Moments of zeta and correlations of divisor-sums: V

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by  Brian Conrey

MOMENTS OF ZETA AND CORRELATIONS OF DIVISOR-SUMS: V
7 BRIAN CONREY AND JONATHAN P. KEATING
1
0
2 Abstract. In this series of papers we examine the calculation of the 2kth moment and
shifted moments ofthe Riemann zeta-functiononthe criticalline using long Dirichletpoly-
n
nomialsanddivisorcorrelations. Thepresentpapercompletesthegeneralstudyofwhatwe
a
J callType II sums whichutilize acirclemethod frameworkandaconvolutionofshifted con-

3 volutionsumsto obtainallofthe lowerordertermsin the asymptoticformulafor the mean
2 square along [T,2T] of a Dirichlet polynomial of arbitrary length with divisor functions as

coefficients.
]
T
N
.
h
t 1. Introduction
a
m
This paper is part V of a sequence devoted to understanding how to conjecture all of the
[ integral moments of the Riemann zeta-function from a number theoretic perspective. The

1 method is to approximate ζ(s)k by a long Dirichlet polynomial and then to compute the
v
mean square of the Dirichlet polynomial (c.f. [GG]). There are many off-diagonal terms and
1
5 it is the care of these that is the concern of these papers. In particular it is necessary to
6 treat the off-diagonal terms by a method invented by Bogomolny and Keating [BK1, BK2].
6
Our perspective on this method is that it is most properly viewed as a multi-dimensional
0
. Hardy-Littlewood circle method.
1
0 In previous papers [CK1, CK3] we have developed a general method to calculate what
7 were called type-I off-diagonal contributions in [BK1, BK2]; these are the off-diagonal terms
1
usually considered in number-theoretic computations, e.g. in [GG]. In parts II [CK2] and
:
v
IV [CK4] we considered the simplest of the type-II off-diagonal terms (in the terminology of
i
X [BK1, BK2]). These, somewhat unexpectedly, give a significant contribution in certain cases.
r They have not previously been analysed systematically. Our purpose here is to develop a
a
general method for computing all type-II off-diagonal terms.
The formula we obtain is in complete agreement with all of the main terms predicted by
the recipe of [CFKRS] (and in particular, with the leading order term conjectured in [KS]).
Date: January 25, 2017.
We gratefully acknowledge support under EPSRC Programme Grant EP/K034383/1LMF: L-Functions
and Modular Forms. Research of the first author was also supported by the American Institute of Mathe-
matics and by a grant from the National Science Foundation. JPK is grateful for the following additional
support: a grant from the Leverhulme Trust, a Royal Society Wolfson Research Merit Award, a Royal So-
ciety Leverhulme Senior Research Fellowship, and a grant from the Air Force Office of Scientific Research,
AirForceMaterialCommand,USAF(numberFA8655-10-1-3088). HeisalsopleasedtothanktheAmerican
Institute of Mathematics for hospitality during two visits when work on this project was conducted.
1
2 BRIAN CONREY ANDJONATHANP. KEATING
2. Shifted moments
We follow the notation of the [CK] papers.
The moment we are interested in is
∞ t
ψ
(1) R (T) = ψ ζ(s+α) ζ(1 s+β) dt
A,B Z (cid:18)T(cid:19) −
0 αY∈A βY∈B
where ψ is a smooth function with compact support, say ψ C∞[1,2], and s = 1/2 + it.
∈
The recipe [CFKRS] tells us how we expect this moment will behave, namely
∞ tT −Pαˆ∈U(αˆ+βˆ)
Rψ (T) = T ψ(t) βˆ∈V (A U +V−,B V +U−,1) dt+o(T)
A,B Z (cid:18)2π(cid:19) B − −
0 X
U⊂A,V⊂B
|U|=|V|
where
∞
τ (n)τ (n)
A B
(A,B,s) := .
B ns
Xn=1
We have used an unconventional notation here; by A U +V− we mean the following: start
−
with the set A and remove the elements of U and then include the negatives of the elements
of V. We think of the process as “swapping” equal numbers of elements between A and B;
when elements are removed from A and put into B they first get multiplied by 1. We keep
−
track of these swaps with our equal-sized subsets U and V of A and B; and when we refer
to the “number of swaps” in a term we mean the cardinality U of U (or, since they are of
| |
equal size, of V).
The technique we are developing in this sequence of papers is to approach the moment
problem (1) through long Dirichlet polynomials, i.e. we consider
∞ t
Iψ (T;X) : = ψ D (s;X)D (1 s;X) dt
A,B Z (cid:18)T(cid:19) A B −
0
τ (m)τ (n)ψˆ T log m
(2) = T A B 2π n
√mn(cid:0) (cid:1)
mX,n≤X
for various ranges of X. We can use the recipe of [CFKRS] to conjecture a formula for Iψ.
We start with
1 Xw
D (s;X) = D (s) dw.
A 2πi Z w Aw
w
Thus,
1 Xz+w
Iψ (T;X) = Rψ (T) dw dz.
A,B (2πi)2 ZZ zw Aw,Bz
z,w
MOMENTS OF ZETA AND CORRELATIONS OF DIVISOR-SUMS: V 3
We insert the conjecture above from the recipe and expect that
∞ 1 Xz+w tT −Pαˆ∈U(αˆ+w+βˆ+z)
Iψ (T;X) = T ψ(t) βˆ∈V
A,B Z (2πi)2 ZZ zw (cid:18)2π(cid:19)
0 z,w X
U⊂A,V⊂B
|U|=|V|
(A U +V−,B V +U−,1) dw dz dt+o(T).
×B w − w z z − z w
We have done a little simplification here: instead of writing U A we have written U A
w
⊂ ⊂
and changed the exponent of (tT/2π) accordingly.
Notice that there is a factor (X/T|U|)w+z in the previous equation. As mentioned above
we refer to U as the number of “swaps” in the recipe, and now we see more clearly the role
| |
it plays; in the terms above for which X < T|U| we move the path of integration in w or z
to + so that the factor (X/T|U|)w+z 0 and the contribution of such a term is 0. Thus,
∞ →
the size of X determines how many “swaps” we must keep track of.
In [CK1] and [CK3] we considered 0 and 1 swap. In [CK2] we considered two swaps but
in a special case. In [CK4] we looked at the general case of two swaps. Now we look at
the general case with any number of swaps. We have also begun to explore the analogous
calculations for averages of ratios of the zeta function, specifically in the context of zero
correlations [CK5, CK6].
3. Type II convolution sums
We make use of the fact that if A = A A A and B = B B B then τ
1 2 ℓ 1 2 ℓ A
∪ ···∪ ∪ ···∪
and τ are convolutions: τ = τ τ τ and τ = τ τ τ . We are thus
B A A1 ∗ A2 ∗···∗ Aℓ B B1 ∗ B2 ∗···∗ Bℓ
interested in
τ (m )...τ (m )τ (n )...τ (n ) T
= A1 1 Aℓ ℓ B1 1 Bℓ ℓ ψˆ( log((n ...n )/(m ...m )).
II 1 ℓ 1 ℓ
O m ...m 2π
X 1 ℓ
m1m2...mℓ≤X
n1n2...nℓ≤X
0<|m1···ℓ−n1...nℓ|<X/τ
Now we embark on a discrete analog of the circle method which basically consists of approx-
imating a ratio, say m /n , by a rational number with a small denominator, say M /N , for
j j j j
j = 1,2,...,ℓ and then sum all of the terms with m /n close to M /N .
j j j j
Now given an M and N we sum over the terms m and n for which m /n ;
j j j j j j ∈ MMj,Nj
for such a pair we define
h := n M m N .
j j j j j
−
ˆ
In general, the rapid decay of ψ governs the range of the h ; see below.
j
We have
n ...n M ...M = (m N +h )...(m N +h )
1 ℓ 1 ℓ 1 1 1 ℓ ℓ ℓ
so that
n ...n h h
1 ℓ 1 ℓ
= 1+ ... 1+
m ...m m N m N
1 ℓ (cid:0) 1 1(cid:1) (cid:0) ℓ ℓ(cid:1)
4 BRIAN CONREY ANDJONATHANP. KEATING
and
n ...n h h h h
1 ℓ 1 ℓ i j
log = +... +O .
m ...m m N m N m ...m N ...N
1 ℓ 1 1 ℓ ℓ (cid:0) 1 ℓ 1 ℓ(cid:1)
The error term is negligible so we can arrange the sum as
τ (m )...τ (m )τ (n )...τ (n ) Th Th
A1 1 Aℓ ℓ B1 1 Bℓ ℓ ψˆ 1 + + ℓ
m ...m (cid:18)2πm N ··· 2πm N (cid:19)
X X X 1 ℓ 1 1 ℓ ℓ
Mj≤Nj≤Q h1,...hℓ m1...mℓ≤X
(Mj,Nj)=1 (∗1),...(∗ℓ)
M1...Mℓ=N1...Nℓ
where
( ) : m N n M = h .
j j j j j j
∗ −
4. h ...h = 0
1 ℓ
6
Let us first look at the situation where none of the h is 0.
j
We replace the convolution sums by their averages, i.e.
Th Th du du
τ (m )τ (n ) (∗1) ... τ (m )τ (n ) (∗ℓ) ψˆ 1 +... ℓ 1 ... ℓ
Z h A1 1 B1 1 im1∼u1 h Aℓ ℓ Bℓ ℓ imℓ∼uℓ (cid:18)2πu N 2πu N (cid:19) u u
u1...uℓ≤X 1 1 ℓ ℓ 1 ℓ
where, for example, with : mN nM = h we expect by the delta-method [DFI] that
∗ −
∞
1
τ (m)τ (n) (∗) r (h) τ (m)e(mN/q) τ (n)e(nM/q)
h A B im∼u ∼ M Xq=1 q h A im∼uh B in∼uMN
where
1 N
τ (m)e(mN/q) = D (w,e )uw−1 dw.
A m∼u A
h i 2πi Z q
|w−1|=ǫ (cid:0) (cid:1)
So, we are led to
1
r (h )...r (h )
M ...M q1 1 qℓ ℓ Z
Mj,NjX,(Mj,Nj)=1 1 ℓ qXj,hj Tℓ≤u1...uℓ≤X
N1...Nℓ=M1...Mℓ
ℓ ℓ
1 N 1 M u N
D (w ,e j )uwj−1dw D (z ,e k ) k k zk−1dz
× 2πi Z Aj j q j j 2πi Z Bk k q M k
Yj=1 |wj−1|=ǫ (cid:0) j (cid:1) kY=1 |zk−1|=ǫ (cid:0) k (cid:1) (cid:0) k (cid:1)
h h du du
ˆ 1 ℓ 1 ℓ
ψ T + + ... .
× (cid:18) (cid:18)2πu N ··· 2πu N (cid:19)(cid:19) u u
1 ℓ ℓ 1 ℓ
We make the changes of variable v = T|hj| and bring the sums over the h to the inside;
j 2πujNj j
u ...u < X implies that
1 ℓ
(2π)ℓXv ...v N ...N
1 ℓ 1 ℓ
h ...h < .
| 1 ℓ| Tℓ
MOMENTS OF ZETA AND CORRELATIONS OF DIVISOR-SUMS: V 5
We detect this condition using Perron’s formula in an integral over s. Then the above is
1 1 r (h )...r (h )
q1 1 qℓ ℓ
Z M ...M 2πi Z (h ...h )s
0<v1,...,vℓ<∞ Mj,NjX,(Mj,Nj)=1 1 ℓ (2)qXj,hj 1 ℓ
N1...Nℓ=M1...Mℓ
ℓ
1 N Th
D (w ,e j ) j wj−1dw
× 2πi Z Aj j q 2πv N j
Yj=1 |wj−1|=ǫ (cid:0) j (cid:1) (cid:0) j j(cid:1)
ℓ
1 M Th
D (z ,e k ) k zk−1dz
B k k
2πi Z k q 2πv M
Yk=1 |zk−1|=ǫ (cid:0) k (cid:1) (cid:0) k k(cid:1)
(2π)ℓXv ...v N ...N s dv dv ds
1 ℓ 1 ℓ ˆ 1 ℓ
ψ(ǫ v + +ǫ v ) ...
×(cid:18) Tℓ (cid:19) 1 1 ··· ℓ ℓ v v s
1 ℓ
where ǫ = sgn(h ). We simplify this a bit. We combine the middle two lines into a single
j j
product over j and gather together all of the like variables:
1 T
LHS := Xs −ℓs ψˆ(ǫ v + +ǫ v )
1 1 ℓ ℓ
2πi Z 2π Z ···
(2) (cid:0) (cid:1) (M1,N1)=X···=(Mℓ,Nℓ)=1 0<v1,...,vℓ<∞
N1...Nℓ=M1...Mℓ
ǫj∈{−1,+1}
ℓ
1 r (h )
M−zjNs+1−wj qj j vs+1−wj−zj
Yj=1(cid:20)(2πi)2 ZZ||wzkj−−11||==ǫǫ j j hXj,qj hjs+2−wj−zj j
N M T ds
D (w ,e j )D (z ,e j ) wj+zj−2dw dz dv
Aj j q Bj j q 2π j j j(cid:21) s
(cid:0) j (cid:1) (cid:0) j (cid:1) (cid:0) (cid:1)
Now we have our key identity:
Theorem 1.
1 ∞ Tt −Pαˆ∈U(ℓ)(αˆ+βˆ+s)
LHS = Xs ψ(t) βˆ∈V(ℓ)
2πi Z Z (cid:18)2π(cid:19)
(2) 0 X
U(ℓ)⊂A
V(ℓ)⊂B
ds
(A U(ℓ) +V(ℓ)−,B V(ℓ)+U(ℓ) −,s+1) dt
s s s
×B − − s
where U(ℓ) denotes a set of cardinality ℓ with precisely one element from each of A ,...,A
1 ℓ
and similarly V(ℓ) denotes a set of cardinality ℓ with precisely one element from each of
B ,...,B .
1 ℓ
5. Preliminary reductions
Using
∞
ˆ
ψ(v + +v ) = ψ(t)e(t(v + +v )) dt
1 ℓ 1 ℓ
··· Z ···
0
6 BRIAN CONREY ANDJONATHANP. KEATING
we see that
∞
ˆ
ψ(ǫ v + +ǫ v ) = ψ(t) e(tv )+e( tv ) ... e(tv )+e( tv ) dt;
1 1 ℓ ℓ 1 1 ℓ ℓ
··· Z − −
ǫj∈{X−1,+1} 0 (cid:0) (cid:1) (cid:0) (cid:1)
also
∞
vs+α+β−1(e(tv)+e( tv)) dv = t−s−α−βχ(1 s α β).
Z − − − −
0
Thus,
∞ ℓ
ψˆ(ǫ v +...ǫ v ) vs+1−wj−zjdv
1 1 ℓ ℓ j j
Z
ǫj∈{X−1,+1} v1,...,vℓ=0 Yj=1
∞ ℓ
= ψ(t) χ(w +z s 1)twj+zj−s−2 dt.
j j
Z − −
0 Yj=1
6. Poles
We have
1 1 u
w−1
τ (m)e(m/q) ζ(w+α)G (w,q) dw,
A m∼u A
h i ∼ q2πi Z q
|w−1|=81 αY∈A (cid:0) (cid:1)
where G is a multiplicative function for which
∞ τ (pj+r)
G (1 α,pr) = A′
A − pj(1−α)
Xj=0
with A′ = A α . With : mN nM = h, this leads to
−{ } ∗ −
∞
1
τ (m)τ (n) (∗) r (h) τ (m)e(mN/q) τ (n)e(nM/q)
h A B im=u ∼ M Xq=1 q h A im∼uh B in∼uMN
1
u−α−βM−1+βN−βZ(A′ )Z(B′ )
∼ −α −β d1−α−β
X X
α∈A d|h
β∈B
µ(q)(qd,M)1−β(qd,N)1−α qd qd
G 1 α, G 1 β, ),
× q2−α−β A − (qd,N) B − (qd,M)
Xq (cid:0) (cid:1) (cid:0)
where
Z(A) = ζ(1+a).
aY∈A
MOMENTS OF ZETA AND CORRELATIONS OF DIVISOR-SUMS: V 7
Inserting this into LHS we have
1 T
LHS = Xs −ℓs ψˆ(ǫ v + +ǫ v )
1 1 ℓ ℓ
2πi Z 2π Z ···
(2) (cid:0) (cid:1) (M1,N1)=X···=(Mℓ,Nℓ)=1 0<v1,...,vℓ<∞
N1...Nℓ=M1...Mℓ
ǫj∈{−1,+1}
ℓ Mβj−1Ns+αj 1 µ(qj)(qjdj,Mj)1−βj(qjdj,Nj)1−αj
Yj=1(cid:20) αXj∈Aj j j hjX,qj,dj (hjdj)s+αj+βj qj2−αj−βj
βj∈Bj
q d q d T ds
vs−1+αj+βjG 1 α , j j G 1 β , j j ) −αj−βjdw dz dv
j Aj − j (q d ,N ) Bj − j (q d ,M ) 2π j j j(cid:21) s
(cid:0) j j j (cid:1) (cid:0) j j j (cid:0) (cid:1)
Now we sum over the h to get factors ζ(s+α +β ); these pair up with the factors χ(w +
j j j j
z s 1) which turned into χ(1 α β s) after collecting the residues w = 1 α
j j j j j
− − − − − −
and z = 1 β that arose from the integral over v . Then using the functional equation for
j j j
ζ we have ζ−(1 s α β ). Thus, the s-integrand without the Xs in LHS becomes
− − j − j s
∞ T ℓ
LHS = ψ(t) −ℓs ζ(1 s α β )t−s−αj−βj
j j
Z 2π − − −
0 (cid:0) (cid:1) (M1,N1)=X···=(Mℓ,Nℓ)=1 Yj=1
N1...Nℓ=M1...Mℓ
Mβj−1Ns+αj 1 µ(qj)(qjdj,Mj)1−βj(qjdj,Nj)1−αj
(cid:20) j j ds+αj+βj q2−αj−βj
αXj∈Aj qXj,dj j j
βj∈Bj
q d q d T
G 1 α , j j G 1 β , j j ) −αj−βjdw dz dt,
Aj − j (q d ,N ) Bj − j (q d ,M ) 2π j j(cid:21)
(cid:0) j j j (cid:1) (cid:0) j j j (cid:0) (cid:1)
and our goal is to prove that this is equal to
∞ Tt −Pαˆ∈U(ℓ)(αˆ+βˆ+s)
ψ(t) βˆ∈V(ℓ)
Z (cid:18)2π(cid:19)
0 X
U(ℓ)⊂A
V(ℓ)⊂B
(A U(ℓ) +V(ℓ)−,B V(ℓ)+U(ℓ) −,s+1) dt.
s s s
×B − −
8 BRIAN CONREY ANDJONATHANP. KEATING
This further reduces to proving
ℓ
ζ(1 s α β )
j j
− − −
(M1,N1)=X···=(Mℓ,Nℓ)=1 Yj=1
N1...Nℓ=M1...Mℓ
Mβj−1Ns+αj 1 µ(qj)(qjdj,Mj)1−βj(qjdj,Nj)1−αj
(cid:20) j j ds+αj+βj q2−αj−βj
αXj∈Aj qXj,dj j j
βj∈Bj
q d q d
j j j j
G 1 α , G 1 β , )dw dz
Aj − j (q d ,N ) Bj − j (q d ,M ) j j(cid:21)
(cid:0) j j j (cid:1) (cid:0) j j j
= (A U(ℓ) +V(ℓ)−,B V(ℓ)+U(ℓ) −,s+1).
s s s
B − −
7. Local considerations
We shall find it convenient to state our main theorem as an identity of the Euler factor
at a prime p. We begin by introducing a set-theoretic notation. First of all, since p is fixed
for this discussion we will often suppress it. In fact we write X for 1/p and mostly consider
power series in X. We take the unusual step of suppressing not only the prime p but the
divisor function and so we write A(n) in place of τ (pn). Also, for a set A we let
A
A = a+α : a A .
α
{ ∈ }
A further piece of notation: A+ = A 0 . We have two important identities. The first is
∪{ }
A+(d) = A(d)+A+(d 1).
−
This is a special case of
(A α )(d 1) = Xα (A α )(d) A(d) .
∪{− } − (cid:18) ∪{− } − (cid:19)
The other identity is
R
A(r +M) = A+(R+M) A+(M 1)
− −
Xr=0
which follows by repeated application of the first identity.
For arbitrary sets A,B we let
∞
C(A,B) := A(M)B(M)XM.
MX=0
Also, we let
∞
Z(A) = A(j)Xj = (1 X1+a)−1.
−
Xj=0 aY∈A
MOMENTS OF ZETA AND CORRELATIONS OF DIVISOR-SUMS: V 9
We begin with sets A ,B and numbers α ,β for j = 1,2,...,ℓ. We consider
j j j j
ℓ
:= Σ (M ,N )XMj(1−βj)−Njαj
Q Aj,Bj,αj,βj j j
min(MXj,Nj)=0 Yj=1
Pℓj=1Mj=Pℓj=1Nj
where
Σ (M,N) = ( 1)qXd(α+β)A (j +q +d min(q +d,N))
A,B,α,β −α
− −
X
d,j,k
q≤1
B (k +q +d min(q +d,M))X2q+d+j+k−min(q+d,M)−min(q+d,N)
−β
× −
Our identity is
Theorem 2.
ℓ
= (1 X1−αj−βj) (A A β ,..., β ,B B α ,..., α ).
1 ℓ 1 ℓ 1 ℓ 1 ℓ
Q − C ∪···∪ ∪{− − } ∪···∪ ∪{− − }
Yj=1
7.1. Some lemmas. Because of the condition min(M ,N ) = 0 we consider Σ (M,0)
j j A,B,α,β
and Σ (0,N). We have
A,B,α,β
Lemma 1.
Σ (M,0) = XMβ (B α )(K)A(K +M)XK B(K)A(K +M)XK
A,B,α,β
(cid:18) ∪{− } −
XK XK
+ B(K)(A β )(K +M)XK
∪{− } (cid:19)
XK
and
Σ (0,N) = XNα (A β )(K)B(K +N)XK A(K)B(K +N)XK
A,B,α,β
(cid:18) ∪{− } −
XK XK
+ A(K)(B α )(K +N)XK .
∪{− } (cid:19)
XK
We defer the proof to later.
The result of the lemma leads us to consider
ℓ
(A ,...,A ;B ,...,B ) = A (K +M )B (K +N )XKj+Mj.
1 ℓ 1 ℓ j j j j j j
F
PMXi=PNi Yj=1
min(Mi,Ni)=0
Ki
We will prove
Lemma 2. We have
(A ,...,A ;B ,...,B ) = (A A ,B B ).
1 ℓ 1 ℓ 1 ℓ 1 ℓ
F C ∪···∪ ∪···∪
10 BRIAN CONREY ANDJONATHANP. KEATING
The right-hand side of Theorem 2 may be expanded. This leads to
Lemma 3. For J 1,...,ℓ let
⊂ { }
β = β : j J α = α : j J .
J j J j
− {− ∈ } − {− ∈ }
We have
ℓ
(1 X1−αj−βj) (A A β ,..., β ,B B α ,..., α )
1 ℓ 1 ℓ 1 ℓ 1 ℓ
− C ∪···∪ ∪{− − } ∪···∪ ∪{− − }
Yj=1
= ( 1)|J1|+|J2| (A β ,B α )
− C ∪− J1 ∪− J2
X
J1,J2⊂{1,...,ℓ}
J1∩J2=∅
where A = A A and B = B B .
1 ℓ 1 ℓ
∪···∪ ∪···∪
The combination of these three lemmas easily leads to a proof of Theorem 2.
7.2. Proof of Theorem 2.
Proof. By Lemma 1 the left side of the identity in Theorem 2 may be written as
ℓ
Σ (M ,N )XMj(1−βj)−Njαj
Aj,Bj,αj,βj j j
min(MXj,Nj)=0 Yj=1
Pℓj=1Mj=Pℓj=1Nj
ℓ
= XM1+...Mℓ (A β )(K +M )B (K +N )XK
j j j j j
(cid:18) ∪{− }
min(MXj,Nj)=0 Yj=1 XK
Pℓj=1Mj=Pℓj=1Nj
A (K +M )B (K +N )XK
j j j j
−
XK
+ A (K +M )(B α )(K +N )XK .
j j j j j
∪{− } (cid:19)
XK
By Lemma 2 this is
= ( 1)ℓ ( 1)|J1|+|J2| (A β ,B α )
− − C ∪− J1 ∪− J2
X
J1,J2⊂{1,...,ℓ}
J1∩J2=∅
and by Lemma 3 this is
ℓ
= (1 X1−αj−βj) (A A β ,..., β ,B B α ,..., α )
1 ℓ 1 ℓ 1 ℓ 1 ℓ
− C ∪···∪ ∪{− − } ∪···∪ ∪{− − }
Yj=1
which is the right side of the identity in Theorem 2.
(cid:3)