How to recover an L-series from its values at almost... integers. Some remarks on a formula of Ramanujan

Proc. Indian Acad. Sci. (Math. Sci.), Vol. 110, No. 2, May 2000, pp. 121±132.
# Printed in India
How to recover an L-series from its values at almost all positive
integers. Some remarks on a formula of Ramanujan
CHRISTOPHER DENINGER
WWU, Mathematisches Institut, Einsteinstrasse 62, D-48149 Munster, Germany
MS received 25 August 1999; revised 15 January 2000
Abstract. We define a class of analytic functions which can be obtained from their
values at almost all positive integers by a canonical interpolation procedure. All the
usual L-functions belong to this class which is interesting in view of the extensive
investigations of special values of motivic L-series. A number of classical contour
integral formulas appear as particular cases of the interpolation scheme. The paper is
based on a formula of Ramanujan and results of Hardy. An approach to the problem
via distributions is also presented.
Keywords. Interpolation formulas; analytic functions; contour integrals; special
values; L-functions
1. Introduction
The purpose of this note is to answer a question, Mazur asked me: Is there an interpolation scheme allowing to recover a complex L-series from its values at almost all
positive integers? This is interesting for example in view of the extensive investigations of
special values of motivic L-series in the last decades, culminating in the Bloch±Kato
conjectures [BK]. Note that p-adic L-functions are determined by their values at these
points because the set in question is dense in Zp .
It turns out that modifying one of Ramanujan's favourite formulas one gets a satisfactory interpolation procedure for a class of analytic functions which in particular comprises
all Dirichlet series and hence all L-series. Incidentally the classical representations of
certain zeta- and L-functions as contour integrals are particular instances of the interpolation scheme.
Ramanujan did not specify exactly to which functions his formula applied. A useful
class F H was singled out however by Hardy in his commentary on Ramanujan's work
[H], ch. XI. We introduce a universal class F of interpolizable functions which is
essentially canonical. Hardy's result then implies that F H F . Apart from the general
setup and a number of examples we also give a short distribution theoretic proof of a
special case of Hardy's result. This uses Schwartz' extension of the Paley±Wiener
theorem to distributions with compact support.
It should be emphasized that this note is essentially a commentary on one aspect of the
work of Hardy and Ramanujan.
2. Preliminaries
To a sequence of complex numbers a ˆ …a †0 , 0 2 Z we associate the Laurent series:
121
122
Christopher Deninger
ˆ
a …x†
:ˆ
X
a x :
0
Extend a to a sequence indexed by the integers by setting a ˆ 0 for < 0. We call
`good' if the following conditions are satisfied:
converges in a punctured neighborhood of the origin;
…1†
0
has a holomorphic continuation to U where U is a neighborhood of
…ÿ1; 0Š and U 0 ˆ U n f0g:
…2†
For some 2 R we have
j …z†j ˆ O…jzj † as jzj ! 1 in U 0 :
If
…3†
is good, the contour integral
Z …0‡†
1
dz
zÿs …z†
I… †…s† :ˆ
2i ÿ1
z
…4†
defines a holomorphic function of s in Re s > for any as in condition (3). Here the
integration is along any path within U n …ÿ1; 0Š starting at ÿ1, encircling the origin
counterclockwise and returning to ÿ1.
The power zÿs is defined via log z ˆ log jzj ‡ i Arg z where ÿ < Arg z .
Note that I… † does not depend on the choice of the path.
Lemma 1.1. In the situation of (4) we have:
I… †…† ˆ a
for > :
Proof. For s ˆ > since zÿ is single valued the integral reduces to
I
1
dz
dz
ÿ
ÿ
z
…z† ˆ Reszˆ0 z
…z†
ˆ a
I… †…† ˆ
2i jzjˆ"
z
z
by taking " sufficiently small.
&
We need another consequence of the residue theorem:
Lemma 1.2. A rational function
ˆ d and we have:
of degree d with no poles on …ÿ1; 0† is good with
X
I… †…s† ˆ ÿ
Resa z
a2Cn…ÿ1;0Š
ÿs
dz
…z†
z
in Re s > d:
Finally we require for later use.
Lemma 1.3. Assume that
Then we have
I… †…s† ˆ ÿ
ˆ
P
0
a x is good with some < 0 in condition (3).
sin s
M… …ÿx††…ÿs† for < Re s < 0 :
Remarks on a formula of Ramanujan
Here
Z
MF…s† ˆ
1
0
xs F…x†
123
dx
x
is the Mellin transform on R‡ .
Proof. For Re s > and every " > 0 small enough we have:
Z ÿ"
Z ÿ1
1
dx
1
dx
ÿs…log jxjÿi†
e
…x† ‡
eÿs…log jxj‡i† …x†
I… †…s† ˆ
2i ÿ1
x 2i ÿ"
x
I
1
dz
zÿs …z† ;
‡
2i jzjˆ"
z
Z 1
I
sin s
dx
1
dz
xÿs …ÿx† ‡
zÿs …z† :
ˆÿ
x
2i
z
"
jzjˆ"
Since j …z†j ˆ O…jzj0 † as jzj ! 0 we have the estimate
I
I
ˆ0
c"0 ÿRe s and hence lim
"!0 jzjˆ"
jzjˆ" for Re s < 0. Hence the formula.
&
Remark. The theory of the Mellin transform is well developed. In [I], Theorem 3.1 for
example two function spaces are defined which are in bijection via the Mellin transform.
Together with Lemma 1.3, Igusa's result leads to information about the interpolation
functional I… †. Unfortunately the class of functions to which we want to apply I in the
next sections is quite different from the one that can be treated in this way.
3. Interpolation
We can now set up the interpolation scheme. Consider the C-algebra:
A0 :ˆ fsequences …a † defined from some 0 onwardsg= where a ˆ …a †0 a0 ˆ …a0 † 0 iff a ˆ a0 for all 0.
0
If sequences a; a0 are equivalent, then a is good iff a0 is good. Hence we can define:
A :ˆ f‰aŠ 2 A0 j
a
is good g:
0
On the other hand let F be the C-vector
P space of holomorphic functions defined on
some half plane Re s > s.t. …x† ˆ > …†x is good. Here the summation is over
all integers > . By the principle of analytic continuation we may identify functions in
F 0 if they agree for Re s 0.
Theorem 2.1. I defines a linear `interpolation' map
I : Aÿ!F 0 via I…‰aŠ† ˆ I…
a †:
It has the `special-values map'
S : F 0 ÿ!A; S…† ˆ ‰……††0 Š
124
Christopher Deninger
as a left-inverse:
S I ˆ id:
Proof. For any sequence a ˆ …a †0 such that a is good the function …s† ˆ I… a †…s†
is holomorphic in some half plane
P Re s > . By Lemma 1.1 it has the property that
……††> a. Hence …x† ˆ > …†x is good as well and thus 2 F 0 .
If a a0 then a ÿ a0 2 C‰x; xÿ1 Š. By Lemma 1.2 we therefore have I… a † ˆ I… a0 † in
0
F . Thus the interpolation map is well defined. As we have seen
…I…
a †…††>
a
and hence S I ˆ id on A.
&
We now define:
F :ˆ Im I F 0 :
Then I and S define mutually inverse C-linear isomorphisms
I
A ÿ!
ÿ F:
S
…5†
This is clear since I was injective having a left-inverse and we have made it surjective.
By construction the functions in F have the property that they are uniquely determined
by their values on any set of integers of the form fj 0 g. Moreover given these values
for 0 there is an explicit formula for the function, valid in some half plane Re s > .
Note that we have a canonical projector:
P ˆ I S : F 0 ÿ!F ; P2 ˆ P:
…6†
In these terms we have:
PROPOSITION 2.2
(1) For A; A0 2 A form A A0 2 A0 . If A A0 2 A, then I…A A0 † ˆ P…I…A† I…A0 †† in F :
(2) S and I are equivariant with respect to the Z-action by shift.
Proof. (1) If A A0 2 A then I…A† I…A0 † 2 F 0 since …I…A† I…A0 ††…† ˆ I…A†…† I…A0 †
…† ˆ a a0 for 0 where …a †0 ; …a0 † 0 are representatives of A; A0 . Hence
0
S…I…A† I…A0 †† ˆ A A0 . Applying I gives the assertion. (2) Shift by one acts on A0 by
T‰…a †Š ˆ ‰…a‡1 †Š. The corresponding is xÿ1 a which is again good. Hence the shift
&
acts on A and by a similar argument also on F 0 . The rest is clear.
Remark. There is a convolution product for sequences but it does not pass to A.
Before we incorporate the Hardy±Ramanujan theory into the picture let us give some
examples. For a sequence a with ‰aŠ in A we set I…a† :ˆ I…‰aŠ†.
= …ÿ1; 0† the class of a is in
Example 2.3. For 2 C consider a ˆ …ÿ †0. Then if 2
A and I…a† ˆ ÿs where arg 2 …ÿ; Š. In particular …s† ˆ ÿs 2 F. The functions
ÿs defined using different normalizations of arg lie in F 0 and are mapped via P to the
principal one.
Remarks on a formula of Ramanujan
125
Proof. For 2
= …ÿ1; 0† the function
…x† ˆ
a …x†
ˆ
1
X
ÿ x ˆ
ˆ0
1
; jxj < 1 ÿ ÿ1 x
is good in our sense. By Lemma 1.2 we have
1
dz
I… †…s† ˆ ÿRe szˆ zÿs
ˆ ÿs :
1 ÿ ÿ1 z z
…7†
&
Example 2.4. a ˆ …1=!†0 defines a class in A and I…a† ˆ ÿ…s ‡ 1†ÿ1 2 F .
Proof.
ˆ ex is clearly good and
Z …0‡†
1
dz
zÿs ez
ˆ ÿ…s ‡ 1†ÿ1
I… a †…s† ˆ
2i ÿ1
z
a …x†
is Hankel's representation of the inverse ÿ-function.
We can also argue as follows: Since
shows that
I…
a †…s†
ˆÿ
a …x†
&
ˆ ex is good for any 2 R, lemma 1.3
sin s
M…eÿx †…ÿs† for Re s < 0:
Now by its definition ÿ…s† equals the Mellin transform of eÿx so that
I…
a †…s†
ˆÿ
sin s
ÿ…ÿs† ˆ ÿ…s ‡ 1†ÿ1
first in Re s < 0 and then for all s by analytic continuation.
Example 2.5. We want to interpolate the values ÿB‡1 =… ‡ 1† of the zeta-function at the
negative integers. Since they grow so quickly that has radius of convergence zero we renormalize them as follows: ÿB‡1 =… ‡ 1†! for ÿ1. We expect them to be
interpolated by the function …ÿs†=ÿ…s ‡ 1† and this is indeed the case. More generally
consider the sequence: …ÿB‡1 …a†=… ‡ 1†!†ÿ1 for 0 < a 1 where Bn …a† is the nth
Bernoulli polynomial. Its -function is
…z† ˆ
1
X
ˆÿ1
ÿB‡1 …a†
1
z
1X
z
eaz
ˆÿ
B …a† ˆ
:
… ‡ 1†!
! 1 ÿ ez
z ˆ0
It is good and j …z†j ˆ O…jzj † for any 2 R. We have
Z …0‡†
1
eaz dz …ÿs; a†
ˆ
zÿs
I… †…s† ˆ
1 ÿ ez z
2i ÿ1
ÿ…s ‡ 1†
…8†
…9†
by
formula from the theory of the Hurwitz zeta function …s; a† ˆ
P1a standard
ÿs
…
‡
a†
c.f.
[EMOT] 1.10. Thus …ÿs;a†
ÿ…s‡1† 2 F is the interpolation of its values
ˆ0 B‡1 …a†
for any 0 ÿ1. It follows that L…;ÿs†
ÿ …‡1†!
ÿ…s‡1† 2 F is the interpolation of its values
0
at the integers 0 for any 0 ÿ1 as well.
126
Christopher Deninger
4. Invoking the Hardy, Ramanujan theory. Further examples
The problem is of course to give good criteria as to when an analytic function defined in
some right half plane belongs to F . For this we take up ideas of Hardy.
We first require a formula of Hardy, [H], (11.4.4) whose proof is omitted in [H]. For the
convenience of the reader we give a proof below. Actually, in the following proposition,
we show a slightly stronger result since this requires no extra effort and may be useful for
extending the theory.
PROPOSITION 3.1
Assume that is holomorphic in Re s > and satisfies an estimate of the form:
j… ‡ it†j f …t†eP‡jtj
for > where P 2 R and f 2 L1 …R† is such that limt!1 f …t† ˆ 0.
Fix an integer 0 > and choose r > such that 0 ÿ 1 < r < 0 . Then for any real
ÿeÿP < x < 0 we have the integral representation:
Z r‡i1
X
1
…x† ˆ
…†x ˆ ÿ
…s†…ÿx†s ds:
2i
sin
s
rÿi1
0
Here the series is absolutely convergent and the integral is in the Lebesgue sense.
Proof. Consider the contour C ˆ C1 ‡ C2 ‡ C3 ‡ C4 :
where L 2 12 ‡ Z. By the residue theorem:
Z
X
1
…†x ˆ
…s†…ÿx†s dx:
2i
sin
s
C
0 <L
We have
eÿjIm sj
sin s
for jIm sj 0:
Using periodicity of sin we get that for R large enough
c1 eÿjIm sj holds on C for all L:
sin s
Hence
Z
C2
c1 …L ÿ r†eÿR emax …Pr;PL†‡R max……ÿx†r ; …ÿx†L † f …R†:
Remarks on a formula of Ramanujan
127
R
R
Thus for fixed L, we have limR!1 C2 ˆ 0. Similarly limR!1 C4 ˆ 0.
Next
Z L‡i1 Z 1
Z
L
ÿjtj PL jtj
L…P‡log…ÿx††
c2
e
e
e
f
…t†…ÿx†
dt
c
e
3
Lÿi1
ÿ1
1
ÿ1
f …t† dt:
Hence the integral exists and tends to zero for L ! 1 by our assumption ÿeÿP < x < 0
i.e. P ‡ log…ÿx† < 0. Similarly the integral from r ÿ i1 to r ‡ i1 exists. Hence the
formula.
&
One now uses the integral representation for of the proposition to show that which a
priori is holomorphic only in 0 < jzj < eÿP extends to a holomorphic function in some
punctured neighborhood U 0 as in (2) above which is bounded by a power of jzj as in
(3). More can be done but let us stay with a class of functions introduced by Hardy. For
A < set:
(
)
's analytic in Re s > for some 2 R such that there
:
F H …A† ˆ
exists P 2 R with j… ‡ it†j eP‡Ajtj in Re s > S
Any such is called allowable for . Set F H ˆ A< F H …A†. Then we have the
following result which follows from the preceeding considerations and those in [H], 11.4:
be the
Theorem 3.2. F H F .PMore precisely, if is allowable for 2 F H let
analytic continuation of > …†x to a punctured neighborhood U 0 of …ÿ1; 0Š. Then
we have j …z†j ˆ O…jzj † as jzj ! 1 in U 0 for every > and the interpolation formula
Z …0‡†
1
dz
zÿs …z† ˆ …s†
I… †…s† ˆ
2i ÿ1
z
therefore holds in Re s > .
Remark. The example of …s† ˆ sin s shows that the condition A < is not unnatural.
Proof. By assumption j… ‡ it†j eP‡Ajtj in > for some P 2 R; A < . Hence
Proposition 3.1 is applicable. Let 0 be the least integer > and choose r > such that
0 ÿ 1 < r < 0 . Then by (3.1) we have for any ÿeÿP < z < 0:
Z r‡i1
1
…s†…ÿz†s ds:
…z† ˆ ÿ
2i rÿi1 sin s
Choose 0 < < ÿ A. Then for ÿ < arg …ÿz† < we have:
j…ÿz†s j ˆ jzj eÿt arg …ÿz† jzj ejtj :
Thus
Z
Z
c1
r‡i1 rÿi1
1
ÿ1
c2 jzj
r
eÿjtj ePr‡Ajtj jzjr ejtj dt
Z
1
ÿ1
e…A‡ÿ†jtj dt ˆ O…jzjr †:
128
Christopher Deninger
Since we know that the series for converges in 0 < jzj < eÿP it follows that extends
to an analytic function in some region U 0 as in (2) where it satisfies …z† ˆ O…jzjr † as
jzj ! 1 for any r > . Thus is good and hence 2 F 0 . Moreover I… †…s† defines a
holomorphic function in Re s > . It remains to prove that I… † ˆ . Unfortunately this
cannot be checked by substituting the above integral representation for into the contour
integral I since the former does not converge for the s on the loop around zero. Instead we
reduce the claim to a formula of Hardy and Ramanujan ± the last equality in [H], 11.4±
which itself is an application of Mellin- or Fourier-inversion:
Formula of Hardy±Ramanujan. For 0 < < 1, let H be holomorphic in Re s ÿ and
satisfy the estimate H …s† eP1 ‡Ajtj there for some P1 and A < . Setting
H …x† ˆ
1
X
H …†…ÿ1† x
ˆ0
we have that
Z
0
1
xw H …x†
dx
ˆ
H …ÿw†
x
sin w
for 0 < Re w < :
By Lemma 1.3 we have for < Re s < 0 :
Z
sin s 1 ÿs
dx
x
…ÿx† :
I… †…s† ˆ ÿ
x
0
Now choose 0 < < 0 ÿ so that in particular < 1. Set H …s† ˆ …s ‡ 0 †. Then the
Hardy±Ramanujan formula applied to H …s† ˆ …s ‡ 0 † gives the equality:
Z 1
dx
…0 ÿ w†:
xwÿ0 …ÿx† ˆ
x
sin
…w
ÿ 0 †
0
Thus for 0 ÿ < Re s < 0 we find that
Z
sin s 1 ÿs
dx
ÿ
x
…ÿx† ˆ …s†:
x
0
Together with the above formula for I… †…s† it follows by analytic continuation that
I… †…s† ˆ …s†
for Re s > as claimed.
&
Remark 3.3. Our interpolation functional I has two advantages over the one of Hardy±
Ramanujan:
Z
sin s ÿ1
dx
IHR : 7ÿ!
…ÿx†ÿs …x†
x
0
which requires convergence at 0 and ÿ1 whereas I needs convergence at ÿ1 only. As a
consequence interpolation formulas involving IHR are valid at most in some region
< Re s < whereas those using I hold in a half plane Re s > . Moreover only in I is
it possible to add to an arbitrary Laurent polynomial without changing its value. This
is crucial for interpolating elements of A i.e. sequences which are only given up to
equivalence.
Remarks on a formula of Ramanujan
129
In the rest of this section we use distributions to give a different and more conceptual
proof of the assertion 2 F in Theorem 3.2 for a restricted class of functions :
Let be an entire function which satisfies an estimate of the form
j…s†j …1 ‡ jsj†N eAjIm sj
in C for some A < . By Schwartz' extension of the Paley±Wiener theorem to
distributions [Y], VI.4 the function is the Fourier±Laplace transform of a distribution T
with compact support in …ÿ; †. Choose some " > 0 such that supp T is disjoint from the
set C" of y in R with jeiy ‡ 1j < ". Let be a smooth function on R which is 0 on C" and
equal to 1 on supp T. We have:
^
…s† ˆ T…s†
ˆ …2†ÿ1=2 hTy ; eÿisy i:
Hence
…x† :ˆ
1
X
…†x ˆ …2†ÿ1=2 hTy ; …1 ÿ xeÿiy †ÿ1 i
ˆ0
for jxj < 1. The formula
…z† ˆ …2†ÿ1=2 hTy ; …y†…1 ÿ zeÿiy †ÿ1 i
gives the analytic continuation of to a neighborhood of …ÿ1; 0Š. Since
X
sup jD h…y†j
jT…h†j C
jjN jyjL
for some constants C; N; L and all smooth functions h on R it follows that
j …z†j ˆ O…jzjÿ1 † as z ! ÿ1:
Hence
is good and thus 2 F 0 . Now
Z …0‡†
1
dz
zÿs …2†ÿ1=2 hTy ; …y†…1 ÿ zeÿiy †ÿ1 i
I… †…s† ˆ
2i ÿ1
z
*
+
Z …0‡†
1
dz
ÿ1=2
ÿ1
Ty ;
zÿs …y†…1 ÿ zeÿiy †
ˆ …2†
2i ÿ1
z
2:3
ˆ …2†ÿ1=2 hTy ; eÿisy i ˆ …s†:
Since supp T …ÿ; †. Hence 2 F .
If more generally T has compact support in R n Z then is still good by the identical
argument, so that 2 F 0 . However we now have, again using (2.3), that:
P ˆ I… †…s† ˆ …2†ÿ1=2 hTy ; …y†eÿisy i;
…10†
where y 2 …ÿ; Š is such that y y mod Z. Note that eÿisy is not smooth but …y†eÿisy
is. Writing T as a finite sum
X
Tˆ
T
130
Christopher Deninger
of distributions T with compact support in …ÿ ‡ ; ‡ † it follows that ˆ
^ ˆ T^ . By (10) we see that P… † ˆ eis and hence
where ˆ T;
X
P ˆ
eis 2 F :
P
Incidentially this is also a consequence of Theorem 3.2 applied to eis .
5. Applications
In this section we illustrate the preceeding theory by interpolating certain interesting
classes of functions. We are mostly interested in L-series and their completed versions by
ÿ-factors.
Set
8
9
< 's analytic in Re s > for some 2 R s.t. for every =
> 0 there exist a and some P 2 R with
F 0‡
H ˆ
:
;
j… ‡ it†j eP ‡jtj in Re s > and
(
F 0H
ˆ
's analytic in Re s > for some `associated' s.t.
j… ‡ it†j eP in Re s > for some P 2 R
)
:
0
0‡
Clearly F 0H F 0‡
H F H. Moreover F H and F H are C-algebras and F H is a module
0
under them. Note that if f 2 F H and 2 F H have f and associated to them, then
max…f ; † is associated to f . P
Clearly every Dirichlet series
an ÿs
n with n > 0 and abscissa of absolute conver0
gence < 1 belongs to F H with being admissible. In particular L-series and their
inverses belong to F 0H .
On the other hand L-series completed by ÿ-factors do not even belong to F 0 since the
associated power series has radius of convergence zero. The reciprocal function however
has a better behaviour if the ÿ-factor is simple. To see this we require the following fact:
PROPOSITION 4.1
For every 0 < a < 2; b 2 R the function ÿ…as ‡ b†ÿ1 belongs to F H with associated
ˆ 1a…12 ÿ b†.
Proof. For given > 0 the complex Stirling asymptotics for ÿ…s† implies that
ÿ…s† ˆ eÿs e…ÿ1=2†log s …2†1=2 …1 ‡ O…sÿ1 ††
in j arg sj ÿ as jsj ! 1. Hence this estimate holds for all s with Re s 12 ; jsj > 1.
Thus we also have
ÿ…s†ÿ1 ˆ es e…1=2ÿs†log s …2†ÿ1=2 …1 ‡ O…sÿ1 †† in Re s 12; jsj > 1
and hence:
jÿ…s†ÿ1 j e e…1=2ÿ† log jsj ejtj=2
e‡jtj=2
Remarks on a formula of Ramanujan
131
in Re s 1=2; jsj > 1 and hence in Re s 1=2. Thus
jÿ…as ‡ b†ÿ1 j ea‡jtja=2
for Re s 1a …12 ÿ b†:
…11†
&
Examples. (1) It follows again that ÿ…s†ÿ1 2 F .
(2) Since …2s ‡ 2†ÿ1 2 F 0H with ˆ ÿ1=2 (abscissa of absolute convergence) and since
ÿ…s ‡ 1†ÿ1 2 F H with ˆ ÿ1=2 by the proposition we find that ÿ…s ‡ 1†ÿ1 …2s ‡ 2†ÿ1 2
F H with ˆ ÿ1=2. Hence theorem 3.2 gives us:
Z …0‡†
1
X
1
z
dz
1
1
zÿs
ˆ
for Re s > ÿ :
!…2
‡
2†
2i ÿ1
z
ÿ…s
‡
1†…2s
‡
2†
2
ˆ0
Note that the series in the integral converges everywhere. Setting
^ ˆ ÿs=2 ÿ s …s†
…s†
2
^ ‡ 2†ÿ1 2 F H with ˆ ÿ1=2 and that:
we get similarly that …2s
1
2i
Z
…0‡†
ÿ1
zÿs
1
X
z
dz
1
ˆ
^
^
z
…2s ‡ 2†
ˆ0 …2 ‡ 2†
1
in Re s > ÿ :
2
^ ÿ1 2 F H with ˆ 1 and hence:
Similarly …s†
Z …0‡†
1
X
1
z dz
1
ˆ
in Re s > 1:
zÿs
^
^
2i ÿ1
z
…s†
ˆ2 …†
(3) A similar formula holds for the completed L-series
^ s† ˆ L…E; s†…2†ÿs ÿ…s†
L…E;
of an elliptic curve E over Q:
Z …0‡†
1
X
1
z dz
1
zÿs
ˆ
^
^
2i ÿ1
z
L…E; s†
ˆ2 L…E; †
3
in Re s > :
2
(4) For …s† ˆ …2s† 2 F 0H F and ˆ 1=2 the corresponding function
p
3.2 is given by …z† ˆ f … ÿz† where f is the even function
1 …1 ÿ w†ew ÿ …1 ‡ w†eÿw
:
f …w† ˆ
ew ÿ eÿw
2
in Theorem
After some calculation which we leave to the reader the formula of theorem 3.2 leads to
the functional equation of …s†. This example was suggested by the discussion of
Ramanujan's formula in [E], 10.10.
Remark. A variant of the first formula was first given by Riesz as mentioned by Hardy:
Z 1
1
X
…ÿ1† x dx
ÿ…s†
ˆ
xÿs
!…2 ‡ 2† x
…2s ‡ 2†
0
ˆ0
valid for ÿ1=2 < Re s < 0.
132
Christopher Deninger
The case ÿ…2s ‡ b†ÿ1 is not covered by the proposition. We close by noting that a direct
computation gives:
Fact 4.2. ÿ…2s ‡ n†ÿ1 2 F for all n 2 Z.
Proof. SinceP
F is shift-invariant we may restrict to ÿ…2s ‡ 1†ÿ1 . The associated function
2
1 w
ÿw
†. The mapping
is …x† ˆ 1
ˆ0 x =…2†! which is entire. We have …w † ˆ 2 …e ‡ e
2
w7!w transforms any strip 0 Re w into a neighborhood U of …ÿ1; 0Š. In 0 Re
is bounded in U. Thus
w the function 12 …ew ‡ eÿw † is bounded and hence
ÿ…2s ‡ 1†ÿ1 2 F 0 . For Re s > 0 we have:
Z …0‡†
Z …0‡†
1
dz
1
dz
1 ÿs ÿ1‡0i
‰z Šÿ1ÿ0i
zÿs …z† ˆ
zÿs … …z† ÿ 1† ÿ
2i ÿ1
z
2i ÿ1
z 2is
Z …0‡†
1
dz
ˆ
zÿs … …z† ÿ 1† :
2i ÿ1
z
Using Lemma (1.3) we see that for 0 < Re s < 1 this equals
Z
Z
sin s 1 ÿs
dx
sin s 1 ÿ2s
dx
x … …ÿx† ÿ 1† ˆ ÿ2
x …cos x ÿ 1†
ÿ
x
x
0
Z 10
sin s
xÿ2s sin x dx
ˆ
s 0
by integration by parts. Substituting the formula in [EMOT], 1.5.1 (38)
Z 1
xÿ1 sin x dx ˆ ÿ…† sin in ÿ 1 < Re < 1:
2
0
We arrive after some calculation at the desired formula:
Z …0‡†
1
dz
zÿs …z† ˆ ÿ…2s ‡ 1†ÿ1 :
2i ÿ1
z
&
Acknowledgements
I would like to thank B Mazur for his question and the Harvard mathematics department
for its hospitality. I would also like to thank the referee for suggestions to improve the
exposition.
References
[BK] Bloch S and Kato K, L-functions and Tamagawa numbers of motives, in: The
Grothendieck Festschrift, vol. 1, Prog. Math. 86 (1990) 333±400
[E] Edwards H M, Riemann's zeta function (Academic Press) (1974)
[EMOT] Erdelyi A et al, Higher transcendental functions. The Bateman Manuscript Project
(McGraw-Hill) (1953) vol. 1
[H] Hardy G H and Ramanujan S, Twelve Lectures on Subjects Suggested by His Life and
Work (Chelsea) (1978)
[I] Igusa J-I, Lectures on forms of higher degree. (Bombay: Tata Institute of Fundamental
research) (1978)
[Y] Yosida K, Grundlehren Bd. 123, (Springer: Functional Analysis) (1971)