Jacob's ladders, the iterations of Jacob's ladder $φ^k_1(t)$ and asymptotic formulae for the integrals of the products ... for arbitrary fixed $n\in\mbb{N}$

0.09 MB·English

by  Jan Mozer

JACOB’S LADDERS, THE ITERATIONS OF JACOB’S LADDER
ϕk(t) AND ASYMPTOTIC FORMULAE FOR THE INTEGRALS
1
OF THE PRODUCTS Z2[ϕn(t)]Z2[ϕn−1(t)]···Z2[ϕ0(t)] FOR
1 1
0
ARBITRARY FIXED n∈N
1
0
2
JANMOSER
n
a
J Abstract. In this paper we introduce the iterations ϕk(t) of the Jacob’s
1
1 ladder. Itisproved,forexample,thatthemean-valueoftheproduct
1 Z2[ϕn1(t)]Z2[ϕn−1(t)]···Z2[ϕ01(t)]

over thesegment [T,T +U]isasymptotically equal tolnn+1T. Northe case
]
A n=1cannotbeobtainedinknowntheoriesofBalasubramanian,Heath-Brown
andIvic.
C
.
h
t
a
1. Results
m
[ Let

1
1 y = ϕ(t)=ϕ (t); ϕ0(t)=t; ϕ1(t)=ϕ (t), ϕ2(t)=ϕ (ϕ (t)),
v 2 1 1 1 1 1 1 1
2 (1.1)
3 ..., ϕk(t)=ϕ (ϕ (...(ϕ (t))...), t∈[T,T +U],
6 1 1 1 1
1 where ϕk(t) denotes the kth iteration of the Jacob’s ladder y = ϕ(t), t ≥ T [ϕ ].
1 0 1
.
1 The following Theorem holds true.
0
Theorem. Let
0
1 (1.2) T ≥T [ϕ ,n]=max{2T [ϕ ],e2(n+1)}, U =T1/3+2ǫ.
: 00 1 0 1
v Then for every fixed n∈N the following is true
i
X T+U n
r (1.3) Z2[ϕk(t)]dt∼Ulnn+1T, T →∞,
a ZT kY=0 1
where
(A) ϕk(t)≥T [ϕ [, k =0,1,...,n+1, t∈[T,T +U],
1 0 1
(B) ϕk(T +U)−ϕk(t)∼U, k=0,1,...,n+1,
1 1
(C) ϕk−1(T)−ϕk(T +U)∼(1−c) T , k=0,1,...,n,
1 1 lnT
T
(D) ρ ϕk−1(T),ϕk−1(T +U) ; ϕk(T),ϕk(T +U) ∼(1−c) ,
¶î 1 1 ó 1 1 © lnT
(cid:2) (cid:3)
and ρ denotes the distance of the corresponding segments.
Key words and phrases. Riemannzeta-function.
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Jan Moser Jacob’s ladder ...
Remark 1. The system of the iterated segments
[ϕn(T),ϕn(T +U)], ϕn−1(T),ϕn−1(T +U) ,...,[T,T +U]
1 1 1 1
(cid:2) (cid:3)
is the disconnected set of segments distributed from right to left (see (C)) and the
neighbouring segments unboundedly recede each from other (see (D), ρ → ∞, as
T →∞, comp. [6], Remark 3).
Remark 2. Let us mention the formula (1.3) especially for the prime numbers of
Fermat-Gauss n=17,257,65537and for the Skewes constant
n=10101034.
It is obvious that nor the formula (n=2)
T+U
Z2[ϕ (ϕ (t))]Z2[ϕ (t)]Z2(t)dt∼Uln3T
Z 1 1 1
T
cannot be reached in known theories of Balasubramanian, Heath-Brown and Ivic
(see [1]).
This paper is a continuation of the series of papers [2]-[7].
2. Consequences of the Theorem
Using the mean-value theorem in (1.3) we obtain
Corollary 1.
n
(2.1) Z2[ϕk(τ)] ∼lnn+1T,
1
Y
k=0
T →∞,ϕk(τ)∈(ϕk(T),ϕk(T +U)),k =0,1,...,n,τ =τ(T,n).
1 1 1
From (2.1) we obtain
Corollary 2.
n 2
(2.2) Z[ϕk(τ)] n+1 ∼lnT,
1
Y(cid:12) (cid:12)
k=0(cid:12) (cid:12)
(cid:12) (cid:12)
n
1 1
(2.3) ln|Z[ϕk(τ)]|∼ lnlnT.
n+1 1 2
X
k=0
Next, by the known inequalities for harmonic, geometric and arithmetic means
we have
Corollary 3.
n
1
(2.4) (1−ǫ)lnT ≤ |Z[ϕk(τ)]|n+21,
n+1 1
X
k=0
n
(2.5) 1 < 1 |Z[ϕk(τ)]|−n+21.
(1+ǫ)lnT n+1 1
X
k=0
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Jan Moser Jacob’s ladder ...
Remark 3. Some new type of the nonlocal interaction of the values
Z2[ϕk(t)] n
1 k=0
(cid:8) (cid:9)
of the signal
1
Z(t)=eiϑ(t)ζ +it
Å2 ã
over the system of disconnected segments
n
ϕk(T),ϕk(T +U)
1 1
k[=0(cid:2) (cid:3)
is expressed by formulae (1.3), (2.1)-(2.5) for the iterated Jacob’s ladder ϕk(t).
1
3. Lemma
We start with the formula (see [2], (3.5), (3.9))
dϕ(t)
Z2(t)=Φ′ [ϕ(t)] , t≥T [ϕ],
ϕ dt 0
where (see [4], (1.5))
1 lnlnt
′
Φ [ϕ(t)]= 1+O lnt.
ϕ 2ß Å lnt ã™
Next we have
dϕ (t) T
(3.1) Z˜2(t)= 1 , t∈[T,T +U], U ∈ 0, ,
dt Å lnTò
by (1.1) we have
Z2(t) Z2(t)
(3.2) Z˜2(t)= = .
2Φ′ [ϕ(t)] 1+O lnlnt lnt
ϕ lnt
(cid:8) (cid:0) (cid:1)(cid:9)
Then we obtain from (3.1) the following lemma (comp. [6], (2.5)).
Lemma. Foreveryintegrablefunction(intheLebesguesense)f(x),x∈[ϕ (T),ϕ (T+
1 1
U)] the following is true
T+U ϕ1(T+U) T
(3.3) f[ϕ (t)]Z˜2(t)dt= f(x)dx, U ∈ 0, .
Z 1 Z Å lnTò
T ϕ1(T)
4. Proof of the Theorem
4.1. From the formula (see (1.1), (1.2), [2], (6.2))
t
(4.1) t−ϕ1(t)∼(1−c)π(t)∼(1−c)
1 lnt
we have
t
ϕ1(t)−ϕ2(t)∼(1−c) ,
1 1 lnt
t
ϕ2(t)−ϕ3(t)∼(1−c) ,
1 1 lnt
.
.
(4.2) .
t
ϕn(t)−ϕn+1(t)∼(1−c) ,
1 1 lnt
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Jan Moser Jacob’s ladder ...
and by (4.1) (4.2) we obtain
(1+ǫ)(1−c)(n+1) n+1 1
ϕn+1(t)>t 1− ≥T 1− ≥ T ≥T [ϕ ],
1 ß lnt ™ Å lnT ã 2 0 1
i.e. (A).
4.2. By comparison of the formula (see [3], (1.5))
T+U
Z2(t)dt∼UlnT, U =T1/3+2ǫ,
Z
T
where 1/3 is the exponent of Balasubramanian, and our formula
T+U
Z2(t)dt∼{ϕ (T +U)−ϕ (T)}lnT,
Z 1 1
T
(see [4], (1.2)) we have
ϕ (T +U)−ϕ (T)∼U,
1 1
by comparison in the cases T →ϕ1(T), T +U →ϕ1(T +U), ... we obtain
1 1
ϕ2(T +U)−ϕ2(T) ∼ ϕ1(T +U)−ϕ1(T),
1 1 1 1
.
.
.
ϕn+1(T +U)−ϕn+1(T) ∼ ϕn(T +U)−ϕn(T),
1 1 1 1
i.e. (B).
4.3. By (4.2), t→T we have
T
ϕ1(T)−ϕ2(T)∼(1−c) ,
1 1 lnT
i.e.
T
(4.3) ϕ1(T)−ϕ2(T +U)+{ϕ2(T +U)−ϕ2(T)}∼(1−c) ,
1 1 1 1 lnT
since (see (B))
ϕ2(T +U)−ϕ2(T)∼U =T1/3+2ǫ
1 1
then from (4.3) the asymptotic formula
T
ϕ1(T)−ϕ2(T +U)∼(1−c)
1 1 lnT
follows. Similarly we obtain all asymptotic formulae in (C). The proposition (D)
follows from (C).
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Jan Moser Jacob’s ladder ...
4.4. From (3.1) by (3.3) we have
T+U n T+U n
Z˜2[ϕk(t)]dt= Z˜2[ϕk(t)]Z˜2(t)dt=
Z 1 Z 1
T kY=0 T kY=0
T+U n ϕ1(T+U) n
= Z˜2[ϕk−1(ϕ (t))]dϕ (t)= 1 Z˜2[ϕk−1(w )]dw =
Z 1 1 1 Z 1 1 1
T kY=1 ϕ11(T) kY=1
ϕ1(T+U) n
= 1 Z˜2[ϕk−2(ϕ (w ))]Z˜2(w )dw =
Z 1 1 1 1 1
ϕ11(T) kY=2
ϕ2(T+U) n
= 1 Z˜2[ϕk−2(w )]dw =···=
Z 1 2 2
ϕ21(T) kY=2
ϕn(T+U)
1
= Z˜2[w ]dw =ϕn+1(T +U)−ϕn+1(T),
Z n n 1 1
ϕn(T)
1
i.e. the following asymptotic formula (see (B))
T+U n
(4.4) Z˜2[ϕk(t)]dt=ϕn+1(T +U)−ϕn+1(T)∼U
Z 1 1 1
T kY=0
holds true. Then, from (4.4) by mean-value theorem (see (3.1), (3.2), (4.1), (4.2);
lnϕk(t)∼lnt) the formula (1.3) follows.
1
IwouldliketothankMichalDemetrianforhelpingmewiththeelectronicversion
of this work.
References
[1] A.Ivic,‘TheRiemannzeta-function’, AWilley-IntersciencePublication,NewYork,1985.
[2] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-
Littlewoodintegral’,(2008).
[3] J.Moser,‘Jacob’sladdersandthetangent lawforshortpartsoftheHardy-Littlewoodinte-
gral’,(2009).
[4] J.Moser,‘Jacob’sladdersandthemultiplicativeasymptoticformulaforshortandmicroscopic
partsoftheHardy-Littlewoodintegral’,(2009).
[5] J.Moser,‘Jacob’sladdersandthequantization oftheHardy-Littlewoodintegral’,(2009).
[6] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the sixth
order|ζ(1/2+iϕ(t)/2)|4|ζ(1/2+it)|2’,(2009).
[7] J. Moser, ‘Jacob’s ladders and the first asymptotic formula for the expression of the fifth
orderZ[ϕ(t)/2+ρ1]Z[ϕ(t)/2+ρ2]Z[ϕ(t)/2+ρ3]Zˆ2(t)forthecollectionofdisconnectedsets‘,
(2009).
[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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