ 1 + ε (1) (α , xˆ ) .
(3.17)
Consider now the error bounds [10, Chap. 1, Eq. (11.07)]. Using the definition of Olver’s errorcontrol function F, and the variational operator [10, Chap. 1, Eq. (11.02)], and recalling that f ( z) is
to be replaced by τ 2 f (α , xˆ ) , we arrive at the following bounds corresponding to the reference point
a1 = 0 ,
(3.18)
where
ε (1) (α, xˆ ) ,
∂ε (1) (α, xˆ )
1

 1 xˆ
1/ 2
≤
exp
 ∫0 Ψ (α , t) f (α , t) d t − 1 ,
1/ 2
ˆ
∂x
τf (α, xˆ )

τ
-12-
(3.19)
Ψ (α , xˆ ) =
4 f (α , xˆ ) f ′′ (α , xˆ ) − 5 f ′ 2 (α , xˆ )
g( xˆ )
+
.
f (α , xˆ )
16 f 3 (α , xˆ )
The bounds (3.18) are valid for Re ξ nonincreasing as x̂ increases from x̂ = 0. Thus they hold
uniformly for 0 ≤ xˆ ≤ xˆ 0 , where x̂ 0 is defined by (3.15) and (3.16): see also Fig. 2.
The second asymptotic solution of (3.2) is to be recessive at x̂ = ∞, and since Re ξ = ∞ in this
case too (see (3.12)) the required asymptotic solution also comes from setting j = 2 in [10, Chap.
6, Eq. (11.05)]: the difference is that the reference point is a2 = ∞ rather than a1 = 0 . Thus, for this
case, let us denote Olver’s w ( z) and ε ( z) by wˆ ( 2) (α , xˆ ) and ε ( 2) (α , x̂ ) , respectively. Then (3.2)
2
2
possesses the following solution which is recessive at x̂ = ∞,
{
}
wˆ ( 2) (α , xˆ ) = f −1/ 4 (α , xˆ )e − τξ 1 + ε ( 2) (α , xˆ ) ,
(3.20)
with the accompanying error bounds
(3.21)
ε ( 2) (α, xˆ ) ,
∂ε ( 2) (α, xˆ )
1

1 ∞
1/ 2
≤
exp
 ∫xˆ Ψ (α , t) f (α , t) d t − 1 .
1/ 2
ˆ
∂x
τf (α, xˆ )

τ
These bounds are valid provided Re ξ is nonincreasing as x̂ decreases from x̂ = ∞, and as such
(3.21) holds uniformly for xˆ 0 ≤ xˆ < ∞: see again Fig. 2.
Having constructed the Liouville-Green solutions, we conclude this section by examining their
error bounds in some more detail. The main issue is convergence of the integrals in (3.18) and
(3.21) at the singular endpoints x̂ = 0 and x̂ = ∞. The chosen partition (3.3) and (3.4) ensures
convergence at x̂ = ∞, since from (3.3)
(3.22)
f (α , xˆ ) ~
1 (1 − α )
( x̂ → ∞),
+
4
2τxˆ
with this relation being twice differentiable and uniformly valid for 0 < δ ≤ τ < ∞ . From [10, Chap.
6, §4], and in particular Eq. (4.06) of that reference, we see from (3.4) and (3.22) that the variation
in the error bound [10, Chap. 6, Eq. (11.07)], and equivalently the integral on the right-hand side of
(3.21), converges at x̂ = ∞, uniformly for 0 < δ ≤ τ < ∞ .
Next, x̂ = 0 is a regular singularity of (3.2), and the relevant information in this case is provided
by [10, Chap. 6, §4.3] (with a2 = 0 in the notation of this reference). Now from (3.3)
-13-
(3.23)
f (α , xˆ ) ~
α2
( x̂ → 0),
4τ 2 xˆ 2
uniformly for 0 < δ ≤ τ < ∞ . Thus f has a double pole, and as Olver remarks, the variation in the
error bound [10, Chap. 6, Eq. (11.07)] converges at a double pole of f provided g also has a double
pole, with leading coefficient − 41 . From (3.4) we see that this indeed is the case, and hence the
integral on the right-hand side of (3.18) converges at x̂ = 0 (uniformly for 0 < δ ≤ τ < ∞ ). We
remark that the scaling (3.1) was necessary to be able to partition the right-hand side of the
(transformed) equation in such a way that the error bounds could converge at both singularities.
Finally, let us estimate the integrals in (3.18) and (3.21). To do so, we find from (3.3) that
(3.24)
 xˆ − (α / τ ) 
f (α , xˆ ) = 



2 xˆ
2

 xˆ  
1 + O τxˆ   ,
e 

(
)
ˆ − τxˆ = τ −1 τxe
ˆ − τxˆ = τ −1O(1) for
uniformly for 0 ≤ x̂ < ∞ and 0 < δ ≤ τ < ∞ . By noting that xe
0 ≤ x̂ < ∞ , and recalling (1.4), we deduce that
2
(3.25)
 xˆ − i  
 1 
f (α , xˆ ) = 
 1 + O   ,
 2 xˆ  
τ
uniformly for 0 ≤ x̂ < ∞ . Moreover, one can readily establish that this is twice differentiable, and
hence inserting (3.25) and its differentiated forms, along with (3.4), into (3.19) we find that
(3.26)
Ψ (α , xˆ ) =
xˆ ( xˆ + 2i) 
 1 
 ,
4 1 + O
 τ  
( xˆ − i) 
uniformly for 0 ≤ x̂ < ∞ , which combined with (3.25) yields the estimate
xˆ 2 + 4 )
(

 1 
1
/
2
∫ Ψ(α, xˆ ) f (α, xˆ ) d xˆ = ∫ ˆ 2 3/2 d xˆ 1 + O τ  
2( x + 1)
1/ 2
(3.27)
uniformly for 0 ≤ x̂ < ∞ . This incidentally confirms the convergence of the integrals in (3.18) and
(3.21) at both singularities.
-14-
4. Liouville-Green Approximations. We shall now identify the asymptotic solutions with the
exact solutions of (3.2). We first consider those that are recessive at x̂ = 0, and we immediately
deduce that there exists a constant C1 (α ) such that
(τxˆ )(1−α ) / 2 (eτxˆ + 1) Ζ 1(α,τxˆ ) = C1(α )wˆ (1) (α, xˆ ) ,
1/ 2
(4.1)
Returning to the original variable x via (3.1), we deduce from (3.17) and (4.1) that
{
}
Ζ 1 (α , x ) = C1 (α ) H (α , x )e − τξ 1 + ε (1) (α , xˆ ) ,
(4.2)
where
(
)
−1/ 2 −1/ 4
(α, x ) ,
H (α , x ) = x (α − 1) / 2 e x + 1
h
(4.3)
and
(
)
h (α , x ) = f α ,τ −1 x =
(4.4)
1 α2
(1 − α )e x
1
+ 2+
−
.
2
4 4x
2x ex + 1 4 ex + 1
(
) (
)
Usually one would determine the constant of proportionality C1 (α ) by comparing both sides of
(4.1) as x approaches a singularity (in this case x = 0). However, we shall take a different approach
which results in a much simpler approximation. The key is that the derivative of Ζ 1 (α , x ) is an
elementary function (see (1.2)). With this in mind, we differentiate both sides of (4.2) to yield
(4.5)
1
[
}]
xα − 1
d
= C1 (α )
H (α , x )e − τξ 1 + ε (1) (α , xˆ ) .
x
d
x
+
e
1
Γ(α )
(1 − 21−α )
{
Then, using
d ξ / d x = τ −1h 1/ 2 (α , x ) ,
(4.6)
(see (3.5) and (4.4)) we have for the right-hand side of (4.5)
(4.7)
C1 (α )
[
{
}]
{
}
d
H (α , x )e − τξ 1 + ε (1) (α , xˆ ) = C1 (α )e − τξ H (α , x ) 1 + ε (1) (α , xˆ )
dx
×h
1/ 2

H ′ (α , x )
∂ε (1) (α , xˆ ) / ∂xˆ
(α, x )
−
1
+
 H (α , x ) h 1/ 2 (α , x )
τf 1/ 2 (α, xˆ ) 1 + ε (1) (α, xˆ )

{
}

.


-15Next, on referring back to (4.2), it follows from (4.5) and (4.7) that
1
xα − 1
= Ζ 1 (α , x ) h 1/ 2 (α , x )
x
+
1
e
Γ(α )
(1 − 21−α )
(4.8)

∂ε (1) (α , xˆ ) / ∂xˆ
H ′ (α , x )

×
−1+
 H (α , x ) h 1/ 2 (α , x )
τf 1/ 2 (α, xˆ ) 1 + ε (1) (α, xˆ )

{
}

.


Solving this for Ζ 1 (α , x ) gives the desired uniform asymptotic approximation. The analysis for
Ζ 2 (α , x ) is similar, and finally using
(
)(
(
)
)
1/ 2

x
x
2
x − α  2x e + α e + 1 − x 
1/ 2
1+
h (α , x ) =
2

2x 
( x − α )2 e x + 1


(4.9)
,
we arrive at our main result.
Theorem 4.1. For j = 1, 2
(4.10)
Ζ j (α , x ) =
(1 − 21−α )Γ(α ) ( x − α )(e x + 1)(1 + χ (α, x))1/2 {
(−1) j
xα
}
j
1 + δ ( ) (α , xˆ )
−1
,
where
χ (α, x ) =
(4.11)
(4.12)
1
δ ( j ) (α, x ) = − −
2
(
)( )
2
( x − α ) 2 (e x + 1)
2x ex + α ex + 1 − x2
,
H ′ (α , x )
∂ε ( j ) (α , xˆ ) / ∂xˆ
−
,
2 H (α , x ) h 1/ 2 (α , x ) 2τf 1/ 2 (α , xˆ ) 1 + ε ( j ) (α , xˆ )
{
}
j
and H (α , x ) , h 1/ 2 (α , x ) are given by (4.3) and (4.4), respectively. The error terms ε ( ) (α , xˆ ) and
their derivatives are bounded by (3.18) and (3.21).
From the error bounds (3.18) and (3.21) we know that the third term on the right-hand side of
(4.12) is O τ −1 uniformly for 0 ≤ x ≤ Ω(σ ) ( j = 1) and Ω(σ ) ≤ x < ∞ ( j = 2), where Ω(σ ) is
( )
defined by (3.16). We now assert that:
-16-
Lemma 4.2.
( )
δ ( j ) (α, x ) = O τ −1 ,
(4.13)
uniformly for 0 ≤ x ≤ Ω(σ ) ( j = 1) and Ω(σ ) ≤ x < ∞ ( j = 2).
To prove this, we shall show that
H ′ (α , x )
1
 1
− −
= O  ,
1
/
2
τ
2 2 H (α , x ) h (α , x )
(4.14)
uniformly for 0 ≤ x < ∞ . To establish this fact, we first find from (4.3) that
H ′ (α , x ) α − 1 1
1
h ′ (α , x )
.
=
− +
−
x
H (α , x )
2x
2 2 e + 1 4 h (α , x )
(
(4.15)
)
Next, from (1.4) and (4.9), it is readily verified that
h 1/ 2 (α , x ) =
(4.16)
x −α 
 1 
1 + O   ,
τ 
2x 
uniformly for 0 ≤ x < ∞ . Therefore, combining (4.15) and (4.16), and after some calculation, we
arrive at the desired estimate
(4.17)
H ′ (α , x )
x
1
 1
 1
− −
=
+ O  = O  ,
1
/
2
2
τ
τ
2 2 H (α , x ) h (α , x ) 2( x − α )
with both O terms being uniform for 0 ≤ x < ∞ . The Lemma now follows.
An extension of our approximations for Ζ 1 (α , x ) and Ζ 2 (α , x ) to 0 ≤ x < ∞ is immediately
achieved from the connection formula (1.5). Thus, for example, Ζ 1 (α , x ) can be represented for
Ω(σ ) ≤ x < ∞ by the relation Ζ 1 (α , x ) = ζ (α ) − Ζ 2 (α , x ) , in which ζ (α ) can be approximated by
the Riemann-Siegel formula (see [1] and [4]), and Ζ 2 (α , x ) approximated by (4.10).
Note that when ζ (α ) = 0 we have from (1.14) and Theorem 4.1 that the eigenfunction φ (α , x )
has the asymptotic approximation
(4.18)
φ (α, x ) =
1
x (1+α ) / 2
1−α
1/ 2
(1 − 2 )Γ(α ) ( x − α )(e x + 1)
(1 + χ (α, x ))
1/ 2

 1 
1 + O   ,
τ 

-17which is uniformly valid for 0 ≤ x < ∞ . Moreover, the O term tends to zero both when x → 0 and
when x → ∞.
5. Numerical Calculations. We shall compute the relative errors η( j ) (α , x ) ( j = 1, 2 ) defined by
(5.1)
Ζ j (α , x ) =
(−1) j
(1 − 2 )Γ(α ) ( x − α )(e
1−α
xα
x
)
+ 1 (1 + χ (α , x ))
{1 + η( ) (α, x)} .
j
1/ 2
We consider two values of α , α 1 and α 2 , say, where
α1 =
(5.2)
1
2
+ 100i ,
for which
(5.3)
ζ (α1 ) = (2.6926L) − (0.020386L)i,
and
(5.4)
α2 =
1
2
+ (101.317851L)i,
(
)
−1/ 2
α −1 /2
φ (α 2 , x ) ,
this being chosen as a zero of ζ (α ) . Thus Ζ 1 (α 2 , x ) = −Ζ 2 (α 2 , x ) = x ( 2 ) e x + 1
where φ (α , x ) is the eigenfunction (recessive at both x = 0 and x = ∞) which has the uniform
asymptotic approximation (4.18).
In (5.1) we computed the exact value of Ζ 1 (α , x ) for small x by a Taylor series expansion, and
the exact value of Ζ 2 (α , x ) for large x by the expansion (2.3): then Ζ 1 (α 1, x ) for large x and
Ζ 2 (α 1, x ) for small x were computed via the connection formula (1.5).
The graph of η(1) (α , x ) for 0 < x ≤ 10 and α = α 1 , is depicted in Fig. 3. This does not show
the divergence as x → ∞ because η(1) (α 1, x ) only becomes large when x > 100: this can be seen
in Table 1 below, where we give values of η(1) (α 1, x ) and η( 2) (α 1, x ) for various values of x. We
j
also give values for η( ) (α 2 , x ) (where j = 1 or j = 2, since both are equivalent on account of
ζ (α 2 ) = 0 ). This error term vanishes at both x = 0 and x = ∞.
-18-
Fig. 3. Graph of the exact relative error η(1) (α 1, x ) .
x
1.0E-135
1.0E-132
1.0E-130
1.0E-125
1.0E-120
1.0E-03
0.25
Ω(0.5)
5
10
50
100
150
155
160
η(1) (α1, x )
η( 2) (α1, x )
η( j ) (α 2 , x )
1.50008E-137
1.50008E-134
1.50008E-132
1.50008E-127
1.50008E-122
1.49335E-05
1.33196E-03
4.60138E-04
3.40746E-05
1.01104E-05
3.40608E-05
2.84513E-05
3.98908E-02
5.96239E+00
8.90997E+02
1.23097E+02
3.89266E+00
3.89266E-01
1.23097E-03
3.89266E-06
1.49335E-05
1.33196E-03
4.60138E-04
3.40746E-05
1.01104E-05
3.40608E-05
2.84513E-05
1.78209E-05
1.69896E-05
1.62027E-05
1.48057E-137
1.48057E-134
1.48057E-132
1.48057E-127
1.48057E-122
1.47392E-05
1.31464E-03
4.54164E-04
3.32888E-05
9.72403E-06
3.30506E-05
2.80105E-05
1.76874E-05
1.68711E-05
1.60974E-05
Table 1. Relative errors for various values of x.
-19We also observe that η( 2) (α 1, x ) only becomes large when x is exponentially small. To see
why this is so in general, we first use (1.5) and (5.1) to derive the identity
(5.5)
(
)
(
)
η( 2) (α, x ) − η(1) (α, x ) = 1 − 21−α Γ(α ) x −α ( x − α ) e x + 1 (1 + χ (α, x )) ζ (α ) .
1/ 2
Now, from Stirling’s formula, we have
(5.6)
Γ(α ) = Γ(σ + iτ ) ~ 2π e
− πi / 4  τ 
iτ
σπi / 2 σ − (1/ 2) − τπ / 2
τ
e
( τ → ∞).
  e
 e
( )
Hence, since χ (α , x ) is bounded (in fact O τ −1 ) for large τ uniformly for 0 ≤ x < ∞ , we deduce
that
(
(5.7) η( 2) (α , x ) − η(1) (α , x ) = ζ (α )τ σ − (1/ 2)e − τπ / 2 x − σ x 2 + τ 2
)
1/ 2 x
e O(1) ( τ → ∞, 0 < x < ∞ ).
Since it is well-known that ζ (σ + iτ ) has at worst algebraic growth as τ → ∞, it follows that
η(1) (α, x ) = η( 2) (α, x ) + o(1)
(5.8)
certainly for
 π (1 − κ )τ 
1
exp−
 ≤ x ≤ 2 π (1 − κ )τ ,
2
σ


(5.9)
where κ is an arbitrary small positive constant: in this interval the o(1) term in (5.8) is exponentially
small in τ. This confirms our assertion above.
( )
Returning to (4.10), we deduce that δ (1) (α , x̂ ) = O τ −1 uniformly for 0 ≤ x̂ ≤ 12 π (1 − κ ) , and
δ ( 2) (α, x̂ ) = O τ −1 uniformly for exp{−τπ (1 − κ ) / (2σ )} ≤ xˆ < ∞ , in both cases κ ∈ (0,1) being
( )
arbitrary.
-20-
References
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remainders, Proc. Roy. Soc. London Ser. A 450 (1995) 439-462.
[2] M. V. Berry and J. P. Keating, The Riemann zeros and eigenvalue asymptotics, SIAM Rev.
41 (1999) 236-266.
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(1958) 484-490.
[4]
H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York and London, 1974.
[5] K. S Kölbig, Complex zeros of an incomplete Riemann zeta function and of the incomplete
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[7] Y. L. Luke, The special functions and their approximations. Vol. II. Mathematics in Science
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R. B. Paris and S. Cang,, An asymptotic representation for ζ
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