Jacob's ladders, $\zeta$-factorization and infinite set of metamorphosis of a multiform

0.13 MB·English

by  Jan Moser

JACOB’S LADDERS, ζ-FACTORIZATION AND INFINITE SET
OF METAMORPHOSIS OF A MULTIFORM
5
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0 JANMOSER
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b Abstract. In this paper we use Jacob’s ladders together with fundamental
e Hardy-Littlewoodformula(1921)toprovetheso-calledζ-factorizationformula
F on the critical line. Simultaneously, we obtain a set of control parameters of
metamorphosisofamultiformconnected withtheRiemann-Siegelformula.
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1

]
A
1. Introduction and main result
C
1.1. Let us remind the unique factorization theorem: any positive integer (= 1)
.
h 6
can be expressed as a product of primes. This expression is unique except of the
t
a order in which the primes occur. Thus, we have
m
n=pm1pm2 pmr, n N, n=1,
[ (1.1) 1 2 ··· r ∈ 6
p <p < <p , m ,m ,...,m N,
1 2 ··· r 1 2 r ∈
2
where
v
5 p ,...,p
1 r
0
are primes. Next, Euler’s identity (1737)
7
7 ∞
1
0 (1.2) = (1 p−x)−1, x>1
. X nx Y −
1 n=1 p
0 may be regarded as an analytical equivalent of the unique factorization theorem.
5
Further, Riemann has defined in 1859 the zeta-function by the formula
1
: ∞ 1
v (1.3) ζ(s)= = (1 p−s)−1, s=σ+it C, σ >1,
i ns − ∈
X nX=1 Yp
r and, consequently, by the method of analytic continuation,he defined the function
a
ζ(s) at all finite points s C except for a simple pole at s = 1. Now, since from
∈
(1.1)), (comp. (1.2), (1.3)) the exponential form of the factorization
1 x r 1 x
(1.4) = , x>1,
n pml
Yl=1 l
Å ã Å ã
follows (as an example), then we can formulate the following.
Remark 1. OnthebasisoftheanalyticcontinuationoftheRiemann’szeta-function
we may suppose that:
(a) there areinheritedforms ofthe factorization(1.4)inthe regionσ 1, s=
≤ 6
1, (together with mutation of these),
Key words and phrases. Riemannzeta-function.
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Jan Moser Jacob’s ladder ...
(b) especially, we may suppose that on the critical line itself
σ =1/2
there are some new species of the factorization property (1.4).
1.2. In this direction the following theorem holds true.
Theorem. Let
k =1,...,k N
0
∈
for every fixed k and
0
lnlnT
1
(1.5) H =H(T) ,TlnlnT
∈ lnT
for every sufficiently big T > 0. ThenÅ for every k anãd for every H that fulfils
condition (1.5) there are the functions
H =H (T,H)>0,
k k
α =α (T,H,k)>0, r =0,1,...,k,
(1.6) r r
1
α =γ : ζ +iγ =0,
r
6 2
such that the following ζ-factorizaÅtion formãula
k
Λ 1
ζ +iα ,
(1.7) ζ 12 +iα0 ∼rY=1(cid:12)(cid:12) 2 r (cid:12)(cid:12)
 (cid:12) (cid:0) (cid:1)(cid:12) (cid:12) Å ã(cid:12)
(cid:12) (cid:12) (cid:12)√H (cid:12)
Λ=Λ(T,H,k)=√2π lnkT
H
k
holds true. Moreover,the sequence
α k
{ r}r=0
has the following properties:
(1.8) T <α <α < <α ,
0 1 k
···
(1.9) α α (1 c)π(T), r =0,1,...,k 1,
r+1 r
− ∼ − −
where
T
π(T) , T
∼ lnT →∞
is the prime counting function and c is the Euler’s constant.
Remark 2. The asymptotic behavior of the set
(1.10) α ,α ,...α
0 1 k
{ }
by(1.9)isasfollows: ifT thenthepointsoftheset(1.10)recedeunboundedly
→∞
each from other and all these points together recede to infinity. Hence, at T
→∞
the set(1.10) behaves like one dimensional Friedmann-Hubble expanding universe.
Remark 3. Consequently, the following holds true: even if the distances
α α q r(1 c)π(T) as T ,
q r
| − |∼| − | − →∞ →∞
q,r=0,1,...,k, q =r
6
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Jan Moser Jacob’s ladder ...
(see (1.9)) of elements of the set (1.10) are gigantic, there is a close constraint
between the values of the set
1 1 1
ζ +iα , ζ +iα ,..., ζ +iα ,
(cid:12) 2 0 (cid:12) (cid:12) 2 1 (cid:12) (cid:12) 2 k (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
ß(cid:12) Å ã(cid:12) (cid:12) Å ã(cid:12) (cid:12) Å ã(cid:12)™
namely the facto(cid:12)rization form(cid:12)ul(cid:12)a (1.7). (cid:12) (cid:12) (cid:12)
2. On metamorphosis of a multiform generated by the
Riemann-Siegel formula
2.1. Let us remind the Riemann-Siegel formula
1
Z(t)=2 cos ϑ(t) tlnn + (t−1/4),
√n { − } O
X
(2.1) n≤τ(t)
t
τ(t)= ,
2π
…
(see [6], p. 60, comp. [7], p. 79) where
t 1 1
ϑ(t)= lnπ+ImlnΓ +i ,
−2 4 2
Å ã
(see ([7]), p. 239). Next, we put in (2.1)
(2.2) Z(t)= a f (t)+R(t),
n n
X
n≤τ(t)
where
1
a = ,
n
√n
(2.3)
f (t)=cos ϑ(t) tlnn ,
n
{ − }
R(t)= (t−1/4),
O
and the functions f (t) are nonlinear. That is, the function Z(t) is nonlinear
n
monoform. Now, we define the following multiform
k
(2.4) G(x ,...,x )= Z(x ), x >T >0, k 2,
1 k r r
| | ≥
Y
r=1
where the monoform Z(t) is the generating function of G.
Further, we define the subset of all random sample points
(x¯ ,...,x¯ ): T <x¯ <x¯ < <x¯ ,
1 k 1 2 k
···
1
x¯ =γ : ζ +iγ =0, r =1,...,k,
r
6 2
and propose the following Å ã
Question. Is there in this subset a point
(y¯ ,...,y¯ )
1 k
of metamorphosis of the multiform (2.4) (that is, the point of significant change of
the structure of the multiform (2.4))?
The answer is given in the following.
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Jan Moser Jacob’s ladder ...
Corollary.
k
(cid:12) a f (α )+R(α )(cid:12)
(cid:12) n n r r (cid:12)∼
rY=1(cid:12)(cid:12)n≤Xτ(αr) (cid:12)(cid:12)
(2.5) (cid:12) (cid:12)
(cid:12) Λ (cid:12)
, T ,
∼ →∞
a f (α )+R(α )
Ã(cid:12) n≤τ(α0) n n 0 0 (cid:12)
(cid:12)P (cid:12)
(cid:12) (cid:12)
i.e. to the infinite subset of the points
α (T),α (T),...,α (T) , T (T , ),
1 2 k 0
{ } ∈ ∞
where T is a sufficiently big, an infinite set of metamorphoses of the multiform
0
(2.4) into quite distinct form on the right-hand side of (2.5) corresponds.
Remark 4. We shall call the elements of the set
(2.6) α (T),α (T),...,α (T) , T (T ,+ )
0 1 k 0
{ } ∈ ∞
as control parameters (functions) of metamorphosis. The reasonto this is that the
parameters
α (T),...,α (T)
1 k
change the old form into the new one (see (2.5), and this last is controlled by the
parameterα (T). That is, the set(2.6) plays a similar roleas shem-ha-m’forashin
0
Golem’s metamorphosis.
3. On Hardy-Littlewood fundamental Lemma 18 from the memoir [2]
3.1. Let us remind the following (see [2], pp. 304, 305): if
T+U
J =J(T,H)= 2dt,
Z J
(3.1) T
t+H
= (t,H)= x(u)du,
J J Z
t
then the following Hardy-Littlewood formula
T+H U
(3.2) J = 2dt=π√2πHU +
Z J O lnT
T
Å ã
holds true, where
1 5
(3.3) U [Ta,Tb], <a<b , 0<H Tǫ,
∈ 2 ≤ 8 ≤
and ǫ is positive and sufficiently small, and (see [2], p. 290)
1 2 1/4 1
(3.4) ζ +it = eiπ/8(2πe)21ite−12itlntx(t) 1+ .
2 − π O t
Å ã Å ã ß Å Pãa™ge 4 of 9
Jan Moser Jacob’s ladder ...
3.2. We use the Riemann function Z(t) instead of the Hardy-Littlewood function
x(t). Namely, the formula
1
(3.5) ζ +it =e−iϑ(t)Z(t)
2
together with formula for ϑ(t)Åfrom [7]ã, p. 329 give that
π 1/4 1
(3.6) x(t)= Z(t) 1+ .
−(cid:16)2(cid:17) O t
Hence, we use the following variant of the Haßrdy-LitÅtlewão™od formula (see (3.6))
T+U0 U
(3.7) J¯=J¯(T,H)= ¯2dt=2πHU + 0 ,
Z J 0 O lnT
T
where (comp. (3.1) – (3.3)) Å ã
t+H
¯= ¯(t,H)= Z(u)du,
J J Z
t
(3.8) U =T0.5001,
0
lnlnT
H ,Tlnl1nT .
∈ lnT
Å ã
Remark 5. Since (see (3.7))
U 1
0
2πHU + =2πHU 1+ ,
0 0
O lnT O HlnT
then we have that the formuÅla (3.7ã) is asymptßotic formÅula forãH™of (3.8).
3.3. Let us remind the following sentences of Hardy and Littlewood (see [2], p.
315): As was observed in 5.5, we do not use the full force of Lemma 18. The
complete lemma, however, seems of considerable interest in itself, and it may prove
to be of service in the future.
Remark 6. We notice explicitly, that our theory of the Jacob’s ladders together
with the Hardy-Littlewood asymptotic formula (3.7) constitute the basis of our
result about metamorphosis of corresponding multiform. Thus, we have obtained,
after94years,theresultthatusethefullforceoftheasymptoticHardy-Littlewood
formula from their Lemma 18.
4. Jacob’s ladders and the Hardy-Littlewood integral (1918)
Letusremindthatwehaveintroduced(see[4],(9.1),(9.2))thefollowingformula
dϕ (t)
(4.1) Z˜2(t)= 1 ,
dt
where
Z2(t) ζ 1 +it 2
Z˜2(t)= = 2 ,
(4.2) 2Φ′ϕ[ϕ(t)] (cid:12)(cid:12) (cid:0)ω(t) (cid:1)(cid:12)(cid:12)
lnlnt
ω(t)= 1+ lnt.
O lnt
The function ß Å ã™
ϕ (t)
1
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Jan Moser Jacob’s ladder ...
is called the Jacob’s ladder (see our paper [3]) according to Jacob’s dream in Chu-
mash, Bereishis, 28:12,has the following properties:
(a)
1
ϕ (t)= ϕ(t),
1
2
(b) the function ϕ(t) is a solution of the nonlinear integral equation (see [3],
[4])
µ[x(T)] T
Z2(t)e−x(2T)tdt= Z2(t)dt,
Z Z
0 0
where each admissible function µ(y) generates the solution
y =ϕ (T)=ϕ(T), µ(T) 7ylny.
µ
≥
Remark 7. The main goal of introducing the Jacob’s ladders is described in [3],
where we have shown, by making use of these Jacob’s ladders, that the Hardy-
Littlewood integral (1918)
T 1 2
ζ +it dt
Z (cid:12) 2 (cid:12)
0 (cid:12) (cid:12)
(see [1], pp. 122, 151 – 156) has - i(cid:12)(cid:12)n aÅdditionãt(cid:12)(cid:12)o the Hardy-Littlewood expression
(and alsoother similar to this one) possessingan unbounded erroratT - the
→∞
following infinite set of almost exact expressions
T 1 2
ζ +it dt=ϕ (T)lnϕ (T)+(c ln2π)ϕ (T)+
Z (cid:12) 2 (cid:12) 1 1 − 1
0 (cid:12) (cid:12)
(cid:12)(cid:12) Å lnTã(cid:12)(cid:12)
+c + , T ,
0
O T →∞
Å ã
where c is the constant from the Titchmarsh-Kober-Atkinson formula (see [7], p.
0
114).
Remark 8. The Jacob’s ladder ϕ (T) can be interpreted by our formula (see [3],
1
(6.2))
T ϕ (T) (1 c)π(T),
1
− ∼ −
where π(T) is the prime-counting function, as an asymptotic complementary func-
tion to the function
(1 c)π(T)
−
in the following sense
ϕ (T)+(1 c)π(T) T, T .
1
− ∼ →∞
5. Proof of Theorem
5.1. First of all we obtain from the formula
T+U0 t+H 2
Z(u)du dt 2πHU
Z Z ∼ 0
T t
® ´
(see (3.7), (3.8)), by mean-value theorem, that
η+H 2
Z(u)du 2πH,
Z ∼
η
® ´ Page 6 of 9
Jan Moser Jacob’s ladder ...
i.e.
η+H
(5.1) (cid:12)(cid:12)(cid:12)Zη Z(u)du(cid:12)(cid:12)(cid:12)∼√2πH, η =η(T,H)∈(T,T +U0).
(cid:12) (cid:12)
Since (see (3.8)(cid:12)) (cid:12)
T
(5.2) H =o ,
lnT
Å ã
then by our lemma (comp. [5], (7.1), (7.2))
k
η+H k−1
(cid:12) (cid:12)
(cid:12) Z[ϕk(t)] Z˜2[ϕr(t)]dt(cid:12) √2πH
(cid:12)Zk 1 1 (cid:12)∼
(cid:12) η rY=0 (cid:12)
(cid:12) (cid:12)
(cid:12) ¯ (cid:12)
(cid:12) (cid:12)
and, by the mean-va(cid:12)lue theorem, we obtain that (cid:12)
√H k−1
Z[ϕk(β)] √2π Z˜−2[ϕr(β)],
| 1 |∼ k 1
Y
(5.3) η+H ηk r=0
−
k
k
β =β(T,H,k) (η,η+H).
∈ ˙
5.2. Now, we make a transformation of the formula (5.3). First of all, we have
˙
(comp. [5], Property 2, (6.4))
k k−r
β (ηk,η+H) ϕr(β) (k−ηr,η+H), r =0,1,...,k,
∈ ⇒ 1 ∈
i.e.
˙ ˙
k
ϕ0(β)=α (ηk,η+H),
1 k ∈
k−1
ϕ1(β)=α (k−η1,η+H),
1 k−1 ∈ ˙
.
.
.
(5.4) ˙
2
ϕk−2(β)=α (η2,η+H),
1 2 ∈
1
ϕk−1(β)=α (η1,η+H),
1 1 ∈ ˙
ϕk(β)=α (η,η+H).
1 0 ∈
˙
Consequently, from (5.3) by (5.4), (4.2) the formula
1 √H k ω(α )
ζ +iα √2π l ,
(5.5) (cid:12)(cid:12)(cid:12)(cid:12) Å2 0ã(cid:12)(cid:12)(cid:12)(cid:12)∼ Hk Yl=1(cid:12)(cid:12)ζ(cid:0)12 +iαl(cid:1)(cid:12)(cid:12)2 k
k
α =α (T,H,k), r =0,1,...,k, H (T,H)=η+H η
r r k
−
follows.
˙ Page 7 of 9
Jan Moser Jacob’s ladder ...
5.3. Next, let us remind the following properties of the disconnected set
k r
r
∆(T,H,k)= [η,η+H].
[
r=0
(see (5.2), comp. [5], (2.5) – (2.7), (2.9)) . If
˙
T
H =o ,
lnT
then Å ã
r r
r r T
[η,η+H] =η+H η =o , r =1,...,k,
| | − lnT
r−1 Å ã
(5.6) r
[η+˙H,η] (˙1 c)π(T),
| |∼ −
1 k
1 k
[η,η+H] [η,η+H] [η,η+H].
˙
≺ ≺···≺
Hence, from (5.4), (5.6) the properties (1.8), (1.9) follow immediately. Further, we
have (comp. [5], (4.3)) ˙ ˙
k
lnt lnη, t (η,η+H),
∼ ∀ ∈
and (see (3.8), (5.1))
˙k
(5.7) lnt lnη lnT, t (η,η+H).
∼ ∼ ∀ ∈
Consequently, we obtain from (5.5) by (4.2), (5.7) the following
1 √H k ˙1 −2
ζ +iα √2π lnkT ζ +iα ,
(cid:12) 2 0 (cid:12)∼ H (cid:12) 2 r (cid:12)
(cid:12) (cid:12) k rY=1(cid:12) (cid:12)
(cid:12) Å ã(cid:12) (cid:12) Å ã(cid:12)
i.e. the formula(cid:12)(1.7) holds t(cid:12)rue. (cid:12) (cid:12)
6. Concluding remarks
If we use the formulae (comp. [7], pp. 221, 329)
t t t π 1
ϑ(t)= ln + ,
2 2π − 2 − 8 O t
1 t 1 Å ã
ϑ′(t)= ln + ,
2 2π O t
1 Å ã
ϑ′′(t) ,
∼ 2t
t
→∞
then we obtain from (2.1), by a little of transformations, the local kind of the
spectral representation
1 τ(x ) x π
r r
Z(x )=2 cos tln +
r
√n n − 2 − 8
X
n≤τ(xr)
ß ™
(6.1) + (x−1/4),
O r
lnlnT A xr
t [xr,xr +H],H ,TlnlnT , τ(xr)= ,
∈ ∈ lnT 2π
Å ã … Page 8 of 9
Jan Moser Jacob’s ladder ...
i.e. the Riemann-Siegel formula (2.1), t=x .
r
Remark 9. Namely, the sequence
τ(x )
r
(6.2) ω , ω =ln , r =1,...,k
{ n,r}n≤τ(xr) n,r n
of the cyclic frequencies ω will be called as the local spectrum of the Riemann-
n,r
Siegel formula (2.1).
Remark 10. Consequently, we have (see (2.4), (6.1), (6.2)) that the multiform
G(x ,x ,...,x )
1 2 k
expresses the complicated oscillating process.Just for this oscillating multiform we
have constructed the set of metamorphosis described by the formula (2.5).
I would like to thank Michal Demetrian for his help with electronic version of
this paper.
References
[1] G.H.Hardy, J.E.Littlewood, ‘Contributiontothe theoryofthe Riemannzeta-function and
thetheoryofdistributionofPrimes‘,ActaMath.41(1918), 119-196.
[2] G.H.Hardy,J.E.Littlewood,‘ZerosoftheRiemannzeta-functiononthecriticalline‘,Math.
Zs.,10(1921), 283-317.ActaMath.41(1918), 119-196.
[3] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic represention of the Hardy-
Littlewoodintegral‘,Math.Notes88,414-422, (2010), arXiv: 0901.3937.
[4] J.Moser,‘Jacob’sladders,thestructureoftheHardy-Littlewoodintegralandsomenewclass
ofnonlinearintegralequations‘,Proc.Stek.Inst. 276,(2011), 208-221,arXiv: 1103.0359.
[5] J. Moser, ‘Jacob’s ladders, reverse iterations and new infinite set of L2-orthogonal systems
generatedbytheRiemannζ 1 +it -function‘,arXiv: 1402.2098.
2
[6] C.L.Siegel, ‘U¨berRiemann’s(cid:0)Nachla(cid:1)ss zeranalytischen Zahlentheorie: QuellenundStudien
zurGeschichtederMathematik, AstronomieundPhysik‘,Abt.B.: Studien,2(1932), 45-80.
[7] 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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