Two complete and minimal systems associated
with the
zerosof the Riemann zeta function
par
JEAN-FRANÇOIS
BURNOLRÉSUMÉ. Nous relions trois thèmes restés
jusqu’alors
distincts:les
propriétés
hilbertiennes des zéros deRiemann,
la "formule duale de Poisson" deDuffin-Weinberger (que
nousappelons
for-mule de
co-Poisson),
les espaces de fonctions entières "de Sonine"définis et étudiés par de
Branges.
Nous déterminons dansquels
espaces de Sonine
(étendus)
les zéros forment unsystème
com-plet,
ou minimal. Nous obtenons des résultatsgénéraux
concer-nant la distribution des zéros des fonctions entières de de
Branges-
Sonine. Nous attirons l’attention sur certaines distributions liées à la transformation de Fourier et qui sont apparues dans nos travaux anterieurs.
ABSTRACT. We link
together
three themes which had remainedseparated
so far: the Hilbert space properties of the Riemann ze-ros, the "dual Poisson formula" of
Duffin-Weinberger (also
namedby
us co-Poissonformula),
and the "Soninespaces"
of entire func-tions defined and studied
by
deBranges.
We determine in which(extended)
Sonine spaces the zeros define acomplete,
or minimal, system. We obtain somegeneral
resultsdealing
with the distri- bution of the zeros of thede-Branges-Sonine
entire functions. We draw attention onto some distributions associated with the Fourier transform and which we introduced in our earlier works.1. The
Duffin-Weinberger
"dualized" Poisson formula(aka co-Poisson)
We start with a
description
of the "dualized Poisson formula" of Duffin andWeinberger ([13, 14]).
We were not aware at the time of[7]
thatthe formula called
by
us co-Poissonformula
had been discovered(much)
earlier. Here is a
(hopefully
not tooinexact)
brief historical account: thestory
starts with Duffin who gave in an innovative 1945 paper[10]
a certainformula
constructing pairs
of functions which arereciprocal
under the sinetransform. As
pointed
outby
Duffin in the conclusion of his paper aspecial
instance of the formula leads to the functional
equation
of the L-functionManuscrit reçu le 28 janvier 2003.
1 - ls + 5S -
...(as
weexplain below,
this goes both ways infact).
Theco-Poisson formula which we discuss later will stand in a similar relation with the zeta function 1 +
29
+ ..., thepole
of zetaadding
its ownspecial
touch to the matter.Weinberger
extended in his dissertation[25]
this work of Duffin and also he found
analogous
formulaeinvolving
Hankeltransforms. Boas
[1]
gave a formalargument allowing
to derive Duffintype
formulae from the Poisson formula. However formalarguments might
be
misleading
and this is whathappened
here: formula[1, 3.(iii)]
whichis derived with the
help
of apurely
formalargument
looks like it is the co-Poissonformula,
but is not in fact correct. It isonly
much later in1991 that Duffin and
Weinberger [13] (see
also[14]) published
andproved
the formula
which,
inhindsight,
we see now is the one to be associated with the Riemann zeta function.They
alsoexplained
its "dual" relation to the so-much-well-known Poisson summation formula. In[7]
we followedlater a different
(esoterically adelic) path
to the same result. Asexplained
in
[7],
there are manifold ways to derive the co-Poisson formula(this
iswhy
we use "co-Poisson" rather than the "dualized Poisson" of Duffin andWeinberger).
In this Introduction we shallexplain
one suchapproach:
are-examination of the Fourier
meaning
of the functionalequation
of theRiemann zeta function.
When
applied
to functions which arecompactly supported
away from theorigin,
the co-Poisson formula createspairs
of cosine-tranformrecipro-
cal functions with the
intriguing
additionalproperty
that each one of thepair
is constant in some intervalsymmetrical
around theorigin. Imposing
two linear conditions we make these constants
vanish,
and this leads us to atopic
which has been inventedby
deBranges
as anillustration,
orchallenge,
to his
general theory
of Hilbert spaces of entire functions([3]), apparently
with the aim to
study
the Gammafunction,
andultimately
also the Rie-mann zeta function. The entire functions in these
specific
deBranges
spacesare the Mellin
transforms,
with a Gammafactor,
of the functions with thevanishing property
for somegeneral
Hankel transform(the
cosine or sinetransforms
being special cases).
Thesegeneral
"SonineSpaces"
were intro-duced in
[2],
and further studied and axiomatizedby
J. and V.Rovnyak
in
[22].
Sonine himself never dealt with such spaces, but in astudy ([23])
of Bessel functions he constructed a
pair
of functionsvanishing
in some in-terval around the
origin
andreciprocal
under some Hankel transform. An account of the Sonine spaces isgiven
in a final section of[3],
additionalresults are to be found in
[4]
and[5].
As the co-Poisson formula has not been available in thesestudies,
the way we have related the Riemann zeta function to the Sonine spaces in[7]
hasbrought
a novel element to thesedevelopments,
a moreintimate,
andexplicit,
web of connections betweenthe Riemann zeta function and the de
Branges
spaces, and their extensionsallowing poles.
Although
this paper ismostly self-contained,
we refer the reader to "OnFourier and
Zeta(s)" ([7])
for themotivating
framework and additionalbackground
and also to our Notes[6, 8, 9]
for our results obtained so far and whose aim isultimately
to reach a betterunderstanding
of someaspects
of the Fourier Transform.Riemann sums
F(n),
or-IF(2), T T
havespecial
connectionswith,
on one hand the Riemann zeta function((s) = (itself
ob-tained as such a summation with
F(x)
=x-S), and,
on the otherhand,
with the Fourier Transform.
In
particular
the functionalequation
of the Riemann zeta function is known to beequivalent
to the Poisson summationformula:
which,
forsimplicity,
weapply
to a function1( x)
in the Schwartz class of smoothquickly decreasing
functions.Note 1. We shall make use
of
thefollowing
conventionfor
the FourierTransform:
With a
scaling-parameter u # 0, (1)
leads to:which,
for the GaussianØ(x)
=gives
the Jacobiidentity
for thetheta function
(a
function ofu2).
Riemann obtains from the thetaidentity
one of his
proofs
of the functionalequation
of the zetafunction,
which werecall here in its
symmetrical
form:But there is more to be said on the Riemann sums
p~( u )
fromthe
point
of view of their connections with the Fourier Transform thanjust
the Poisson summation formula
(2);
there holds the co-Poisson intertwin-("dualized
Poisson formula"of Duffin-Weinberger [13]),
whichreads:
, ., B
We show in
[7]
that it isenough
to suppose for itsvalidity
that the inte-grals f~
andJJRg(y)dy
areabsolutely convergent.
The co-Poisson formula thencomputes
the Fourier Transform of alocally integrable
func-tion which is also
tempered
as adistribution,
the Fourier transformhaving
the
meaning given
to itby
Schwartz’stheory
oftempered
distributions. In the case wheng(x)
issmooth, compactly supported
away from x =0,
thenthe
identity
is anidentity
of Schwartz functions. It is afunny thing
thatthe easiest manner to prove for such a
g(x)
that the sides of(4) belong
tothe Schwartz class is to use the Poisson formula
(2)
itself. So the Poisson formulahelps
us inunderstanding
the co-Poisson sums, and the co-Poisson formula tells usthings
on the Poisson-sums.A most
interesting
case arises when the functiong(x)
is anintegrable function, compactly supported
away from x =0,
which turns out to have theproperty
that the co-Poisson formula is anidentity
in Theauthor has no definite
opinion
on whether itis,
or isnot,
an obviousprob-
lem to decide which
(compactly supported
away from x =0)
will besuch that
(one, hence)
the two sides of the co-Poissonidentity
are square-integrable.
Theonly thing
one can say so far is that has to be itselfsquare-integrable.
Note 2. Both the Poisson summation
formulae (1), (2),
and the co-Poissonintertwining formula (4)
tell us 0 = 0 whenapplied
to oddfunctions (tak- ing
derivatives leads tofurther
identities whichapply non-trivially
to odd-functions.)
In all thefollowing
we dealonly
with evenfunctions
on thereal line. The square
integrable
among them will beassigned squared-norm
dt. We let K =
L2(0,
oo;dt),
and we letF+
be the cosine trans-form
on K:The elements
of
K are alsotacitly
viewed as evenfunctions
on R.Let us return to how the functional
equation (3)
relates with(2)
and(4).
The left-hand-side of
(3) is,
forexpression
which is valid in the criticalstrip
is:/ ,
More
generally
we have the Muntz Formula[24, 11.11]:
We call the
expression
inside theparentheses
themodified
Poisson sum(so
the summation isaccompanied
with the substractedintegral). Replac- ing 0(t)
with withg(t) smooth, compactly supported
away from t =0, gives
a formulainvolving
a co-Poisson sum:Let us now write
f (s)
= theright
MellinTransform,
as
opposed
to theleft
Mellin Transform dt. These transformsare
unitary
identifications of K =L2(0, oo; dt)
with£2(s =-1
+iT;
Let 7 be the
unitary operator 1(f)(t)
=f (1 It) / I t I -
Thecomposite F+.1
isscale invariant hence
diagonalized by
the MellinTransform,
and thisgives,
on the critical line:
with a certain function
X(s)
which we obtaineasily
from the choicef (t)
=to be
~rs-1/2r(12s)~r(2),
hence alsoX(s) _ s).
The co-Poisson formula
(4)
follows then from the functionalequation
inthe form
together
with(7)
and(6).
And the Poisson formula(2) similarly
followsfrom
(8) together
with(5).
We refer the reader to[7]
for further discussion andperspectives.
The
general
idea of theequivalence
between the Poisson summation for- mula(2)
and the functionalequation (3),
with an involvement of theleft
Mellin
Transforrrc f(t)tS-1 dt,
has been familiar andpopular
for many decades.Recognizing
that theright
MellinTransform f (t)t-S dt
allowsfor a distinct Fourier-theoretic
interpretation
of the functionalequation
emerged only recently
with ouranalysis [7]
of the co-Poisson formula.2. Sonine spaces of de
Branges
and co-Poissonsubspaces
Let us now discuss some
specific aspects
of the co-Poisson formula(4) (for
an evenfunction):
, - , -
We are
using
theright
Mellin transform =f °° g(t)t-S
dt. Let us takethe
(even) integrable
functiong(t)
to be with itssupport
in[a, A] (and,
aswill be omitted from now on, also
[-A, -a]
ofcourse),
with 0 a A. Letus assume that the co-Poisson sum
F(t) given by
theright
hand sidebelongs
to K =
L2(0,
oo;dt).
It has theproperty
ofbeing equal
to the constant-g(1)
in(0, a)
and with its Fourier(cosine)
transformagain
constant in(0,1/A).
Afterrescaling,
we mayalways
arrange that aA =1,
which wewill assume
henceforth,
so thatI /A
= a(hence, here,
0 a1).
So we are led to associate to each a > 0 the sub-Hilbert space
La
of Kconsisting
of functions which are constant in(0, a)
and with their cosinetransform
again
constant in(0, a). Elementary arguments (such
as theones used in
[7, Prop. 6.6]),
prove that theLa’s
for 0 a oo compose astrictly decreasing
chain of non-trivial infinite dimensionalsubspaces
of Kwith K =
Ua>oLa, {0}
=La
=Ub>aLb (one
may also show thatUb>aLb,
while dense inLa,
is a propersubspace).
This filtration is aslight
variant on the filtration of K which is
given by
the Sonine spacesKa,
a >0,
defined and studied
by
deBranges
in[2].
The Sonine spaceKa
consists ofthe functions in K which are
vanishing identically,
as well as their Fourier(cosine) transforms,
in(0, a).
Theterminology
"Soninespaces",
from[22]
and
[3],
includes spaces related to the Fourier sinetransform,
and also tothe Hankel
transforms,
and is used to refer to some isometric spaces ofanalytic functions;
we will also callKa
andLa
"Soninespaces" .
In thepresent
paper we useonly
the Fourier cosine tranform.Theorem 2.1
(De Branges [2]).
Let 0 a oo. Letf (t) belong
toKa.
Then its
completed right
Mellintransforms
is anentire
function.
The evaluations atcomplex
numbers w are continuouslinear
forms
onKa.
We gave an
elementary proof
of this statement in[6].
See also[8,
Th6or6me
1]
for a useful extension. Someslight change
of variable is nec-essary to recover the
original
deBranges formulation,
as he ascribes to the real axis the roleplayed
hereby
the critical line. Thepoint
of view in[2]
is to start with a direct characterization of the entire functions
M( f ) (s).
Indeed a
fascinating discovery
of deBranges
is that the space of functionsM(f)(s), f
EKa
satisfies all axioms of hisgeneral theory
of Hilbert spacesof entire functions
[3] (we
use the critical line where[3] always
has the realaxis).
It appears to be useful not to focusexclusively
on entirefunctions,
and to allow
poles, perhaps only finitely
many.Proposition
2.2([7, 6.10]).
Letf (t) belong
toLa.
Then itscompleted right
Mellintransform M( f )(s) _
ameromorphic func-
tion in the entire
complex plane,
with at mostpoles
at 0 and at 1. Theevaluation
f H for 0, ~ ~ 1,
orf H Ress=o(M( f )), f H Ress-1(M( f ))
are continuous linearforms
onLa.
One has thefunc-
tional
equations M(,~+( f ))(s)
=M( f )(1 - s).
We will write
a
for the vector inLa
withThis is
0,1.
For w = 0 we haveYo
whichcomputes
the residue at0,
andsimilarly Y#
for the residue at 1. We areusing
the bilinear forms[f, g]
=f (t)g(t)
dt and not the Hermitian scalarproduct ( f , g) = f (t)g(t)
dt in order to ensure that thedependency
ofYw,k
withrespect
to w is
analytic
and notanti-analytic.
There are also evaluators in thesubspace Ka,
which are(for w # 0,1 ) orthogonal projections
fromLa
to
Ka
of the evaluatorsDefinition 3. We let
Ya
CLa
be the closedsubspace
ofLa
which isspanned by
the vectors 1 0 1~ mp, associated to the non-trivialzeros p of the Riemann zeta function with
multiplicity
mp.Definition 4. We let the "co-Poisson
subspace"
be the
subspace
ofsquare-integrable
functionsF(t)
which are co-Poissonsums of a
function g
EL 1 (a, A; dt) (A
=The
subspace
ofKa
definedanalogously
toYa
is denotedZa (rather ZA)
in
[7].
Thesubspace
ofKa analogous
to the co-Poissonsubspace Pa
ofLa
is denoted
Wa (rather WA)
in[7].
One has nKa.
It may beshown that if the
integrable
functiong(t), compactly supported
away from t =0,
has its co-Poisson sum inLa, then g
issupported
in[a, A]
and issquare-integrable
itself.3. Statements of
Completeness
andMinimality
It is a non-trivial fact that
Pa (and Wa also,
as is proven in[7])
is closed.This is
part
of thefollowing
two theorems.Theorem 3.1. The vectors
ya P, ki
0 k mp, associated with the non- trivial zerosof
the Riemann zetafunction,
are a minimalsystem
inLa if
and
only if a
1.They
are acomplete system if
andonly if
a > 1. Fora 1 the
perpendicular complement
toYa is
the co-Poissonsubspace Pd.
For a > 1 we may omit
arbitrarily (finitely)
manyof
the and stillhave a
complete system
inLa.
Theorem 3.2. The vectors
ZP ~, 0 k
mp, are aminimal,
but not com-plete, system for
a 1.They
are not minirraalfor
a =1~
but thesystem
obtainedfrom omitting
2arbitrarily
chosen among them(with
the conven-tion that one either orrtits and
ZP,,",,P_2
or andZP,,~~~P~_1)
is
again
a minimalsystem,
which is alsocomplete
in In the case a > 1 the vectors arecomplete
inKa,
evenafter omitting arbitrarily (finitely)
many among them.
Remark 5. This is to be contrasted with the fact that the evaluators associated with the non-trivial zeros of a Dirichlet L-function
x) (for
an even
primitive
character of conductorq)
are acomplete
and minimalsystem
inCompleteness
was proven in[7, 6.30],
andminimality
isestablished as we will do here for the Riemann zeta function.
We use the
terminology
that an indexed collection of vectors in aHilbert space K is said to be minimal if no ua is in the closure of the linear span of the
u(3 ’s, (3 ~
a, and is said to becomplete
if the linear span of theua’s
is dense in K. To each minimal andcomplete system
is associateda
uniquely
determined dualsystem (va)
with(v(3, ua) _ (actually
inour
Ld’s,
we use rather the bilinear form~f,g~
= Such adual
system
isnecessarily minimal,
butby
no meansnecessarily complete
in
general (as
anexample,
one may take ~n = 1 -zn,
n >1,
in theHardy
space of the unit disc. Then Vm = for m >
1,
andthey
are notcomplete).
For
simplicity sake,
let us assume that the zeros are allsimple. Then,
once we know that
((S)/(8 - p),
for p a non-trivial zero,belongs
to thespace
Li
of(right)
Mellin transforms of elements ofL1,
we thenidentify
the
system
dual to the asconsisting
of(the
inverse Mellin transformsof)
the functions((S)/((8 - ~(p)7r’~r(~/2)).
Without anysimplifying assumption,
we still have that the dualsystem
is obtained from suitable lin-ear combinations
(it
does not seem very useful tospell
them outexplicitely)
of the functions
((S)/(8 - p)~,
1 l p a non-trivial zero.The
proofs
of 3.1 and 3.2 are a furtherapplication
of thetechnique
of[7, Chap.6],
which uses a Theorem of Krein on Nevanlinna functions[19, 17].
Another
technique
is needed to establish thecompleteness
inLi
of the functions((8)/(8 - p)i,
1::; 1
?7~:Theorem 3.3. The
functions ((8)/(8 - p)l, for
p a non-trivial zero and1 ::; 1
mpbelong
toL1. They
are minimal andcomplete
inLi.
The dualsystem
consistsof
vectorsgiven for
each pby triangular
linear combinationsof
the evaluators0 k
mp.There appears in the
proof
of 3.3 somecomputations
of residues which arereminiscent of a theorem of
Ramanujan
which is mentioned in Titchmarsh[24, IX.8.].
The last section of the paper deals with the zeros of an
arbitrary
Soninefunctions,
and with theproperties
of the associated evaluators. We obtain inparticular
adensity
result on the distribution of its zeros, with thehelp
of the
powerful
tools from the classicaltheory
of entire functions[20].
4.
Aspects
of Sonine functionsNote 6. We let
La
be the vector spaceof right
Mellintransforms of
ele-ments
of La (and similarly for Ka). They
aresquare-integrable functions
on the critical
line, which,
as weknow from
2.2 are alsomeromorphic
in theentire
complex plane.
We are notusing
here theGamma-completed
Mellintransform,
but the bare Mellintransform f (s),
whichaccording to
2.2 hastrivial zeros at
-2n,
n >0,
andpossibly
apole
at s =1~
andpossibly
doesnot vanish at s = 0.
Definition 7. We
let 1HI2
be theHardy
space of theright half-plane Re(s)
>1/2.
Wesimultaneously view]HI2
as asubspace
ofL2(Re(s) = 2,
andas a space of
analytic
functions in theright half-plane.
We also use self-explanatory
notations such asThe
right
Mellin transform is an isometric identification ofL2(1, oo; dt)
with this is one of the famous theorems of
Paley-Wiener [21],
af-ter a
change
of variable.Hence,
for 0 a and A =1 /a,
theright
Mellin transform is an isometric identification of with
ASIHf2.
Furthermore,
theright
Mellin transform is an isometric identification of C .loto,
+L2( a,
oo;dt)
with This leads to thefollowing
char-acterization of
La:
Proposition
4.1. Thesubspace La of L2(Re(s)
= consistsof
the measurable
functions F(s)
on the critical line whichbelong to
and are such that
s)
alsobelongs to
Such afunction F(s) is
the restriction to the critical lineof
ananalytic function,
meromor-phic
in the entirecomplex plane
with at most apole
at s =1,
and withtrivial zeros at s =
-2n,
n EI~,
n > 0.Proof.
We knowalready
from 2.2 that functions inLa
have the statedproperties.
If a functionF(s) belongs
to viewed as a space of(equivalence
classesof)
measurable functions on the criticalline,
thenit is
square-integrable
and is the Mellin transform of an elementf (t)
ofC.
10ta
+L2(a,
oo;dt).
We know that the Fourier cosine transform off
has
x(s)F(1- s)
as Mellintransform,
so the second condition on F tells usthat
f belongs
toLa.
0We recall that
X(s)
is the function(expressible
in terms of the Gammafunction)
which is involved in the functionalequation
of the Riemann zetafunction
(8),
and is in fact thespectral multiplier
of the scale invariantoperator F+ - I,
for theright
Mellin transform.Note 8.
Abusively,
we will say thatX(s)F(1- s)
is the Fouriertransform of F(s),
and will sometimes even insteadof x(s)F(l - s).
It is
zcseful
to take note thatif
we writeF(s) = ~’(s)B(s)
we then haveX(s)F(l - s) = ((s)O(l - s).
’
Proposition
4.2. Thefunctions (( s) / (s - p)’,
1 1 mp associated with the non-trivial zerosof
the Riemann zetafunction belong
toL1.
Proof.
The functionF(s) = ((s)/(s- p)l
issquare-integrable
on the critical line. AndX (s) F (1 - s) = 1) 1 ( (s) / (s - (1- p)) 1.
So weonly
need to prove thats S 1 ~(s)/(s - p)l belongs
to This is well-known to be true ofs-/((8)/8 (from
the formula((8)/8
=11(s - 1) - It .rp:rSdt,
valid for 0
Re(s)),
hence it holds also for If we exclude aneigborhood
of p thenp)l
isbounded,
sogoing
back to the definitionof]HI2
as a space ofanalytic
functions in theright half-plane
with a uniformbound of their
L2
norms on vertical lines we obtain the desired conclusion.D The
following
will be useful later:Proposition
4.3.If G(s) belongs
toLa
and vanishesat s = w then
G(s)/(s - w) again belongs
toLa. If G(s) belongs
toKa
and7r-S/2r()G(s)
2 vanishes at s = w thenG(s)/(s - iv) again belongs
toKa.
Proof.
We could prove this in the"t-picture",
but will do it in the "s-picture".
We see as in thepreceding proof
thatG(s)/(s - w)
stillbelongs
The entire function
s(s-
vanishes at s = wso
s(s-1)~r’-s~2I~(2).~+(G)(s)
vanishes at s = 1-w and the sameargument
then shows that
X(s)G(l - s)/(l -
s -~v) belongs
to We thenapply Proposition
4.1. The statement forKa
is provenanalogously.
0Note 9. It is a
general
truth in all deBranges’ spaces
that such a statementholds
for zeros
wsymmetry
axis(which
is here the criticalline).
Thisis,
infact,
almost oneof
the axiomsfor
deBranges’
spaces. Thepossibility
to divide
by (s - w) if
w is on thesymmetry
axisdepends
on whether the structurefunction
E(on this,
werefer
to[3])
is notvanishing
orvanishing
at w. For the Sonine spaces, the
proposition 4.3
proves that the structurefunctions Ea(z)
have no zeros on thesymmetry
axis. For more on theEa(z)’s
and alliedfunctions,
see[8]
and[9].
A variant on this
gives:
Lemma 4.4.
If F(s) belongs
toka
thenF(s)/s belongs
toL~,.
Proof.
The functionF(s)/s (which
isregular
at s =0) belongs
to the spaceAsH2, simply
fromRe(s) > 2.
Itsimage
under the Fourier transform is.~’+(F)(s)/(1 - s)
whichbelongs
to 0Proposition
4.5. One hasdim(La/Ka)
= 2.Proof.
This isequivalent
to the fact that the residue-evaluatorsYo
andYl
are
linearly independent
inLa,
which may be established in a number ofelementary
ways; wegive
twoproofs.
Evaluators off thesymmetry
axisare
always
non-trivial in deBranges
spaces so there isF(s)
EKa
withF’(0) ~
0(one
knows further From[6,
Th6or6me2.3.]
that any finitesystem
of vectorsZa.,k
inKa
is alinearly independent system).
So we haveF(s)/s
=G(s)
ELa
notvanishing
at 0 but with nopole
at 1. Its "Fourier transform"s)
vanishes at 0 but has apole
at 1. This provesdim(La/Ka) >
2 and the reverseequality
follows from the fact that thesubspace Ka
is definedby
two linear conditions.For the second
proof
we go back to theargument
of[6]
which identifies theperpendicular complement
toKa
inL2(0,
oo;dt)
to be the closed spaceL2(0, a)+F+(L2(0, a)).
It is clear thatLa
is theperpendicular complement
to the
(two dimensions)
smaller space(L2 (0, a)
f1 +.~+(L2(~, a)
nand this proves 4.5. D
The
technique
of the secondproof
has the additional benefit:Proposition
4.6. The unionUb» Kb is
dense inKa,
andLb
is densein
Ld.
Proof. Generally speaking
+Bb) = (nb>aAb)
+ when wehave vector spaces indexed
by
b > a withAbi
CA62
andBbl
CBb2
forb1 b2
andAb n Bb
=101
for b > a. Weapply
this toAb
=L2 (0, b; dt)
andBb
=+L2 b, lt,
asKa
=(Aa
+Aa
=nb>aab, Ba
=nb>aBb,
and
Aa
+Ba
is closed as asubspace
ofL2(0,
oo;dt).
0Proposition
4.7. The vector spaceKb
isproperly
included inKa
andthe same holds
for
therespective subspaces of
Fourierinvariant,
orskew, functions (and similarly for La).
Proof. Let g
EKb,
with b > aand g having
the leftmostpoint
of itssupport
at b. Then has the leftmostpoint
of itssupport
at a.If g
is invariant under Fourier then we use +
.jfg(atjb) to
obtainagain
an invariantfunction,
with leftmostpoint
of itssupport
at a. D Definition 10. We say that a functionF(s), analytic
in C with at mostfinitely
manypoles,
has theL-Property
if the estimatesF(Q
+iT) _
+ hold
(away
from thepoles),
for -oo a ab oo, E > 0.
-
Theorem 4.8. The
functions
inLa
have theL-Property.
Proof.
Letg(t)
be a function inLa
and letG(s)
=Jooo g(t)t-S
dt be itsright
Mellin transform. The functiong(t)
is a constanta(g)
on(0, a).
Anexpression
forG(s)
as ameromorphic
function(in -1 Re(s) 1, hence)
in the
right half-plane
is:We established in
[6]
a few results of anelementary
nature about the func- tions which are denoted thereCa(u,1 - s) (in particular
we showed that these functions are entire functions of
s).
ForRe(s)
1one has
according
to[6,
eq.1.3.]:
hence for 0
Re(s)
1:This is bounded
on 4 Re(s) 4 (using
the well-known uniform esti- mateIIm(s)/27rI-Re(s)+1/2
aslim(s)l
- oo in verticalstrips [24, IV.12.3.~).
We also have fromintegration by parts
andanalytic
continua-tion to
Re(s
> 0 theexpression:
-- ..
which is
O(lsl/u) on 1 Re(s) 4,
0 u.Combining
all this we find theestimate: -1 -
This
(temporary)
estimatejustifies
the use of thePhragmen-Lindelof prin- ciple
from bounds onRe(s) = -1 ±,E.
On anyhalf-plane Re(s) > 2
+ 6> 2
(excluding
of course aneighborhood
of s =1)
one hasG(s)
=O(ARe(s))
from the fact that
(s - 1)G(s)ls belongs
toAsH 2
and that elements ofH 2
are bounded inRe(s) > 2
+ E> 2.
And the functionalequation
2 2
G(1 - s)
=X(1 - gives
us estimates on the lefthalf-plane.
This shows that the
L-Property
holds forG(s).
Inparticular,
the Lindel6f exponentsJIG (a)
are at most 0 and atmost 2 -
Q for Q2.
DRemark 11. In
fact,
theproof given
above establishes theL-Property
for
G(s)
in astronger
form than stated in the definition 10. One has forexample G(s)
=0,7((l
+ foreach q
>0,
on thestrip! -1}
G(s)
= for each c > 0on 1 Re(s)
1(away
from theallowed
pole
at s =1),
andG(s)
= onRe(s) > 2
+ > 0.Definition 12. We let
£1
to be the sub-vector space ofL1 containing
thefunctions
g(t)
whoseright-Mellin
transformsG(s)
are on allvertical
strips a Re(s) b,
and for allintegers
N > 1(away
from thepole,
and theimplied
constantdepending
on g, a,b,
andN).
Theorem 4.9. The sub-vector space
Ll
is dense inL1.
Proof.
Fromproposition
4.6 weonly
have to show that any functionG(s)
ina
Lb,
b > 1 is in the closureof Z-1.
For this let0(s)
be the Mellin transform of a smooth function withsupport
insatisfying 0(i)
= 1. Thefunction
0(s)
is an entire function which decreases faster than any(inverse)
power
of Isl
asIIm(s)1
-> oo in anygiven strip
a ::; a G b. Let us considerthe functions =
0(,E(s - 1) + !)G(s)
as e ~ 0. On the critical linethey
are dominated
by
a constantmultiple
of sothey
aresquare-integrable
and converge in
L2-norm
toG(s).
We prove that for 1exp(-E)b
bthese functions all
belong
toG1.
Theirquick
decrease in verticalstrips
isguaranteed by
the fact thatG(s)
has theL-Property.
The functionon the critical line is the Mellin transform of a
multiplicative
convolutionon
(0, oo)
of an element inL2(b, oo)
with a smooth functionsupported
inThe
support
of thismultiplicative
convolution will be included in if 1exp(-E)b.
So for those E > 0 one hasG,(s)
EIts
image
under.~’+
is0(-E(s - 1)
+])7+ (G) (s)
=OT(E(S - 2)
+2).~’+(G)(s)
where07’(w)
=0 (1 - w)
has the sameproperties
as0 (w) (we
recall our abusive notation =
X(s)F(l - s).)
Hencealso
belongs
tos H2
and thiscompletes
theproof
thatG, (s)
E£i .
0Lemma 4.10. The
subspace Ll
is stable underJc*+.
Proof.
Clear fromthe estimates of x(s)
in verticalstrips ([24, IV. 12.3.]).
05.
Completeness
of thesystem
of functions((s)/(s - p)
We will use a classical estimate on the size of
((s)-l:
Proposition
5.1(from [24, IX.7.]).
There is a real number A and astrictly increasing
sequenceTn
> n such that1((s)l-l IslA
onIIm(s)1
=Tn,
-1
Re(s)
+2.Note 13. From now on an
infinite
sum¿pa(p) (with complex
numbers orfunctions
or Hilbert space vectorsa(p)’s
indexedby
the non-trivial zerosof
the Rierraann zeta
function)
meansa(p),
where the limitmight be, if
we aredealing
withfunctions,
apointwise
almosteverywhere limit,
or a Hilbert space limit. When we say that thepartial
sums arebounded
(as complex numbers,
or as Hilbert spacevectors)
weonly refer
tothe
partial
sums as written above. When zve say that the series isabsolutely convergent
it means that we grouptogether
the contributionsof
thep’s
withTn lim(p) I Tn+l before evaluating
the absolute value or Hilbert norm.When
building
seriesof
residues we write sometimesthings
asif
the zeroswere all
simple:
this isjust
to make the notationeasier,
but nohypothesis
is made in this paper on the
multiplicities
mp, and theformula
usedfor writing a(p)
is asymbolic representation,
validfor
asimple
zero,of
the morecomplicated expression
which wouldapply
in caseof multiplicity,
which wedo not
spell
outexplicitely.
Theorem 5.2. Let
G(s)
be afunctions
inL1
whichbelongs
to the densesubspace G1 of functions
withquick
decrease in verticalstrips.
Then theseries
of
residuesfor
afixed Z =f. 1,
not a zero:converges
absolutely pointwise
toG(Z)
on(C ~ 111.
It also converges abso-Lutely
inL2-norm
toG(Z)
on the critical line.This is a series of residues for
f
where s is the variable 1Z-s
is a
parameter (with
theexception
of the residue at s =Z).
We havewritten the contribution of p as if it was
simple (Titchmarsh
uses a simlarconvention in
[24, IX.8.]).
In fact the exactexpression
is a linear combina- tion of( (Z) / (Z - p)l,
1 L mp . We note that the trivial zeros and s = 1are not