Jacob's ladders, heterogeneous quadrature formulae, big asymmetry and related formulae for the Riemann zeta-function

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by  Jan Moser

JACOB’S LADDERS, HETEROGENEOUS QUADRATURE
FORMULAE, BIG ASYMMETRY AND RELATED FORMULAE
FOR THE RIEMANN ZETA-FUNCTION
4
1
0
JANMOSER
2

n
Abstract. In this paper weobtain as our mainresultnew class of formulae
a
expressing correlation integrals of the third-order in Z on disconnected sets
J
G˚1(x),G˚2(y) by means of an autocorrelative sum of the second order in Z.
3 Moreover, the distance of the sets G˚1(x),G˚2(y) from the set of arguments of
1
autocorrelative sum isextremely big, namely ∼Aπ(T), T →∞,where π(T)

istheprime-countingfunction.
]
A
C
.
h 1. Introduction
t
a 1.1. Let us remind the following formulae
m
2T 2π
[ Z4(t)dt Z4(t ), T ,
1 ZT ∼ lnT T≤tνX≤T+2T ν →∞
(1.1)
v T+U 2π
6 Z2(t)dt Z2(t ), U =√T lnT, T
Z ∼ lnT ν →∞
5 T T≤tXν≤T+U
8
(see [5], (4.4), (4.7)), where
2
. 1
1 Z(t)=eiϑ(t)ζ +it ,
0 Å2 ã
4 (1.2)
t 1 t
1 ϑ(t)= lnπ+ImlnΓ +i
: −2 Å4 2ã
v
(see [9], pp. 79, 329), and t is the Gram sequence (comp. [9], p. 99). The
i ν
X { }
formulae (1.1) were proved by us in connection with the Kotelnikov-Whittaker-
r Nyquist theorem. Namely, by these formulae we have expressed the biquadratic
a
and quadratic effects for the continuous signals
Z(t), t [T,2T]; t [T,T +U]
∈ ∈
from the point of view of information theory.
1.2. The formulae (1.1) are:
(a) asymptotic quadrature formulae (from the left to the right),
(b) asymptotic summation formulae (from the right to the left).
Next, for the second formula in (1.1) (for example) we have
t [T,T +U] t [T,T +U],
ν
(1.3) ∈ → ∈
Z2(t) Z2(t ),
ν
→
Key words and phrases. Riemannzeta-function.
1
Jan Moser Jacob’s ladder ...
i. e. the segment [T,T +U] and the exponent 2 are conserved.
Remark 1. By(1.3)itisnaturaltocallthe formulaeofthekind(1.1)homogeneous
formulae.
On the contrary, we obtain in this paper some heterogeneous asymptotic quad-
rature formulae – such formulae that the properties (1.3) are not fulfilled.
2. Heteregeneous quadrature formulae
2.1. Let (see [3], (3))
G (x)=G (x;T,H)=
1 1
π
= t: t ( x)<t<t (x) , 0<x ,
2ν 2ν
{ − } ≤ 2
[
T≤t2ν≤T+H
(2.1) G2(y)=G2(y;T,H)=
π
= t: t ( y)<t<t (y) , 0<y ,
2ν+1 2ν+1
{ − } ≤ 2
[
T≤t2ν+1≤T+H
H =T1/6+2ǫ,
where the collection of sequences
t (τ) , τ [ π,π], ν =1,2,...
ν
{ } ∈ −
is defined by the equation (see [3], (1))
ϑ[t (τ)]=πν+τ; t (0)=t ,
ν ν ν
and (see [3], (8))
x x y y
(2.2) m G (x) = H + , m G (y) = H + ,
1 2
{ } π O(cid:16)lnT(cid:17) { } π O(cid:16)lnT(cid:17)
where m G (x) , ...is the measure of the set G , ....
1 1
{ }
Let
ϕ G˚ (x) =G (x), ϕ G˚ (y) =G (y).
1 1 1 1 2 2
{ } { }
The following Theorem holds true.
Theorem 1.
ω(t)Z[ϕ (t)]Z2(t)dt=
Z 1
G˚1(x)
m G (x) x H
1
= { } Z(t )Z t + + ,
Q lnP ν Å ν lnP ã OÅlnTã
1 0 X 0
T≤tν≤T+U1
(2.3) ω(t)Z[ϕ (t)]Z2(t)dt=
Z 1
G˚2(y)
m G (y) y H
2
= { } Z(t )Z t + + ,
− Q lnP ν Å ν lnP ã OÅlnTã
1 0 X 0
T≤tν≤T+U1
π
x,y 0, , T ,
∈(cid:16) 2 i →∞
where
1 lnlnt
(2.4) ω(t)= 1+ ,
lntß OÅ lnt ã™
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Jan Moser Jacob’s ladder ...
and (see [1], (38); H U )
1
→
1 U2
Q = 1= U lnP + 1 ,
1 π 1 0 OÅ T ã
X
(2.5) T≤tν≤T+U1
T
U1 =√T lnP0, P0 =… .
2π
Furthermore we have
(2.6) m G˚ (x) , m G˚ (y) <T1/3+ǫ.
1 2
{ } { }
2.2.
Remark 2. Every of the correlation integrals in (2.3) contains the product
(2.7) Z[ϕ (t)]Z2(t)
1
and, as usually, we have (see [6], (6.2); [7], (8.4))
t
(2.8) t ϕ (t) (1 c) (1 c)π(t), t ,
1
− ∼ − lnt ∼ − →∞
i. e. wehavebigdifferenceofargumentsin(2.7),wherecistheEulerconstantand
π(t) is the prime-counting function.
Next, in the case
t=T˚; ϕ (T˚)=T T T˚
1
⇒ →∞ ⇔ →∞
we obtain from (2.8)
T˚ T 1
T˚ T (1 c) 1 (1 c)
− ∼ − lnT˚ ⇒ − T˚∼ − lnT˚ ⇒
T˚ T lnT˚ lnT,
⇒ ∼ ⇒ ∼
i. e.
T
T˚ T (1 c) ,
− ∼ − lnT
and (see (2.5))
T T
T˚ (T +U ) (1 c) U (1 c) .
1 1
− ∼ − lnT − ∼ − lnT
Consequently we have
˚
ρ [T˚,T +H];[T,T +U ] (1 c)π(T);
1
(2.9) { }∼ −
˙ ˚
[T,T +U ] [T˚,T +H],
1
≺
where ρ stands for the distance of the˙corresponding segments. We may, of course,
put
(2.10) G (x) [T,T +H]=G¯ (x) G (x),...
1 1 1
∩ →
if necessary.
Remark 3. We have the following properties
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Jan Moser Jacob’s ladder ...
(a)
˚
G˚ (x),G˚ (y) [T˚,T +H] [T,T +U],
1 2
∈ →
(comp. (1.3), (2.10)), where G˚ (˙x),G˚ (y) are disconnected sets,
1 2
(b) if
=G˚ (x),G˚ (y)
1 2
G
then extremely big distance occurs, namely (comp. (2.9), (2.10))
ρ ;[T,T +U ] (1 c)π(T), T ,
1
{G }∼ − →∞
(c) for the corresponding orders of Z (comp. (1.3))
1+2 1+1
→
then the formulae (2.3) are strongly heterogeneous (comp. Remark 1).
Remark 4. Moreover we explicitly notice the following:
(a) the formulae (2.3) are not accessible by the current methods in the theory
of the Riemann zeta-function,
(b) small improvements of the exponents
1 1
, , ...
6 2
are irrelevant for main direction of this paper (comp. our paper [2], Ap-
pendex A: On I.M. Vinogradov’ scepticism on possibilities of the method
of trigonometric sums).
3. Big asymmetry and related formulae
3.1. Let (see [4], p. 29)
G (x)=G (x;T,U )=
3 3 2
π
= t: g ( x)<t<g (x) , 0<x ,
2ν 2ν
{ − } ≤ 2
[
T≤g2ν≤T+U2
(3.1) G4(y)=G4(y;T,U2)=
π
= t: g ( y)<t<g (y) , 0<y ,
2ν+1 2ν+1
{ − } ≤ 2
[
T≤g2ν+1≤T+U2
U =T5/12+2ǫ,
2
where the collection of sequences
g (τ) , τ [ π,π], ν =1,2,...
ν
{ } ∈ −
is defined by the equation (see [4], (6))
π τ
ϑ [g (τ)]= ν+ ; g (0)=g ,
1 ν ν ν
2 2
where (comp. (1.2))
1 t t t π
ϑ(t)=ϑ (t)+ , ϑ (t)= ln ,
1 OÅtã 1 2 2π − 2 − 8
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and (see [4], (13))
x x
m G = U + ,
3 2
(3.2) { } π O(cid:16)lnT(cid:17)
y y
m G = U + .
4 2
{ } π O(cid:16)lnT(cid:17)
Let
ϕ G˚ (x) =G (x), ϕ G˚ (y) =G (y).
1 3 3 1 4 4
{ } { }
The following Theorem holds true.
Theorem 2.
ω(t)Z2[ϕ (t)]Z2(t)dt
ZG˚3(x) 1 ∼
4
(3.3) U sinx+ ω(t)Z2[ϕ (t)]Z2(t)dt,
∼ π 2 ZG˚4(x) 1
π
x 0, , T .
∈(cid:16) 2i →∞
Remark 5. By the asymptotic formula (3.3) is expressedthe property ofbig asym-
metry in the distribution of the values of Z on disconnected sets G˚ (x),G˚ (x).
3 4
Namely, the correlation integral (comp. (2.8)) on the disconnected set G˚ (x) es-
3
sentially exceeds of that on the set G˚ (x). For example
4
4
U , T ,
ZG˚3(π/2)−ZG˚4(π/2) ∼ π 2 →∞
where
m G˚ (π/2) +m G˚ (π/2) =U
3 4 2
{ } { }
if (comp. (2.10))
G (x) [T,T +U ] G (x),...
3 2 3
∩ →
3.2. Finally, the following Theorem holds true.
Theorem 3.
ω(t)Z[ϕ (t)]Z2(t)dt=
Z 1
G˚3(x)
m G (x)
= { 1 } ω(t)Z2[ϕ (t)]Z2(t)dt
2m{G3(x)} ZG˚3(x) 1 −
m G (x)
{ 1 } ω(t)Z2[ϕ (t)]Z2(t)dt+ (HT−ǫ),
− 2m{G4(x)}ZG˚4(x) 1 O
(3.4) ω(t)Z[ϕ (t)]Z2(t)dt=
Z 1
G˚2(y)
m G (y)
= { 2 } ω(t)Z2[ϕ (t)]Z2(t)dt
2m{G4(y)}ZG˚4(y) 1 −
m G (y)
{ 2 } ω(t)Z2[ϕ (t)]Z2(t)dt+ (HT−ǫ),
− 2m{G3(y)}ZG˚3(y) 1 O
π
x,y 0, , T .
∈(cid:16) 2i →∞
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Remark 6. For the disconnected sets
G˚ (x),G˚ (y),G˚ (x),G˚ (x),G˚ (y),G˚ (y);G˚ G˚ = ,G˚ G˚ =
1 2 3 4 3 4 1 2 3 4
∩ ∅ ∩ ∅
we have the following property: the correlation integrals of the order 1 + 2 on
G˚ (x),G˚ (y) are expressed as the linear combinations of the correlation integrals
1 2
of the order 2+2 on G˚ (x),G˚ (x);G˚ (y),G˚ (y) correspondingly.
3 4 3 4
4. Proof of Theorem 1
4.1. In the paper [1] (see (10)) we have proved the following autocorrelative for-
mula
Z(t )Z(t +β)=
ν ν
X
(4.1) T≤tν≤T+U1
2sin(βlnP )
= 0 U lnP + (√T ln2T),
1 0
π βlnP O
0
where
1
(4.2) β = , U =√TlnP .
OÅlnTã 1 0
In the case
x π
βlnP =x β = , x 0,
0
⇒ lnP0 ∈(cid:16) 2 i
we obtain from (4.1), (4.2)
x
Z(t )Z t + =
ν Å ν lnP ã
X 0
(4.3) T≤tν≤T+U1
2sinx
= U ln2P + (√T ln2T).
1 0
π x O
Consequently we have (see (2.5), (4.2))
1 x
Z(t )Z t + =
Q lnP ν Å ν lnP ã
1 0 X 0
(4.4) T≤tν≤T+U1
sinx 1 π sinx 2
=2 + ; x 0, ,1 .
x OÅlnTã ∈(cid:16) 2i ⇒ x ∈ïπ ã
4.2. Next, in the paper [3], (5),(9) we have obtained the following mean-value
formula on the disconnected sets G (x),G (y) (see (2.1))
1 2
2
Z(t)dt= Hsinx+ (xT1/6+ǫ),
Z π O
G1(x)
(4.5) Z(t)dt= 2Hsiny+ (yT1/6+ǫ),
Z −π O
G2(y)
H =T1/6+2ǫ; T1/6ψ2ln5T T1/6+ǫ.
→
Hence, from (4.4) by (2.3) we obtain
1 sinx
Z(t)dt=2 + (T−ǫ),
m G (x) Z x O
(4.6) { 1 } G1(x)
1 siny
Z(t)dt= 2 + (T−ǫ).
m G (y) Z − y O
{ 2 } G2(y)
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Jan Moser Jacob’s ladder ...
4.3. In the paper [7], (9.2), (9.5) we have proved the following Lemma: if
˚
ϕ [T˚,T +U] =[T,T +U],
1
{ }
˙
then for every Lebesgue-integrable function
f(x), x [T,T +U]
∈
we have
˚
T+U T+U
f[ϕ (t)]Z˜2(t)dt= f(x)dx,
(4.7) Z ¯ 1 Z
T˚ T
T
T T [ϕ ], U 0, ,
≥ 0 1 ∈Å lnTò
where
Z2(t)
Z˜2(t)= =ω(t)Z2(t);
1+ lnlnt lnt
(4.8) O lnt
(cid:8) (cid:0)1 (cid:1)(cid:9) 1 lnlnt
ω(t)= = 1+ .
1+ lnlnt lnt lntß OÅ lnt ã™
O lnt
(cid:8) (cid:0) (cid:1)(cid:9)
Consequently, we have (see (4.7), (4.8))
˚
T+U T+U
ω(t)f[ϕ (t)]Z2(t)dt= f(x)dx,
(4.9) Z ¯ 1 Z
T˚ T
T
U 0, .
∈Å lnTò
4.4. Hence, by (4.6), (4.9) we obtain following formulae
1
ω(t)Z[ϕ (t)]Z2(t)dt=
m{G1(x)}ZG˚1(x) 1
sinx
=2 + (T−ǫ),
x O
(4.10)
1
ω(t)Z[ϕ (t)]Z2(t)dt=
m{G2(y)}ZG˚2(y) 1
siny
= 2 + (T−ǫ).
− y O
Finally, simple elimination of the values
sinx siny
2 , 2
x y
from (4.4), (4.9) gives (2.3).
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Jan Moser Jacob’s ladder ...
4.5. In the case
f(x)=1
in (4.7) we obtain (see (2.1), (2.10), (4.8))
˚
1 2 T+H 1 2
ζ +it dt< ζ +it dt
ZG˚1(x)(cid:12)(cid:12) Å2 ã(cid:12)(cid:12) ZT˚¯ (cid:12)(cid:12) Å2 ã(cid:12)(cid:12) ∼
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) T+H (cid:12) (cid:12) (cid:12)
lnT 1 dt,
∼ Z ·
T
i. e.
˚
(4.11) T+H 1 2
ζ +it dt<AHlnT, T .
ZT˚¯ (cid:12)(cid:12) Å2 ã(cid:12)(cid:12) →∞
(cid:12) (cid:12)
Let (cid:12) (cid:12)
˚
(4.12) T +H T˚ T1/3+ǫ,
− ≥
(comp. [6], (2.5); where 1 is th˙e Balasubramanianexponent). Then
3
˚
T+H 1 2 ˚ ˚
(4.13) ζ +it dt (T +H T˚)lnT >B(T +H T˚)lnT,
ZT˚¯ (cid:12)(cid:12) Å2 ã(cid:12)(cid:12) ∼ − −
(cid:12) (cid:12) ˙ ˙
and (see (4.11), (4.(cid:12)13)) (cid:12)
˚
(4.14) T +H T˚<CH =CT1/6+ǫ.
−
˙
Now, (4.14) contradicts (4.12). Consequently we have that
˚
m G˚ (x) <T +H T˚<T1/3+ǫ
1
{ } −
˙
and we obtain the second inequality in (2.6) by the similar way.
5. Proofs of Theorem 2 and Theorem 3
5.1. Inthepaper[4],(14),(15)wehaveprovedthefollowingmean-valueformulae
Z2(t)dt=
Z
G3(x)
x T 2
= U ln + (cx+sinx)U + (T5/12ln2T),
2 2
π 2π π O
(5.1)
Z2(t)dt=
Z
G4(y)
y T 2
= U ln + (cy siny)U + (T5/12ln2T)
2 2
π 2π π − O
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Jan Moser Jacob’s ladder ...
on the disconnected sets G (x),G (y), (comp. (3.1)). From (5.1) we have in the
3 4
case x=y
Z2(t)dt Z2(t)dt=
Z −Z
G3(x) G4(x)
4
= U sinx+ (xT5/12ln2T).
2
π O
Consequently,we obtainfrom (5.1) by (4.7) – (4.9) with f(x)=Z2(x) the formula
(3.3).
5.2. Next, from the formula (see [4], (16))
1 1
Z2(t)dt Z2(t)dt
m G (x) Z − m G (x) Z ∼
{ 3 } G3(x) { 4 } G4(x)
sinx
4 , T
∼ x →∞
we obtain by (4.7) – (4.9) (f(x)=Z2(x)) the following formula
1
ω(t)Z2[ϕ (t)]Z2(t)dt
(5.2) m{G3(x)} ZG˚3(x) 1 −
1 sinx
ω(t)Z2[ϕ (t)]Z2(t)dt 4 , T
− m{G4(x)}ZG˚4(x) 1 ∼ x →∞
and, by the similar way, we obtain the formula for y (0,π/2]. Hence, the elimi-
∈
nation of
sinx siny
2 , 2
x y
from (4.10), (5.2) implies the formula (3.4).
I would like to thank Michal Demetrian for his help with electronic version of
this paper.
References
[1] J.Moser,‘Onautocorrelative sum inthetheory of theRiemannzeta-function‘, ActaMath.
Univ.Comen.,37(1980), 121-134, (inRussian).
[2] J.Moser,‘OntherootsoftheequationZ′(t)=0‘,ActaArith.40(1981),97-107,(inRussian),
arXiv: 1303.0967.
[3] J. Moser, ‘New consequences of the Riemann-Siegel formula‘, Acta Arith., 42 (1982), 1-10,
(inRussian),arXiv: 1312.4767.
[4] J. Moser, ‘New mean-value theorems for the function |ζ 1 +it |2‘, Acta Math. Univ.
2
Comen.,46-47(1985), 21-40. (cid:0) (cid:1)
[5] J.Moser,‘OntheorderofTitchmarshsuminthetheoryoftheRiemann’szeta-functionand
onthebiquadraticeffectintheinformationtheory‘,CzechoslovakMath.J.,41(116),(1991),
663-684,(inRussian),arXiv: 1112.9548.
[6] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-
Littlewoodintegral’,Math.Notes88,(2010) 414-422,arXiv: 0901.3937.
[7] J.Moser,‘Jacob’sladders,thestructureoftheHardy-Littlewoodintegralandsomenewclass
ofnonlinearintegralequations‘,Proc.Stek.Inst. 276,(2011), 208-221,arXiv: 1103.0359.
[8] E.C. Titchmarsh, ‘On van der Corput ’s method and the zeta-function of Riemann (IV)‘,
Quart.J.Math.,(Oxford),5,(1934), 98-105.
[9] 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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