An efficient algorithm for computing Dirichlet L -functions
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An efficient algorithm for computingDirichlet L-functionsRasa Šleževičienėa Department of Physics and Mathematics, Šiauliai University, P.Višinskio Str. 19, LT-77156, Šiauliai, LithuaniaPublished online: 26 Jan 2007.
To cite this article: Rasa Šleževičienė (2004) An efficient algorithm for computingDirichlet L-functions, Integral Transforms and Special Functions, 15:6, 513-522, DOI:10.1080/1065246042000272072
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Integral Transforms and Special FunctionsVol. 15, No. 6, December 2004, pp. 513–522
AN EFFICIENT ALGORITHM FOR COMPUTINGDIRICHLET L-FUNCTIONS
RASA ŠLEŽEVICIENE∗
Department of Physics and Mathematics, Šiauliai University, P. Višinskio Str. 19,LT-77156 Šiauliai, Lithuania
(Received 3 June 2003)
We present a very simple class of efficient algorithms for computing Dirichlet L-functions to arbitrary precision.
Keywords: Dirichlet L-functions; Computation; Algorithm; Precision
AMS Subject Classification Numbers: 11M06
1 INTRODUCTION
Let s = σ + it be a complex variable. The Riemann zeta-function is for σ > 1 defined by
ζ(s) =∞∑
m=1
1
ms=
∏p
(1 − 1
ps
)−1
,
and by analytic continuation elsewhere except for a simple pole at s = 1; here and in the sequel,the product is taken over all prime numbers p. Borwein [1] proposed an efficient algorithmfor the arbitrary precision calculation of the Riemann zeta-function. Using the representation
ζ(s) = 1
1 − 21−s
∞∑m=1
(−1)m−1
ms, (1)
which is valid for σ > 0, Borwein proved the following result.
THEOREM 1 Let σ > 0 and let pn(x) = ∑nk=0 akx
k be an arbitrary polynomial of degree n
which does not vanish at −1. Let
cj = (−1)j
(j∑
k=0
(−1)kak − pn(−1)
).
∗ E-mail: [email protected]
ISSN 1065-2469 print; ISSN 1476-8291 online © 2004 Taylor & Francis LtdDOI: 10.1080/1065246042000272072
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514 R. ŠLEŽEVICIENE
Then
ζ(s) = −1
(1 − 21−s)pn(−1)
n−1∑j=0
cj
(1 + j)s+ ξn(s),
where
ξn(s) = 1
(1 − 21−s)pn(−1)�(s)
∫ 1
0
pn(x)| log x|s−1
1 + xdx.
Here, �(s) is the gamma-function.Theorem 1 leads to an algorithm for computing ζ(s) in σ > 0. The trick is to choose pn
such that the error term ξn is as small as possible. An obvious choice for pn are the Chebychevpolynomials. An even simpler algorithm, though not quite as fast, can be based on takingpn(x) = xn(1 − x)n. These algorithms can improve the standard algorithms based on theEuler–Maclaurin method, which are implemented in the major computer-algebra packages.However, they do not compete with algorithms based on the Riemann–Siegel formula forcomputations concerning the zeros on the critical line. In this paper, we present a similarresult for Dirichlet L-functions.
2 STATEMENT OF RESULTS
Let (Z/qZ)∗ denote the multiplicative group of prime residue classes modulo q and let
ψ : (Z/qZ)∗ −→ {z ∈ C: |z| = 1}
be a homomorphism. A Dirichlet character χ mod q is defined by
χ(m) ={
ψ(m mod q) if (m, q) = 1,
0 if otherwise.
The character χ0(m) identically equal to 1 for all m, coprime with q, is called principal. Forfurther details on characters see for example [2]. The Dirichlet L-function L(s, χ) attachedto a character χ mod q is for σ > 1 given by
L(s, χ) =∞∑
m=1
χ(m)
ms=
∏p
(1 − χ(p)
ps
)−1
. (2)
Note that if χ is non-principal, then the series in Eq. (2) converges for σ > 0. If χ is principalthen
L(s, χ0) = ζ(s)∏p|q
(1 − 1
ps
). (3)
Hence, Dirichlet L-functions L(s, χ0) associated to principal characters can be computed viaBorwein’s algorithm.
We start by presenting the algorithm for computing Dirichlet L-functions in its genericform.
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ALGORITHM FOR COMPUTING DIRICHLET L-FUNCTIONS 515
THEOREM 2 Let σ > 0, χ modq be a non-principal character and let pn(x) = ∑nk=0 akx
kq
be an arbitrary polynomial of degree nq which does not vanish at 1. Let
cj = pn(1) −j∑
k=0
ak. (4)
Then
L(s, χ) = 1
pn(1)
q∑a=1
χ(a)
n−1∑j=0
cj
(a + qj)s+ ξn(s), (5)
where
ξn(s) = −1
pn(1)�(s)
∫ 1
0
pn(x)| log x|s−1
1 + x + · · · + xq−2 + xq−1
q−1∑�=1
(x − 1)�−1
�!q∑
a=�+1
χ(a)
a(a)l−1 dx.
(6)
Here, (a)i denotes the Pochhammer symbol, i.e. (a)0 = 1, (a)i = a(a − 1)(a − 2) · · · (a −i + 1) for i ≥ 1.
The algorithm of Theorem 2 can be based on taking pn(x) = xqn(1 + xq)n. Then, we getthe following theorem.
THEOREM 3 With
ej = 2n −j−n∑k=0
(n
k
)
for σ > 0
L(s, χ) = 1
pn(1)
q∑a=1
χ(a)
2n−1∑j=0
ej
(a + qj)s+ γn(s),
where
|γn(s)| ≤ q1−σ exp(q/e + (|t |π)/2)(1 + |t |/σ)
nσ.
The right-hand side in the latter expression gives the correct order of growth with respect to n.We see that we need a large n to get a small error term, therefore, the convergence is not veryfast. A certain restriction on q allows to use a similar representation for the L-functions asEq. (1) in the case of the Riemann zeta-function and leads to a better result.
We assume that q is odd. In this case χ(2) �= 0, and with the same argument that provesEq. (1), we get
L(s, χ) = 1
1 − χ(2)21−s
∞∑m=1
(−1)m−1χ(m)
ms. (7)
What can we say about the convergence of this series? When we have (−1)m−1χ(m) �=χ0(m), we obtain by partial summation and Weierstrass’ theorem that series (7) convergesin the half-plane σ > 0. As L(s, χ) is regular for all s the singularities coming fromzeros of 1 − χ(2)21−s are removable. In the case (−1)m−1χ(m) = χ0(m), we can applyrepresentation (3) and use Borwein’s result.
The following theorem gives an algorithm for computing Dirichlet L-functions attached toa character with odd modulus.
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516 R. ŠLEŽEVICIENE
THEOREM 4 Let q be odd, χ mod q be a non-principal character and let pn(x) =∑nk=0 akx
kq be an arbitrary polynomial of degree nq which does not vanish at −1. Further, let
cj = (−1)j
(j∑
k=0
(−1)kak − pn(−1)
). (8)
Then for σ > 0,
L(s, χ) = −1
pn(−1)(1 − χ(2)21−s)
q∑a=1
χ(a)(−1)a−1n−1∑j=0
cj
(a + qj)s+ ξn(s),
where
ξn(s) = 1
pn(−1)�(s)(1 − χ(2)21−s)
∫ 1
0
pn(x)| log x|s−1
1 + xq
q∑a=1
χ(a)xa−1 dx.
We realise this algorithm by taking pn(x) = xqn(1 − xq)n. Without loss of generality, let n beeven. Then, we get the following theorem.
THEOREM 5 Let q be odd. For σ > 0,
L(s, χ) = 1
2n(1 − χ(2)21−s)
q∑a=1
(−1)a−1χ(a)
2n−1∑j=0
ej
(a + qj)s+ γn(s),
where
ej = (−1)j
(j−n∑k=0
(n
k
)− 2n
)
and
|γn(s)| ≤ ϕ(q)
8n
(1 + |t |/σ) exp((|t |π)/2)
|1 − χ(2)21−s | .
Here, ϕ(m) denotes the Euler totient.
FIGURE 1 L(s,
( ·11
))for s ∈ [0, 1] when n = 2 ( ), n = 4 (- - -), n = 6 (◦ ◦ ◦), n = 10 (+ + +) in the
algorithm of Theorem 3.
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ALGORITHM FOR COMPUTING DIRICHLET L-FUNCTIONS 517
FIGURE 2 L(s,
( ·11
))for s ∈ [0, 1] when n = 2 ( ), n = 4 (- - -), n = 6 (◦ ◦ ◦), n = 10 (+ + +) in the
algorithm of Theorem 5.
Borwein’s algorithm is nearly optimal in the following sense. There is no sequence of n-termexponentional polynomials that converge to ζ(s) on an interval [a, b], a > 1 much faster thanthose of the algorithms. Similarly, one can also show that the algorithm of Theorem 5 is nearlyoptimal. Figures 1 and 2 give an impression on the speed of convergence of the algorithms ofTheorems 3 and 5, respectively.
3 PROOFS OF THE RESULTS
We start with the proofs of Theorems 2 and 4.
Proof of Theorem 2 In view of the well-known formula
�(s) =∫ ∞
0us−1 exp(−u) du,
it is not difficult to show that
n−s = 1
�(s)
∫ 1
0xn−1| log x|s−1 dx; (9)
both identities are valid for σ > 0. Therefore, using Eq. (2), we obtain
L(s, χ) = 1
�(s)
∫ 1
0| log x|s−1
∞∑n=1
χ(n)xn−1 dx
= 1
�(s)
∫ 1
0
| log x|s−1
1 − xq
q∑a=1
χ(a)xa−1 dx. (10)
The polynomial Fχ(x) = ∑q
a=1 χ(a)xa is a generalized Fekete polynomial [see Ref. 3 fordetails]. By the orthogonality relation for characters, we have Fχ(1) = 0 for non-principal χ .
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518 R. ŠLEŽEVICIENE
Therefore, the singularity at x = 1 is removable. Out of technical reasons, we will remove itlater. Now, we write
ξn(s) := 1
pn(1)�(s)
∫ 1
0
pn(x)| log x|s−1
1 − xq
q∑a=1
χ(a)xa−1 dx
= 1
pn(1)�(s)
∫ 1
0
pn(1)| log x|s−1
1 − xq
q∑a=1
χ(a)xa−1 dx
− 1
pn(1)�(s)
∫ 1
0
(pn(1) − pn(x))| log x|s−1
1 − xq
q∑a=1
χ(a)xa−1 dx. (11)
The first term of Eq. (11) gives L(s, χ) by Eq. (10). The second term can be rewritten as
1
pn(1)�(s)
∫ 1
0
∑nk=1 ak(1 − xkq)
1 − xq| log x|s−1
q∑a=1
χ(a)xa−1 dx
= 1
pn(1)�(s)
∫ 1
0
n∑k=1
ak
k−1∑j=0
xqj | log x|s−1q∑
a=1
χ(a)xa−1 dx.
Interchanging the summations and the integration, this equals
1
pn(1)�(s)
∫ 1
0
n−1∑j=0
n∑k=j+1
akxqj | log x|s−1
q∑a=1
χ(a)xa−1 dx
= 1
pn(1)
q∑a=1
χ(a)
n−1∑j=0
n∑k=j+1
ak
1
�(s)
∫ 1
0xqj+a−1| log x|s−1 dx.
Applying Eq. (9) and taking into account that∑n
k=j+1 ak = pn(1) − ∑j
k=0 ak , we get theseries expansion in Eq. (5). In the neighbourhood of x = 1, we have the Taylor expansion
q∑a=1
χ(a)xa−1 =q−1∑l=1
∑q
a=l+1 χ(a)∏l
k=1(a − k)
l! (x − 1)l .
Substituting this in Eq. (11) yields Eq. (6). This proves Theorem 2. �
Proof of Theorem 4 Similar as above, we find that
L(s, χ) = 1
(1 − χ(2)21−s)�(s)
∫ 1
0| log x|s−1
q∑a=1
χ(a)(−1)a−1xa−1 dx
1 + xq. (12)
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ALGORITHM FOR COMPUTING DIRICHLET L-FUNCTIONS 519
Notice that the integrand has no singularities. We write
ξn(s) := 1
pn(−1)(1 − χ(2)21−s)�(s)
∫ 1
0
pn(x)| log x|s−1
1 + xq
q∑a=1
χ(a)(−x)a−1 dx
= 1
pn(−1)(1 − χ(2)21−s)�(s)
∫ 1
0
pn(−1)| log x|s−1
1 + xq
q∑a=1
χ(a)(−x)a−1 dx
+ 1
pn(−1)(1 − χ(2)21−s)�(s)
∫ 1
0
(pn(x) − pn(−1))| log x|s−1
1 + xq
×q∑
a=1
χ(a)(−x)a−1 dx. (13)
By Eq. (12) the first term gives L(s, χ). We have
pn(x) − pn(−1) =n−1∑j=0
cjxjq(1 + xq),
where the coefficients cj are given by Eq. (8). Thus,
pn(x) − pn(−1)
1 + xq=
n−1∑j=0
cjxjq
and Eq. (13) proves the assertion of Theorem 4. �
Now, we give the remaining proofs of Theorems 3 and 5.
Proof of Theorem 3 The expression for L(s, χ) follows from Theorem 2. To estimate theerror, we observe that by Theorem 2
γn(s) = 1
2n�(s)
∫ 1
0
xn(1 + xq)n| log x|s−1
xq−1 + xq−2 + · · · + x + 1
q−1∑l=1
q∑a=l+1
χ(a)(a)l−1
al! (x − 1)l−1 dx.
Obviously,
|γn(s)| ≤ 1
2n
q−1∑l=1
q∑a=l+1
(a)l−1
al!1
|�(s)|∫ 1
0| log x|σ−1xqn(1 + xq)n|x − 1|l−1 dx
≤ 1
2n
q−1∑l=1
q∑a=l+1
(a)l−1
al!1
|�(s)|n∑
k=0
(n
k
) ∫ 1
0| log x|σ−1xq(k+n) dx.
Using Eq. (9) this equals
1
2n
�(σ )
|�(s)|q−1∑l=1
q∑a=l+1
(a)l−1
al!n∑
k=0
(n
k
) 1
(q(k + n) + 1)σ
≤ 1
(qn)σ
�(σ )
|�(s)|q−1∑l=1
1
l!q∑
a=l+1
(a − 1)!(a − l − 1)! (14)
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520 R. ŠLEŽEVICIENE
By Stirling’s formula,
n! ∼ √2πn
(n
e
)n
,
we findq−1∑l=1
1
l!q∑
a=l+1
(a − 1)!(a − l − 1)! ≤ q exp
(q
e
). (15)
Using the product representation, we get for σ > 0
∣∣∣∣ �(σ)
�(σ + it)
∣∣∣∣2
=∞∏
n=0
(1 + t2
(σ + n)2
).
This gives
∣∣∣∣ �(σ)
�(σ + it)
∣∣∣∣ =(
1 +(
t
σ
)2)1/2 ( ∞∏
n=1
(1 + t2
(σ + n)2
))1/2
≤(
1 + (t/σ )2
|t |π)1/2
(sinh(|t |π))1/2
≤(
1 + |t |σ
)exp
( |t |π2
). (16)
Substituting Eqs. (15) and (16) in Eq. (14), we obtain the estimate for γn(s). Theorem 3 isproved. �
Asn∑
k=0
(n
k
) 1
(q(k + n) + 1)σ≥ 2n
(2qn)σ,
we have the right order with respect to n.
Proof of Theorem 5 The expression for L(s, χ) follows from Theorem 4. To give an estimatefor the error, we observe that by Theorem 2
γn(s) = 1
2n�(s)(1 − χ(2)21−s)
∫ 1
0
xqn(1 − xq)n| log x|s−1
1 + xq
q∑a=1
χ(a)xa−1 dx.
As xqn(1 − xq)n is bounded on [0, 1] by 1/4n, application of Eq. (9) yields
|γn(s)| ≤ 1
8n|�(s)(1 − χ(2)21−s)|q∑
a=1
|χ(a)|∫ 1
0xa−1| log x|σ−1 dx
≤ ϕ(q)
8n|(1 − χ(2)21−s)|�(σ)
|�(s)| . (17)
Formula (16) together with Eq. (17) completes the proof of the theorem. �
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ALGORITHM FOR COMPUTING DIRICHLET L-FUNCTIONS 521
4 APPLICATIONS
We conclude with several graphics obtained by the algorithm of Theorem 5, which illustratecertain topics from the theory of Dirichlet L-functions.
A Dirichlet L-function attached to a real character may have a real zero β ∈ [1/2, 1]. Sucha hypothetical zero is called exceptional zero or Siegel zero in honour of Siegel, who provedthat for any positive ε, there is a constant c(ε) such that
β ≤ 1 − c(ε)
qε,
if β is a Siegel zero of L(s, χ), where χ is a character mod q. Unfortunately, the constantc(ε) is ineffective for ε < 1/2 (which makes a lot of trouble). Big progress concerning thisproblem was made in recent years, e.g. by Granville and Stark [4], who showed, that theabc-conjecture implies the non-existence of Siegel zeros, and also by Iwaniec and Sarnak [5],who found an interesting approach via L-functions associated with cusp forms. However, theexistence problem is still open. Figures 3 and 4 show the absence of real zeros for the DirichletL-functions attached to the Legendre symbols mod 317 and mod 227, respectively.
As in the case of the Riemann zeta-function, it is widely expected that Dirichlet L-functionsL(s, χ) do not vanish in the half-plane σ > 1/2, or equivalently, that all non-real zeros of anyL(s, χ) associated to a primitive character χ lie on the so-called critical line σ = 1/2; this is
FIGURE 3 L(s,
( ·317
))for s ∈ [0, 1] with n = 100 in the algorithm of Theorem 5.
FIGURE 4 L(s,
( ·227
))for s ∈ [0, 1] with n = 100 in the algorithm of Theorem 5.
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522 R. ŠLEŽEVICIENE
FIGURE 5 |L ((1/2) + it,
( ·5
))| for t ∈ [0, 20] with n = 10 in the algorithm of Theorem 5.
the famous yet unproved generalized Riemann hypothesis. Figure 5 shows the first non-realzeros of the Dirichlet L-function attached to the Legendre symbol mod 5 on the critical line.
In Ref. [6], Yildirim generalized some well-known results on the zero-distribution of thederivatives of the Riemann zeta-function to the derivatives of Dirichlet L-functions. Assumingthe generalized Riemann hypothesis, Yildirim proved that if χ is a character mod q withχ(−1) = 1 and q ≥ 216, then L′(s, χ) has exactly one real zero in the left half of the criticalstrip, and that, if χ(−1) = −1 and q ≥ 23, then L′(s, χ) has no zeros in the left half of thecritical strip. This behaviour can also be found in Figures 3 and 4.
The splitting of the Dedekind zeta-function into L-functions is intimately connected withthe Dedekind conjecture. It is widely believed that the Dedekind zeta-function to any fieldsplits into a product of the Riemann zeta-function and Dirichlet L-functions. This is true,if the underlying field extension is Galois. Therefore, in a similar manner as for DirichletL-functions, we can illustrate the behaviour of the derivative of the Dedekind zeta-functionsof quadratic number fields, which is described in Ref. [7].
Acknowledgements
Part of this work was done during a visit to Frankfurt University, which was kindly supportedby ‘Šiauliu universiteto Tarptautiniu komandiruociu fondas’. The author would like to thankProf. W. Schwarz and Prof. J. Wolfart for their kind hospitality and interest.
References
[1] Borwein, P. (2000). An efficient algorithm for the Riemann zeta function. Canadian Math. Soc. ConferenceProceedings, 25, 29–34.
[2] Davenport, H. (2000). Multiplicative Number Theory, 3rd ed. Springer.[3] Fekete, M. and Pólya, G. (1912). Über ein problem von Laguerre. Rend. Circ. Mat. Palermo, 34, 89–120.[4] Granville, A. and Stark, H. M. (2000). Abc implies no ‘Siegel zeros’ for L-functions of characters with negative
discriminant. Invent. Math., 139(3), 509–523.[5] Iwaniec, H., Sarnak, P. (2000). The non-vanishing of central values of automorphic L-functions and landau-Siegel
zeros. Israel J. Math., 120 part A, 155–177.[6] Yildirim, C. Y. (1996). Zeros of derivatives of Dirichlet L-functions. Turkish J. Math., 20, 521–534.[7] Šleževiciene, R. (2002). On the zeros of the derivative of Dedekind zeta-functions. Proc. of XLIII Conf. of Lith.
Math. Soc., Vilnius (spec. issue of Liet. matem. rink. 42). MII, Vilnius, pp. 107–112.
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