Jacob's ladders and the asymptotically approximate solutions of a nonlinear diophantine equation

0.12 MB·English

by  Jan Moser

JACOB’S LADDERS AND THE ASYMPTOTICALLY
APPROXIMATE SOLUTIONS OF A NONLINEAR
DIOPHANTINE EQUATION
0
1
0
JANMOSER
2

n
Abstract. The nonlinear equation which is connected with the main term
a
of the Hardy-Littlewood formulaforζ2(1/2+it) isstudied. In this direction
J
I obtain the fine results which cannot be reached by published methods of
8
Balasubramanian,Heath-BrownandIvicinthefieldoftheHardy-Littlewood
1
integral.

]
A
C
.
h 1. Formulation of the results
t
a
1.1. Let
m
[
d(n)sin U lnP U P π
1 (1.1) S(T,U)=2 2 n cos 2πP + ln 2πP ,
v n<P √n U2(cid:0) lnPn (cid:1) ßÅ 2ã n − − 4™
9 X
1 where P =T/2π and d(n) is the number of divisors of n. In this paper I consider
0
the nonlinear diophantine equation
3
1. S(T,U)
(1.2) τ = , τ [η,1 η] 1
0 lnT ∈ − { }
0 [
1 in two variables T,U with the parameter τ where
:
v (1.3) T [T ,T +U ], U 0,T1/6−ǫ/2 , U =T1/3+2ǫ,
i ∈ 0 0 0 ∈ 0 0 0
X
Ä ó
and 0<η is a sufficiently small fixed number and 0<T is a sufficiently big fixed
r 0
a number.
Definition. Let for τ¯ [η,1 η] 1 there be a sequence T (τ¯) , T and
0 0
∈ − ∪{ } { } →∞
the values T˜ =T˜(T ,τ¯), U˜ =U˜(T ,τ¯) for which
0 0
(1.4) T˜ [T ,T +1.1U ], U˜ 0,T1/6−ǫ/2 ,
∈ 0 0 0 ∈ 0
S(T˜,U˜) Ä ó
τ¯ , T
∼ lnT˜ 0 →∞
is fulfilled. Then the pair [T˜,U˜] is called the asymptotically approximate solution
(AAS) of the equation (1.2) for τ =τ¯.
Key words and phrases. Riemannzeta-function.
1
Jan Moser Jacob’s ladder ...
1.2. The method of parallel and rotating chords (see [4]-[6]) leads to the proof of
the following theorems.
Theorem 1. For τ =1 there is the continuum AAS of the equation
S(T,U)
1= .
lnT
The structure of the set of these solutions is such as follows: to each sufficiently
big T continuum of AAS corresponds.
0
Theorem 2. Let γ denote the sequence of the zeroes of ζ(1/2+it). Then for each
τ [η,1 η] there is a continuum of the AAS of the equation
∈ −
S(T,U)
τ = .
lnT
The structure of the set of these solutions is such as follows: to each sufficiently
big γ continuum of AAS corresponds.
Remark 1. It is clear that these results cannot be reached by published methods
of Balasubramanian, Heath-Brown and Ivic in the field of the Hardy-Littlewood
integral.
This paper is a continuation of the series of works [4]-[11].
2. Lemmas
2.1.
Lemma 1.
T+U
(2.1) cos 2ϑ(t) tlnn dt=
{ − }
ZT
sin U lnP U P π
=U 2 n cos 2πP + ln 2πP +
U2(cid:0) lnPn (cid:1) ßÅ 2ã n − − 4™
U +U3 T
+ , P = ,
O T 2π
Å ã
for U >0, where
1 1 1
ϑ(t)= tlnπ+Im lnΓ + it .
−2 4 2
ß Å ã™
Proof. Following the formulae (see [7], pp. 221, 329)
1 t 1 1 1
ϑ(t)= tln t π+ ,
2 2π − 2 − 8 O t
Å ã
(2.2)
1 t 1
ϑ′(t)= ln +
2 2π O t
Å ã
we have (t=T +x, x [0,U])
∈
(2.3) 2ϑ(T +x) (T +x)lnn=
−
P P π 1+U +U3
xln +2πP ln 2πP + .
n n − − 4 O T
Å ã
Then we obtain (2.1) by (2.3). (cid:3)
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Jan Moser Jacob’s ladder ...
2.2.
Lemma 2.
1 T+U 1
(2.4) Z2(t)dt=S(T,U)+ ,
U O T1/6
ZU Å ã
for U T1/6−ǫ/2.
≤
Proof. Let us remind the Hardy-Littlewood formula (see [12], p. 80)
d(n)
(2.5) Z2(t)=2 cos 2ϑ(t) tlnn + (t−1/6)
√n { − } O
nX≤2tπ
with the Motohashi error term (see [3], p. 125). Since
d(n) 1 U
= UTǫ = ,
2Tπ≤Xn≤T2+πU √n OÅ √Tã OÅT1/2−ǫã
then
d(n)
Z2(t)=2 cos 2ϑ(t) tlnn +
√n { − }
n<P
X
1 U
+ ,
O T1/6 O T1/2−ǫ
Å ã Å ã
and
T+U d(n) T+U
(2.6) Z2(t)dt=2 cos 2ϑ(t) tlnn dt+
√n { − }
ZT n<P ZT
X
U U2
+ + .
O T1/6 O T1/2−ǫ
Å ã Å ã
Since
U +U3 d(n) U +U3 U +U3
= Tǫ√T = ,
T √n O T O T1/2−ǫ
Å ã Å ã
n<P
X
then we obtain (2.4) by (2.1), (2.6). (cid:3)
3. Proofs of the Theorems
3.1. Proof of Theorem 1. Let us remind that for every sufficiently big T there
0
is a continuum of pairs
(3.1) [T˜,U˜]: T˜ [T ,T +U ], U˜ 0,T1/6−ǫ/2
0 0 0
∈ ∈
Ä ó
for which the formula
1 T˜+U˜ lnlnT˜
(3.2) Z2(t)dt=lnT˜ 1+
U˜ ZT˜ ® OÇ lnT˜ å´
is true (see [6], Corollary 2, Remark 4). Then we obtain
lnlnT˜ S(T˜,U˜) 1
(3.3) 1+ = + , T
OÇ lnT˜ å lnT˜ OÅT˜1/6lnT˜ã 0 →∞
by (2.4), (3.2). Finally, we obtain the assertion by (3.1), (3.3).
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Jan Moser Jacob’s ladder ...
3.2. Proof of Theorem 2. First of all the formula
γ+U(γ,α) lnlnγ
(3.4) Z2(t)dt=τUlnγ 1+ ,
O lnγ
Zγ ß Å ã™
τ =tan[α(γ,U)] [η,1 η], U(γ)=γ1/3+2ǫ+∆(γ)<1.1γ1/3+2ǫ
∈ −
forrotatingchordistrue((3.4)isthedualasymptoticformulawhichcorrespondsto
[5],(4.4)by[6],(1.2)). Itisclearthatforeveryfixeddirectionoftherotatingchord
(i.e. for the fixed value τ [η,1 η]) a continuum of parallel chords corresponds.
∈ −
From this set we choose a continual subset such that the condition
(3.5) U˜ <γ1/6−ǫ/2
(compare with [6], Remark 4) is fulfilled. For this continuum set the formula
1 T˜+U˜ lnlnT˜
(3.6) Z2(t)dt=τlnT˜ 1+
U˜ ZT˜ ® OÇ lnT˜ å´
is true (see (3.4)), where
(3.7) γ <T˜ <γ+1.1U (γ); U (γ)=γ1/3+2ǫ.
0 0
Next, from (3.6) by (2.4) we obtain
S(T˜,U˜) lnlnT˜
(3.8) τ = + , γ .
lnT˜ OÇ lnT˜ å →∞
Finally, we obtain, by (3.5), (3.7), (3.8) the assertion.
4. Discussion on necessity of a new theory for short and microscopic
parts of the Hardy-Littlewood integral
Let us remind the results of Balasubramanian, Heath-Brown and Ivic for the
Hardy-Littlewood integral
T
Z2(t)dt
Z0
and for the parts of this integral.
4.1. First of all the Balasubramanianformula
T
(4.1) Z2(t)dt=T lnT +(2c 1 ln2π)T +R(T), R(T)= (T1/3+ǫ)
− − O
Z0
is true (see [1]).
Remark 2. The Good’s Ω-theorem (see [2]) implies for the Balasubramanian’sfor-
mula (4.1) that
(4.2) limsup R(T) =+ ,
| | ∞
T→∞
i.e. the error term in (4.1) is unbounded at T .
→∞
For the short interval the Balasubramanian’s formula implies
T+U0
(4.3) Z2(t)dt=U lnT +(2c ln2π)U + (T1/3+ǫ), U =T1/3+2ǫ.
0 0 0
− O
ZT
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Jan Moser Jacob’s ladder ...
4.2. Furthermore, let us remind the Heath-Brown’s estimate (see [3], (7.20), p.
178)
T+G
(4.4) Z2(t)dt= GlnT +G (TK)−14 (S(K)+
ZT−G O( K | |
X
K
+K−1 S(x)dx e−GT2K
| | å ´
Z0
(for definition of used symbols see [3], (7.21)-(7.23)), uniformly for Tǫ G
≤ ≤
T1/2 ǫ. And, finally, we add the Ivic’ estimate ([3], (7.26))
−
T+G 1
(4.5) Z2(t)dt= (Gln2T), G T1/3−ǫ0, ǫ = 0.009.
0
O ≥ 108 ≈
ZT−G
Remark 3. Itisquiteevidentthattheintervals[T G,T+G],G (0,1),forexam-
− ∈
ple, cannotbe reachedin theories leading to the formula (4.3)of Balasubramanian
or to the estimates (4.4) and (4.5) of Heath-Brown and Ivic, respectively.
4.3. Inthis situationI developedthe new theorybasedongeometricpropertiesof
the Jacob’s ladders. Let us remind the basic formulae of our theory.
Titchmarsh-Kober-Atkinson(TKA) formula (see [12], p. 141)
∞ c ln(4πδ) N
(4.6) Z2(t)e−2δtdt= − + c δn+ (δN+1)
n
2sinδ O
Z0 n=0
X
remained as an isolated result for the period of 56 years. We have discovered (see
[4]) the nonlinear integral equation
µ[x(T)] T
(4.7) Z2(t)e−x(2T)tdt= Z2(t)dt,
Z0 Z0
in which the essence of the TKA formula is encoded. Namely, we have shown in
[4]thatthe followingalmost-exactformulafortheHardy-Littlewoodintegral(after
the period of 90 years)
T ϕ(T) ϕ(T) ϕ(T) lnT
(4.8) Z2(t)dt= ln +(c ln2π) +c +
0
2 2 − 2 O T
Z0 Å ã
takes place, where ϕ(T) is the Jacob’s ladder, i.e. an arbitrary solution to the
nonlinear integral equation (4.7).
Remark 4. In the case of our result (4.8) the errorterm tends to zero as T goes to
infinity, namely
lnT
lim r(T)=0, r(T)= ,
T→∞ OÅ T ã
(compare with (4.2)).
4.4. In the papers [4],[5] I obtained the following additive formula
T+U ϕ(T) 1
(4.9) Z2(t)dt=Uln e−a tan[α(T,U)]+
2 O T1/3−4ǫ
ZT Å ã Å ã
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Jan Moser Jacob’s ladder ...
thatholdstrueforshortpartsoftheHardy-Littlewoodintegral. Next,inthepaper
[6] I proved the multiplicative asymptotic formula (µ[ϕ]=7ϕlnϕ)
T+U lnlnT T
(4.10) Z2(t)dt=UlnT tan[α(T,U)] 1+ , U 0,
O lnT ∈ lnT
ZT ß Å ã™ Å ò
for short and microscopic parts of the Hardy-Littlewood integral.
Remark 5. The formulae (4.7)-(4.10) - and all corollaries from these (see [5], [6])
- cannot e derived within complicated methods of Balasubramanian,Heath-Brown
andIvic. Thisprovesthenecessityofanewmethod-whichisbasedonelementary
geometric properties of Jacob’s ladders - to study short and microscopic parts of
the Hardy-Littlewood integral.
IwouldliketothankMichalDemetrianforhelpingmewiththeelectronicversion
of this work.
References
[1] R. Balasubramanian, ‘An improvement on a theorem of Titchmarsh on the mean quare of
|ζ(1/2+it)|‘,Proc.Lond.Math.Soc.3,36(1978)540-575.
[2] A. Good, ‘Ein Ω-Resultat fu¨r quadratische Mittel der Riemannschen Zetafunktion auf der
kritischeLinie’,Invent. Math.41(1977), 233-251.
[3] A.Ivic,‘TheRiemannzeta-function‘, AWilley-IntersciencePublication,NewYork,1985.
[4] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-
Littlewoodintegral’,(2008), arXiv:0901.3973.
[5] J.Moser,‘Jacob’sladdersandthetangent lawforshortpartsoftheHardy-Littlewoodinte-
gral’,(2009), arXiv:0906.0659.
[6] J.Moser,‘Jacob’sladdersandthemultiplicativeasymptoticformulaforshortandmicroscopic
partsoftheHardy-Littlewoodintegral’,(2009), arXiv:0907.0301.
[7] J. Moser, ‘Jacob’s ladders and the quantization of the Hardy-Littlewood integral’, (2009),
arXiv:0909.3928.
[8] 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), arXiv:0911.1246.
[9] 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), arXiv:0912.0130.
[10] J. Moser, ‘Jacob’s ladders, the iterations of Jacob’s ladder ϕk1(t) and asymptotic formulae
fortheintegralsoftheproductsZ2[ϕn1(t)]Z2[ϕn−1(t)]···Z2[ϕ01(t)]forarbitraryfixedn∈N‘
(2010), arXiv:1001.1632.
[11] J. Moser, ‘Jacob’s ladders and the asymptotic formula for the integral of the eight order
expression|ζ(1/2+iϕ2(t))|4|ζ(1/2+it)|4‘,(2010), arXiv:1001.2114.
[12] 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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