Table Of ContentJACOB’S LADDERS AND THE ALMOST EXACT ASYMPTOTIC
REPRESENTATION OF THE HARDY-LITTLEWOOD INTEGRAL
9
0 JANMOSER
0
2
Abstract. Inthispaperweintroduceanonlinearintegralequationsuchthat
n thesystemofglobalsolutiontothisequationrepresentsaclassofaverynarrow
a beam at T →∞ (an analogue to the laser beam) and this sheaf of solutions
J leadstoanalmost-exactrepresentationoftheHardy-Littlewoodintegral. The
6 accuracyofourresultisessentiallybetterthantheaccuracyofrelatedresults
2 ofBalasubramanian,Heath-BrownandIvic.
]
A
C
1. Introduction
.
h
Letus remindthatHardyandLittlewoodstartedtostudythe followingintegral
t
a in 1918:
m
[ T 1 2 T
(1.1) ζ +it dt= Z2(t)dt,
1 2
v Z0 (cid:12) (cid:18) (cid:19)(cid:12) Z0
(cid:12) (cid:12)
3 where (cid:12) (cid:12)
(cid:12) (cid:12)
7 1 1 1 t
9 Z(t)=eiϑ(t)ζ +it , ϑ(t)= tln(π)+Imln Γ +i ,
2 −2 4 2
3 (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
. and they have derived the following formula (see [4], page 122, 151-156)
1
0 T
9 (1.2) Z2(t)dt T ln(T), T .
0 Z0 ∼ →∞
:
v In this paper we show that except the asymptotic formula (1.2) that posses an
i unbounded error there is an infinite family of other asymptotic representations of
X
theHardy-Littlewoodintegral(1.1). Eachmemberofthisfamilyisanalmost-exact
r
a representation of the given integral. The proof of this will be based on properties
of a kind of functions having some canonical properties on the set of zeroes of the
function ζ(1/2+it).
(A) Let us remind further that in 1928 Titchmarsh has discovered a new treat-
menttotheintegral(1.1)bywhichtheTitchmarsh-Kober-Atkinson(TKA)formula:
∞ N
c ln(4πδ)
(1.3) Z2(t)e−2δtdt= − + c δn+ δN+1 ,
n
2sin(δ) O
Z0 n=0
X (cid:0) (cid:1)
where δ 0, C is the Euler constant, c are constant depending upon N,was de-
n
→
rived.
Key words and phrases. Riemannzeta-function.
1
Jan Moser Jacob’s ladder ...
The TKA formula has been published for the first time in 1951 in the funda-
mental monograph by Titchmarsh (see [8], [1],[6],[7]). It was thought for about 56
years that the TKA formula is a kind of curiosity (see [5], page 139). However, in
this paper we show that the TKA formula itself contains new kinds of principles.
(B) Namely, in this work we introduce a new class of curves, which are the
solutions to the following nonlinear integral equation:
µ[x(T)] T
(1.4) Z2(t)e−x(2T)tdt= Z2(t)dt,
Z0 Z0
∞
wheretheclassoffunctions µ isspecifiedas: µ C ([y , ))isamonotonically
0
{ } ∈ ∞
increasing (to + ) function and it obeys µ(y) 7yln(y) . The following holds
∞ ≥
true: for any µ µ it exists just one solution to the equation (1.4):
∈{ }
ϕ(T)=ϕ (T), T [T , ), T =T [ϕ], ϕ(T) as T .
µ 0 0 0
∈ ∞ →∞ →∞
Let us denote by the symbol ϕ the system of these solutions. The function ϕ(T)
{ }
is related to the zeroes of the Riemann zeta-function on the critical line by the
following way. Let t = γ be a zero of the function ζ(1/2+it) of the order n(γ),
where n(γ)= (ln(γ)), (see [3], page 178). Then the points [γ,ϕ(γ)], γ >T (and
0
O
only these points) are the inflection points with the horizontal tangent. In more
details, it holds true the following system of equations:
(1.5) ϕ′(γ)=ϕ′′(γ)= =ϕ(2n)(γ)=0, ϕ(2n+1)(γ)=0,
··· 6
where n=n(γ).
With respectto this propertyanelement ϕ ϕ is to be namedas the Jacob’s
∈{ }
ladder leading to [+ ,+ ] (the rungs of the Jacob’s ladder are the segments of
∞ ∞
the curve ϕ lying in the neighborhoods of the points [γ,ϕ(γ)], γ >T [ϕ]). Finally,
0
also the composition of the functions G[ϕ(T)] is to be named the Jacob’s ladder if
∞
the following conditions are fulfilled: G C ([y ,+ )), G grows to + and G
0
∈ ∞ ∞
has a positive derivative everywhere.
Let us mention that the mapping (the operator)
Hˆ : µ ϕ
{ }→{ }
can be named the Z2-mapping of the functions of the class µ .
{ }
(C) Jacob’s ladder implies the following results:
(a) AnalmostexactasymptoticformulafortheHardy-Littlewoodintegral. Let
us mention that our new formula makes more exact also the leading term
in (1.2):
T ϕ(T) ϕ(T)
Z2(t)dt ln , T + , ϕ ϕ ,
∼ 2 2 → ∞ ∀ ∈{ }
Z0 (cid:18) (cid:19)
i.e. theleadingterminthisformulaisalsotheJacob’sladder,andtherefore
we can say that the leading term has a very fine structure.
(b) The system ϕ has the property of an ”infinitely close approach” of any
{ }
two Jacob’s ladders at T + .
→ ∞
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Jan Moser Jacob’s ladder ...
(c) Our new formula for the Hardy-Littlewood integral is stable with respect
to the choice of the elements from some subset ϕ ⋆ in ϕ .
{ } { }
2. Results
The following theorem holds true:
Theorem 1. The TKA formula implies:
(A)
T
(2.1) Z2(t)dt=F[ϕ(T)]+r[ϕ(T)], T T [ϕ],
0
≥
Z0
where
(2.2)
y y y ln(ϕ(T)) ln(T)
F[y]= ln +(c ln(2π)) +c , r[ϕ(T)]= = .
0
2 2 − 2 O ϕ(T) O T
(cid:16) (cid:17) (cid:26) (cid:27) (cid:18) (cid:19)
(B) For all ϕ (T),ϕ (T) ϕ we have
1 2
∈{ }
1
(2.3) ϕ (T) ϕ (T)= , T max T [ϕ ],T [ϕ ] .
1 2 0 1 0 2
− O T ≥ { }
(cid:18) (cid:19)
(C)Iftheset µ(y ) isboundedthenforallϕ(T)in ϕ andforanyfixedϕ (T)
0 0
{ } { } ∈
ϕ :
{ }
A A
(2.4) ϕ (T) <ϕ(T)<ϕ (T)+ , T T =sup µ(y ) .
0 0 0 0
− T T ≥ { }
µ
Letusmentionthattheconstantsinthe -symbolsdonotdependuponthechoice
O
of ϕ(T).
Let us remind the Balasubramanian’s formula (see [2]):
T
(2.5) Z2(t)dt=T ln(T)+(2c 1 ln(2π))T + T1/3+ǫ ,
− − O
Z0 (cid:16) (cid:17)
and Ω-theorem of Good (see [3]):
T
(2.6) Z2(t)dt T ln(T) (2c 1 ln(2π))T =Ω T1/4 .
− − − −
Z0 (cid:16) (cid:17)
Remark 2. Combining the formulae (2.1), (2.2) and (2.5) one obtains that:
Formula(2.5)possessesquitelargeuncertaintysincethedeviationfromthe
•
value of (1.1) is given by R(T)= (T1/3+ǫ), and (see (2.6)) since
O
lim R(T) =+ ,
T→∞| | ∞
this cannot be removed.
Following (2.2) we have
•
lim r[ϕ(T)]=0,
T→∞
and this means that formula (2.1) seems to be almost exact.
Remark 3. WehavefoundanewfactthattheleadingtermintheHardy-Littlewood
integral is a ladder (see (2.1)), i.e. it has a fine structure. There is no analogue of
this in the formulae (1.2) or (2.5).
Remark 4. We say explicitly that:
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Jan Moser Jacob’s ladder ...
Formula (2.3) contains a new effect, namely, any two Jacob’s ladders ap-
•
proach each other at T .
→∞
(2.4) implies that the representations (2.1) and (2.2) of the integral of
•
Hardy-Littlewood (1.1) are stable under the choice of ϕ(T) ϕ in the
∈ { }
case of a bounded set µ(y ) .
0
{ }
Remark 5. Following the second part of Remark 3 the representations (2.1) and
(2.2) of the Hardy-Littlewoodintegral(1.1) is microscopically unique in sense that
any two Jacob’s ladders ϕ ,ϕ , T T (see (2.4)) cannot be distinguished at
1 2 0
≥
T .
→∞
InthefifthpartofthisworkweestablishtherelationbetweentheJacob’sladder
and the prime-counting function π(T):
1 ϕ(T)
π(T) T , T .
∼ 1 c − 2 →∞
− (cid:26) (cid:27)
3. Existence of the Jacob’s ladder
The following lemma holds true:
Lemma1. Letµ(y) µ befixed. Thenthereexistsanuniquesolutionϕ(T), T
∈{ } ≥
T [ϕ] to the integral equation (1.4) that obeys the property (1.5).
0
Proof. By the Bonnet’s mean-value theorem in the case y >0 and µ>0 we have
µ M
(3.1) Z2(t)e−y2tdt= Z2(t)dt, M >0,
Z0 Z0
where e−y2t, t [0,µ] is decreasing and equals 1 at t=0.
∈
First of allwe will show that the formula (3.1) maps to any fixed µ>0 just one
M >0, since the case M =M is impossible because it would mean that
1 2
6
M2
(3.2) Z2(t)dt=0.
ZM1
Let µ(y) µ . Then the following formula
∈{ }
µ(y) M(y)
(3.3) Z2(t)e−y2tdt= Z2(t)dt
Z0 Z0
defines a function M(y)=M (y), y y . Letthe symbol M denote the class of
µ 0
≥ { }
the images of the elements µ(y) µ .
∈{ }
The function M(y) is positive and increases to + . In fact, let
∞
µ(y)
(3.4) Φ(y)= Z2(t)e−y2tdt.
Z0
Then
(3.5) Φ′(y)= 2 µ(y)tZ2(t)e−y2tdt+Z2[µ(y)]e−y2µ(y)dµ(y) >0
y2 dy
Z0
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Jan Moser Jacob’s ladder ...
(the first term is obviously positive and the second one is non-negative). Thus,
we see that the function Φ(y), y [y ,+ ) is increasing. Subsequently, for any
0
∈ ∞
y,∆y >0,y y one has
0
≥
M(y+∆y)
0<Φ(y+∆y) Φ(y)= Z2(t)dt M(y+∆y)>M(y).
− ⇒
ZM(y)
The function M(y), y y is continuous. In fact, for any fixed yˆ [y ,+ ) we
0 0
≥ ∈ ∞
have
M (yˆ)=limsupM(y), M (yˆ)=liminfM(y).
1 2
y→yˆ+ y→yˆ+
And with respect to the continuity of the left-hand side of eq. (3.3) we obtain (see
(3.2)):
M2(yˆ)
Z2(t)dt=0 M (yˆ)=M (yˆ)=M(yˆ+0).
1 2
ZM1(yˆ)
The existence ofM(yˆ 0)andthe identities: M(yˆ)=M(yˆ 0)=M(yˆ+0)can
− −
be shown by analogy.
The function M(y), y y obeys the following properties:
0
≥
(a) It has a continuous derivative at any point y such that M(y)=γ, where γ
6
is a zero of the function ζ(1/2+it).
(b) It has a derivative equal to + at any point y such that M(y) = γ, γ
∞
mentioned above.
Thesepropertiesofthe M functioncanbeprovedasfollows. By(3.3)and(3.4)we
have
(3.6)
Φ(y+∆y) Φ(y) M(y+∆y) M(y)
− = − Z2 M(y)+θ [M(y+∆y) M(y)] ,
∆y ∆y { · − }
′
whereθ (0,1). UsingthefactthatΦ(y)>0(see(3.5))wecandeducefrom(3.6)
∈
that:
(i) for any values of y suchthat Z2[M(y)]>0 there exists a continuousderiv-
ative, and
(ii) for any values of y such that Z2[M(y)]=0 we have
M(y+∆y) M(y)
(3.7) lim − =+ .
∆y→0 ∆y ∞
Since our function T = M(y), y y is continuous and increasing (to + )
0
≥ ∞
thereexistsuniqueinversefunctionthatisalsocontinuousandincreasing(to+ ):
∞
(3.8) y =ϕ(T)=ϕ (T), T T [ϕ]=M (y ).
M 0 µ 0
≥
As a consequence of eqs. (3.5) and (3.6) we have
dϕ(T) Z2(T)
′ ′
(3.9) = , Φ =Φ [ϕ(T)]>0, T T [ϕ],
dT Φ′[ϕ(T)] y ≥ 0
′ ∞
(ϕ(γ) = 0, see (3.7)). Eq. (3.9) implies that ϕ(T) C ([T [ϕ],+ )) and also
0
∈ ∞
that the property (1.5) holds true. Inserting (3.8) into (3.3) one obtains that
X(T)=ϕ (T), T T [ϕ] is a solution to the integral equation (1.4). (cid:3)
µ 0
≥
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Jan Moser Jacob’s ladder ...
4. Consequences from the TKA formula
The following Lemma holds true:
Lemma 2. Let M(y) M be arbitrary, then
∈{ }
M(y)
(4.1) Z2(t)dt=F(y)+r(y),
Z0
where
y y y ln(y)
(4.2) F(y)= ln +E +c , r(y)= , E =c ln(2π),
0
2 2 2 O y −
(cid:16) (cid:17) (cid:18) (cid:19)
and the constant within the -symbol is an absolute constant.
O
Proof. We start with the formula (1.3) and N =1:
∞
c ln(4πδ)
(4.3) Z2(t)e−2δtdt= − +c +c δ+ (δ2).
0 1
2sin(δ) O
Z0
As long as Z(t) <At1/4, t t , we have
0
| | ≥
1 1
f(t,δ)=t1/2e−δt f ,δ = ,
≤ 2δ √2eδ
(cid:18) (cid:19)
∞ e−δU A2
Z2(t)e−2δtdt<B , B = , U t .
δ3/2 √2e ≥ 0
ZU
The value U =µ(1/δ) is to be chosen by the following rule:
1 7 1 1 B
Bδ−3/2e−δU δ2 µ ln > ln .
≤ ⇒ δ ≥ δ δ δ δ7/2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Now, (4.3) implies:
µ(1/δ) c ln(4πδ) 1 7 1
(4.4) Z2(t)e−2δtdt= − +c +c δ+ δ2 , µ ln ,
0 1
2sin(δ) O δ ≥ δ δ
Z0 (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
(seetheintroduction,part(B)-theconditionforµ(y)),andfortheremainderterm
we have:
∞
(4.5) Z2(t)e−2δtdt= δ2 .
− O
Zµ(1/δ)
(cid:0) (cid:1)
Let δ (0,δ ] with δ being sufficiently small, then
0 0
∈
1 1 δ2
= 1+ + δ4 ,
sin(δ) δ 6 O
(cid:26) (cid:27)
(cid:0) (cid:1)
and
c ln(4πδ) 1 1 D 1
− = ln + + δln ,
2sin(δ) 2δ δ 2δ O δ
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
and (see (4.4))
µ(1/δ) 1 1 D 1
(4.6) Z2(t)e−2δtdt= ln + +c + δln , D =c ln(4π).
0
2δ δ 2δ O δ −
Z0 (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
Putting δ = 1/y, y = 1/δ into eq. (4.6) and using eq. (3.3) we obtain the
0 0
formulae (4.1) and (4.2), respectively. Since the constants in the eqs. (4.3), (4.5)
are absolute, the constant entering the -symbol in (4.2) is absolute, too. (cid:3)
O
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Jan Moser Jacob’s ladder ...
5. Proof of the theorem
Putting T =y/2 into eq. (2.5) and comparing with the formula (4.1) we obtain
y
<M(y), y .
2 →∞
Furthermore, putting into eq. (2.5)
y A
T = 1+ , A>1 c,
2 ln y ! −
2
and comparing with eq. (4.1) we have (cid:0) (cid:1)
y A
M(y)< 1+ , y .
2 ln y ! →∞
2
Subsequently (see (3.8)), (cid:0) (cid:1)
y A y ϕ(T)
(5.1) 0<M(y) < 0<2T ϕ(T)<B ,
− 2 2 ln y ⇒ − ln[ϕ(T)]
2
i.e. the following equation holds t(cid:0)ru(cid:1)e
(5.2) 1.9T <ϕ(T)<2T.
Insertingy =ϕ(T) ϕ into eq. (4.1) (see (3.8)) we obtainthe formula(2.1) and
∈{ }
the estimate (2.2), (see (5.2)).
′
The relation (2.3) follows from F (y) = 1/2ln(y/2)+E +1 and from the eq.
(2.1) written for ϕ and ϕ , respectively, with help of (5.2).
1 2
Since µ(y)>M(y), y y (see (3.3)) and in the case of boundedness of the set
0
≥
µ(y ) the choice of values (see (2.4)):
0
{ }
T =sup µ(y ) µ(y )>M (y ), µ(y) µ ,
0 0 0 µ 0
{ }≥ ∀ ∈{ }
µ
is regular, i.e. the interval [T ,+ ) is the common domain of the functions
0
∞
ϕ (T), µ(T) µ .
µ
∈{ }
6. Relation between the Jacob’s ladders and prime-counting
function π(T)
Comparing the formulae (2.1) and (2.5) we obtain
ϕ
ω ω(T)=(1 c)T + T1/3+ǫ , ω(t)=tln(t)+(c ln(2π))T.
2 − − O −
(cid:16) (cid:17) (cid:16) (cid:17)
Let us consider the power series expansion in the variable ϕ/2 T of the previous
−
formula. We obtain the following nonlinear equation
∞ xk T ϕ
(6.1) x(ln(T) a) =1 c+ T−2/3+ǫ , x= − 2,
− − k(k 1) − O T
Xk=2 − (cid:16) (cid:17)
where a=ln(2π) 1 c. Because of (see (5.1) and (5.2))
− −
1
x= ,
O ln(T)
(cid:18) (cid:19)
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Jan Moser Jacob’s ladder ...
we obtain from (6.1):
1 c 1 ϕ(T) T 1
x= − + T = 1+ ,
ln(T) a O ln3(T) ⇒ − 2 ln(T) O ln(T)
− (cid:18) (cid:19) (cid:26) (cid:18) (cid:19)(cid:27)
and furthermore, by using the Selberg-Erdo¨s theorem, we obtain the formula:
1 ϕ(T)
(6.2) π(T) T , T , ϕ ϕ .
∼ 1 c − 2 →∞ ∀ ∈{ }
− (cid:26) (cid:27)
Remark 6. As aconsequenceoftheabovewrittenwehavethatthe Jacob’sladders
are connected (along to the zeroes of the function ζ(1/2+it)) also to the prime-
counting function π(T), see (6.2).
More interesting information can be deduced from the nonlinear equation (6.1).
Namely, inserting
(6.3) Te−a =τ, e−aϕ(eaτ)=ψ(τ)
into (6.1) we obtain:
∞ xk τ ψ(τ)
(6.4) xln(τ) =1 c+ τ−2/3+ǫ , x= − 2 .
− k(k 1) − O τ
Xk=2 − (cid:16) (cid:17)
The following statement holds true: if in the equation:
A A A A
1 3 n n+1
(6.5) x= + + + + +... ,
ln(τ) ln3(τ) ··· lnn(τ) lnn+1(τ)
the coefficients A ,A ,...,A are already known, then the coefficient A is de-
1 3 n n+1
termined by (6.4). We obtain:
1 1
A =1 c, A = (1 c)2, A = (1 c)3,
1 3 4
− 2 − 6 −
1 1
A = (1 c)3+ (1 c)4, ... .
5
2 − 12 −
Changing the variables in (6.5) into the initial ones (see (6.3)) we obtain the
following asymptotic formula
(6.6)
1 ϕ(T) A A A B B
1 3 1 2 3
T + + + + +... , T ,
T − 2 ∼ ln(T) a (ln(T) a)3 ···∼ ln(T) ln2(T) ln3(T) →∞
(cid:26) (cid:27) − −
where B =aA , B =a2A +A , ....
2 1 3 1 3
Remark 7. Let us remark that the asymptotic formula (6.6) is an analogue to the
following asymptotic formula
1 T dt 1 1 2!
+ + +... ,
T ln(t) ∼ ln(T) ln2(T) ln3(T)
Z2
for the Gauss logarithmic integral.
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Jan Moser Jacob’s ladder ...
7. Fundamental properties of Z2-transformation
Let us mention explicitly that the key idea of the proof of the theorem was to
introduce the new integral transformation: Z2-transformation.
By using the appropriate terminology from optics (see example: Landau& Lif-
shitz, Field theory, GIFML, Moscow 1962,page 167):
the elements µ(y) µ are called the rays and the set µ itself is called
• ∈{ } { }
the beam,
thebeamscrossingeachotherinagivenpointarecalledhomocentricbeams.
•
The fundamental property of the Z2-transformation lies in the following:
iftheset µ(y ) isboundedthenthebeam µ (and,atthesametime,also
0
• { } { }
anyotherhomocentricbeam)istransformedintothehomocentricbeam ϕ
{ }
oftheJacob’sladders,withrespecttothepoint[+ ,+ ](see(2.3),(2.4)),
∞ ∞
thetransformedbeam ϕ isverynarrowinsenseof(2.4),i.e. anZ2-optical
• { }
system generates an analogue of a laser beam.
Let us consider for example the homocentric (with respect to the point [y ,y2])
0 0
sheaf of rays:
(7.1) u(y;ρ,n)=y2[1+ρ(y y )n], ρ [0,1], n N.
0
− ∈ ∈
Following the equation u (y+∆;0,n)=u (y;1,n) we obtain that
0 1
y(y y )n
0
∆= − , at y ,
1+(y y )2 →∞ →∞
0
−
i.e. (7.1) is a diverging beapm. Anyway, the Z2-mapping transforms (7.1) into an
analogue of a laser beam ϕ .
u
{ }
8. On intervals that cannot be reached by estimates of
Heath-Brown and Ivic
We will show the accuracy of our formula (2.1) in comparison with known esti-
mates of Heath-Brown and Ivic (see [5], (7.20) page 178, and (7.62) page 191)
T+G 1
(8.1) Z2(t)dt= Gln2(T) , G=T1/3−ǫ0, ǫ = .
0
O 108
ZT
(cid:0) (cid:1)
First of all, one can easily obtain the tangent law from both (2.1) and (8.1):
T+U ϕ(T) 1
(8.2) Z2(t)dt=Uln e−a tan[α(T,U)]+
ZT (cid:18) 2 (cid:19) O(cid:18)T1/3+2ǫ0(cid:19)
for 0 < U < T1/3+ǫ0, where α = α(U,T) is the angle of the chord of the curve
y = 1ϕ(T) crossing the points [T,1ϕ(T)] and [T +U,1ϕ(T +U)]. Further, from
2 2 2
(2.1) and (2.5) we have
1
(8.3) tan(α )=tan[α(T,U )]=1+ , U =T1/3+2ǫ.
0 0 0
O ln(T)
(cid:18) (cid:19)
And finally, considering the set of all chords of the curve y = 1ϕ(T) which are
2
paralleltoourfundamentalchordjoiningpoints[T,1ϕ(T)]and[T+U,1ϕ(T+U)],
2 2
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Jan Moser Jacob’s ladder ...
we obtain the continuum of formulae:
b 1
Z2(t)dt=(b a)ln(T)+ (b a)+ ,
Za − O − O(cid:18)T1/3+2ǫ0(cid:19)
(8.4) 0<b a<1,(a,b) T,T +T1/3+ǫ0 .
− ⊂
(cid:16) (cid:17)
Remark 8. It is quite evident that the interval (0,1) cannot be reached in known
theories leading to estimates of Heath-Brown and Ivic.
References
[1] F. V. Atkinson, ‘The mean value of the zeta-function on critical line’, Quart. J. Math. 10
(1939) 122–128..
[2] R. Balasubramanian, ’An improvement on a theorem of Titchmarsh on the mean square of
|ζ(1/2+it)|’,Proc. London. Math. Soc.336(1978)540–575.
[3] A. Good, ‘Ein Ω-Resultat fu¨r quadratische Mittel der Riemannschen Zetafunktion auf der
kritischeLinie’,Invent. Math. 41(1977) 233–251.
[4] G. H.HardyandJ. E.Littlewood, ‘Contributiontothe theoryofthe Riemannzeta-function
andthetheorypfthedistributionofPrimes’,Acta. Math. 41(1918) 119–195.
[5] A.Ivic,‘TheRiemannzeta-function’,AWilley-IntersciencePublication,NewYork,1985.
[6] H. Kober, ‘Eine Mittelwertformeln der Riemannschen Zetafunktion’, Composition Math. 3
(1935) 174–189.
[7] E. C. Titchmarsh, ‘The mean-value of the zeta-function on the critical line’, Proc. London.
Math. Soc.(2)27(1928) 137–150.
[8] E.C.Titchmarsh,‘ThetheoryoftheRiemannzeta-function’,ClarendonPress,Oxford,1951.
Department of Mathematical Analysis and Numerical Mathematics, Comenius Uni-
versity,MlynskaDolina M105,84248Bratislava,SLOVAKIA
E-mail address: [email protected]
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