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Discrete crop circles drawn by Riemann's zeta function and its further remote properties

Authors:
Yuri Matiyasevich at St. Petersburg Department of Steklov Institute of Mathematics
Yuri Matiyasevich
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Abstract

Abstract: We consider a particular way to calculate approximations to the alternating zeta function via the determinants of certain matrices, and numerically examine characteristic polynomials, eigenvalues and eigenvectors of these matrices. It turns out that the ratios of entries of such eigenvectors lay inside very thin annuluses.
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ζ(s) =
∞
X
n=1
n−s.
Re(s)>1
s= 1
Re(s) = 1/2
η(s) =
∞
X
n=1
(−1)n+1n−s= (1 −2×2−s)ζ(s).
Re(s)>0
η(s)
ηN(s) =
N
X
n=1
aN,nn−s.
ηN(s)η(s)
ξ(1 −s)−ξ(s)=0
ξ(s) = h(s)η(s),
h(s) = π−s
2(s−1)Γ(1 + s
2)
1−2×2−s.
t
Im (ξ(1/2+it)) = 0.
τ ξ(1/2+it)t=τ
ξ(1/2+it) =
∞
X
k=0
ξk(τ)(τ−t)k
ξk(τ) = (−1)k
k!
dk
dtkξ(1/2+it)t=τ
.
Im (ξk(τ)) = 0
k
η(s)τ
ηN(τ, s) =
N
X
n=1
aN,n(τ)n−s.
ξN(τ, s) = h(s)ηN(τ , s),
ξN(τ, 1/2+it) =
∞
X
k=0
ξN,k (τ)(τ−t)k
ξN,k (τ) = (−1)k
k!
dk
dtkξN(τ, 1/2+it)t=τ
.
aN,1(τ)=1
Im (ξN,k (τ)) = 0.
aN,n(τ)N= 40 τ= 15
N τ
±1
n aN,n
±1
η(s)
3
τ s
η(s) = lim
N→∞ ηN(τ, s).
ηN(τ, s)
ηN(τ, s)
η(s)s1/2+ iτ
1/2+iτ
ηN(τ, s)
τ
bN,n(τ) = n−iτaN,n(τ).
ηN(τ, s) =
N
X
n=1
bN,n(τ)n−(s−τ).
bN,1(τ)=1,
bN,1(τ) = bR
N,1(τ), bN,n(τ) = bR
N,n(τ)+ibI
N,n(τ), n > 1,
bR
N,1(τ), . . . , bR
N,N (τ), bI
N,2(τ), . . . , bI
N,N (τ).
ImξN(τ, s)=
N
X
n=1
Imh(s)n−(s−iτ)bR
N,n(τ)+
N
X
n=2
Reh(s)n−(s−iτ)bI
N,n(τ).
BN(τ) = [bI
N,N (τ), . . . , bI
N,2(τ), bR
N,1(τ), . . . , bR
N,N (τ)]T.
MN(τ)BN(τ) = VN
VN(2N−1) ×(2N−1) MN(τ)
MN(τ)
−N+ 1 N−1VN
−1 2N−3
N
MN(τ) = µk,l(τ)2N−3
k=−1
N−1
l=−(N−1)
,
VN= [υ−1, . . . , υ2N−3]T
µk,l(τ)υk
−1MN(τ)
µ−1,l(τ) = (1, l = 0,
0, ,
υ−1= 1.
MN(τ)
k= 0,...,2N−3
µk,l(τ) =
(−1)k
k!Re dk
dtkh(1/2+it)(1 −l)−(1/2+i(t−τ)) t=τ!, l < 0,
(−1)k
k!Im dk
dtkh(1/2+it)(1 + l)−(1/2+i(t−τ)) t=τ!, l ≥0,
υk= 0.
MN(τ)
MN,l(τ)
MN(τ)
l
(−1)ldet MN,l(τ)
det MN,0(τ)=(bI
N,l, l < 0,
bR
N,l, l ≥0.
2N
ηN(τ, s)
MN(τ, s) = µk,l(τ , s)2N−3
k=−1
N−1
l=−(N−1)
MN(τ)
µk,l(τ, s) =
i(1 −l)−(s−τ), k =−1, l < 0,
(1 + l)−(s−τ), k =−1, l ≥0,
µk,l(τ), k ≥0
MN(τ)MN(τ, +∞)
ηN(τ, s) = det MN(τ, s)
det MN(τ).
τ s
η(s) = lim
N→∞
det MN(τ, s)
det MN(τ).
µk,l(τ)k≥0
l < 0
µk,l(τ) = (1 −l)−1
2
(−1)kk!
k
X
m=0 k
m(−ln(1 −l))k−mRe ik−mhhmi(1/2+iτ);
l≥0
µk,l(τ) = (1 + l)−1
2
(−1)kk!
k
X
m=0 k
m(−ln(1 + l))k−mIm ik−mhhmi(1/2+iτ)
hhmi(1/2+iτ) = dm
dtmh(1/2+it)t=τ
.
n−it
MN(τ, s)
pN,0(τ, s)qN,0(τ)
PN(τ, s, λ) = det MN(τ, s)−λE2N−1=
2N−1
X
l=0
pN,l(τ, s)λl
QN(τ, λ) = det MN(τ)−λE2N−1=
2N−1
X
l=0
qN,l(τ)λl
E2N−1(2N−1) ×(2N−1)
η(s) = lim
N→∞
pN,0(τ, s)
qN,0(τ).
pN,l(τ, s)
qN,l(τ)
η(s)l s 1/2 + iτ
l τ
s
η(s) = lim
N→∞
pN,l(τ, s)
qN,l(τ).
ξ(s)s= 1/2
ξ(1 −σ−it)−ξ(σ+ it)=0
σ τ
ξN(τ, 1−σ−it)−ξN(τ , σ + it)≈0.
ξN(τ, s)ξ(s)
s1/2 + iτ
ξ(1 −σ−it) = ξ(1 −σ+ it).
Re(s) = 1/2
ξ(1 −σ+ it)−ξ(σ+ it)=0.
ξN(τ, 1−σ+ it)−ξN(τ , σ + it)≈0.
f(1 −σ+ it)−f(σ+ it)≈0,
f(1 −σ+ it)f(σ+ it)
f(1 −σ+ it)
f(σ+ it)
1
η(s)
h(1 −σ+ it)η(1 −σ+ it)
h(σ+ it)η(σ+ it)= 1.
N l
h(1 −σ+ it)pN,l(τ, 1−σ+ it)
h(σ+ it)pN,l(τ, σ + it)≈1;
λN,1(τ, s), . . . , λN,m (τ, s),...λN,2N−1(τ, s)
MN(τ, s)
|λN,1(τ, s)| ≤ · · · ≤ |λN,m(τ, s)| ≤ · · · ≤ |λN,2N−1(τ, s)|.
ηN(τ, s) = det MN(τ, s)
det MN(τ)=Q2N−1
m=1 λN,m(τ, s)
det MN(τ).
λN,1(τ, s)
η(s)
ηN(τ, s)s0ηN(τ , s)
λN,1(τ, s0)=0
h(1 −σ+ it)Q2N−1
m=1 λN,m(τ, 1−σ+ it)
h(σ+ it)Q2N−1
m=1 λN,m(τ, σ + it)=
h(1 −σ+ it)ηN(τ, 1−σ+ it)
h(σ+ it)ηN(τ, σ + it)≈1.
λN,m(τ, σ + it)
PN,N0(τ, σ + it, λ) =
N0
X
l=0
pN,l(τ, σ + it)λl.
h(σ+ it)PN,N0(τ, σ + it, λ).
λN,m(τ, 1−σ+ it)
h(1 −σ+ it)PN,N0(τ, 1−σ+ it, λ).
λN,m(τ, σ + it)λN,m(τ, 1−σ+ it)
λN,m(τ, 1−σ+ it)
λN,m(τ, σ + it)≈1,
σ+ it1−σ+ it h(s)σ= 1/2
λN,m(τ, 1/2+it)
m τ σ t σ + it
1−σ+ it h(s)
λN,m(τ, 1−σ+ it)
λN,m(τ, σ + it)−→ 1
N→ ∞
WN,m(τ, s)=[ωN ,m,−N+1(τ , s), . . . , ωN,m,0(τ, s), . . . , ωN,m,N−1(τ , s)]T
λN,m
MN(τ, s)WN ,m(τ , s) = λN,m(τ, s)WN,m(τ, s).
WN,m(τ, s)
1
ωN,m,0(τ, s) = 1.
LN(τ, λN ,m(τ , s))
LN(τ, λ) = MN(τ)−λE2N−1.
LN,k (τ, λ)k
ωN,m,k (τ, s) = (−1)kdet(LN,k (τ, λN,m(τ, s)))
det(LN,0(τ, λN,m (τ, s))).
λN,1(τ, s0) = 0
s0LN(τ, λN ,0(τ, s0)) MN(τ)
WN,1(τ, s0)
s
WN,1(τ, s)
MN(τ)
LN,k (τ, λN,m(τ, 1−σ+ it)LN ,k(τ, λN,m(τ, σ +it)
WN,m(τ, 1−σ+ it)
WN,m(τ, σ + it)
ωN,m,k (τ, 1−σ+ it)≈ωN,m,k (τ, σ + it).
ωN,m,k (τ, 1−σ+ it)
ωN,m,k (τ, σ + it)
1
ωN,m,k (τ, 1−σ+ it)
ωN,m,k (τ, σ + it)
|z|= 1
ρN,m,k (τ, s0, s00) = ωN,m,k(τ, s0)
ωN,m,k (τ, s00), k =−N+ 1, . . . , N −1.
Re(s0) + Re(s00) = 1,Im(s0) = Im(s00 ).
s0s00
1/2+iτ
1
N= 50 m= 1 τ= 14
s0= 1/3 + 15i s00 = 6/7 + 14i
C1= 0.3708842799394989375085843404586974710326419033382 +
1.0618744177199283671338046020549367658223350300371i.
ρN,m,k (τ, s0, s00)
R1<|ρN,m,k (τ, s0, s00)−C1|< R1+ 10−47
R1=1.2342462753580744181860242203074444592191737722398.
2N−1
˜ρN,m,k (τ, s0, s00) = ωN,m,k (τ, s0)
ωN,m,k (τ, s00), k =−N+ 1, . . . , N −1.
R2<|˜ρN,m,k (τ, s0, s00)−C2|< R2+ 10−47
C2= 1 −C1
R2=1.1247810578571624993362791499949324350181486428418.
Re(C1) + Re(C2) = 1,Im(C1) = Im(C2)
C2
a
1
ϕ
¨ρN,m,k (τ, s0, s00) = ρN,m,k (τ, s0, s00)−1
˜ρN,m,k (τ, s0, s00)−1, k =−N+1,...,−1,1, . . . , N −1
aeiϕ
k= 0 ρN,m,0(τ, s0, s00) =
˜ρN,m,0(τ, s0, s00)=1
¨ρN,m(τ, s0, s00 ) = ¨ρN,m,2(τ, s0, s00).
¨ρN,m(τ, s0, s00 ) = 1.075702732456057626198949106... +
0.216743602505064201542441468...i
k6= 0
|¨ρN,m,k (τ, s0, s00)−¨ρN,m(τ, s0, s00)|<10−49.
m= 2,3,
4
Re(¨ρN,m,k (τ, s0, s00)) + i Im(¨ρN,m,k (τ, s0, s00))
=ρN,m,k (τ, s0, s00)−1
˜ρN,m,k (τ, s0, s00)−1
=
ωN,m,k (τ,s0)
ωN,m,k (τ,s00 )−1
ωN,m,k (τ,s0)
ωN,m,k (τ,s00 )−1
=ωN,m,k (τ, s0)−ωN,m,k (τ, s00)
ωN,m,k (τ, s0)−ωN,m,k (τ, s00)
=Re(ωN,m,k (τ, s0)) + i Im(ωN,m,k(τ, s0)) −Re(ωN,m,k (τ, s00)) −i Im(ωN,m,k(τ, s00))
Re(ωN,m,k (τ, s0)) −i Im(ωN,m,k (τ, s0)) −Re(ωN,m,k (τ, s00)) −i Im(ωN,m,k(τ, s00 )).
Re(ωN,m,k (τ, s0)),Im(ωN,m,k (τ, s0)),Re(ωN,m,k (τ, s00)),Im(ωN,m,k(τ, s00 )),
|¨ρN,m,k (τ, s0, s00)|= 1
Re(ωN,m,k (τ, s0)) = Re(ωN,m,k (τ, s00)) +
2Im(¨ρN,m,k (τ, s0, s00))
1− | ¨ρN,m,k(τ, s0, s00 )|2Im(ωN,m,k (τ, s00)),
Im(ωN,m,k (τ, s0)) = |1−¨ρN,m,k (τ, s0, s00)|2
1− | ¨ρN,m,k(τ, s0, s00 )|2Im(ωN,m,k (τ, s00)).
¨ρN,m,k (τ, s0, s00) ¨ρN,m(τ, s0, s00)
αN,m(τ, s0, s00 )) = 2Im(¨ρN,m (τ, s0, s00))
1− | ¨ρN,m,k(τ, s0, s00 )|2,
βN,m(τ, s0, s00 )) = |1−¨ρN,m (τ, s0, s00)|2
1− | ¨ρN,m,k(τ, s0, s00 )|2.
Re Im
Re(WN,m(τ, s0)) ≈Re(WN ,m(τ , s00)) + αN,m(τ , s0, s00)Im(WN,m(τ, s00 )),
Im(WN,m(τ, s0)) ≈βN ,m(τ , s0, s00)Im(WN,m(τ , s00)).
s
1/2+iτ
αN,m(τ, s)βN ,m(τ)
WN,m(τ, s)≈WR
N,m(τ)+(αN ,m(τ , s)+iβN,m(τ, s))WI
N,m(τ)
WR
N,m(τ)WI
N,m(τ)
WR
N,m(τ)=[ωR
N,m,−N+1(τ, s),...,ωR
N,m,0(τ, s),...,ωR
N,m,N −1(τ, s)]T,
WI
N,m(τ)=[ωI
N,m,−N+1(τ, s),...,ωR
N,m,0(τ, s),...,ωI
N,m,N −1(τ, s)]T.
˜s
WR
N,m(τ) = Re(WN,m(τ, ˜s)),
WI
N,m(τ) = Im(WN,m(τ, ˜s)).
αN,m(τ, s) = αN ,m(τ , s, ˜s),
βN,m(τ, s) = βN ,m(τ , s, ˜s).
m= 1
WR
N,1(τ)WI
N,1(τ)
WR
N,1(τ)WN,1(τ, s0)s0ηN(τ, s)
s0
λN,1(τ, s0)=0 WN,1(τ, s0)
LN(τ, 0) WR
N,1(τ) =
WN,1(τ, s0)
WI
N,1(τ)WN,1(τ, +∞)
N= 40 τ= 14
WI
N,1(τ)
ωI
N,1,k (τ) = ωN,1,k (τ, +∞)
PN−1
n=−N+1 ωN,1,n(τ, +∞)21/2.
αN,1(τ, s)βN,1(τ, s)
α β
N+1
X
k=−N+1 Re(ωN,1,k (τ, s)) −ωR
N,1,k (τ)−αωI
N,1,k (τ)2
N+1
X
k=−N+1 Im(ωN,1,k (τ, s)) −βωI
N,1,k (τ)2
.
s0ηN(τ, s)αN ,1(τ, s0) = βN,1(τ, s0)=0
s= 1/3 + 15i
α=−4.439188393651894483835125620... ·1023,
β=−6.256317129726259611895862483... ·1022,
ωN,1,k (τ, s)
ωR
N,1,k (τ)+(α+ iβ)ωI
N,1,k (τ)−1
<1.288 ·10−25
k=−N+ 1, . . . , N −1
WN,1(τ, s)
s
WN,1(τ, s0)
λN,1(τ, s0) = 0
WN,1(τ, s)
WR
N,1(τ)WN,1(τ, s0)
WR
N,1(τ)WN,1(τ, s)s
WR
N,1(τ)≈Re(WN,1(τ, s)) + γIm(WN,1(τ, s))
γ
−1 = Re(ωN,1,1(τ, s)) + γIm(ωN,1,1(τ , s)).
1−2×2−s
1−n×n−s
ξN(τ, 1/2+it) =
∞
X
k=0
ξN,k (τ)(t−τ)k,
(−1)k
ηN(s) =
N
X
n=1
(−1)n+1aN,nn−s.
b
a b
bI
N,1(τ)MN(τ)
bI
N,1(τ) = 0
a
b k k!
a b
b
BN(τ) = [bR
N,1(τ), bI
N,2(τ), . . . , bI
N,k (τ), bR
N,k (τ), . . . , bI
N,N (τ)bR
N,N (τ)]T.
MN(τ)
ξ(s) = f(s)ζ(s)
f(s) = π−s
2(s−1)Γ(1 + s
2),
N= 50 m= 1 τ= 14 s0= 1/3 + 15i
s00 = 6/7 + 15i
0.737520 <|z−0.265863 + 0.070807i|<0.737544
0.275111 <|z−0.734137 + 0.070807i|<0.275150.
ζ
ζ
z
-0.5 0.5 1.0 1.5
0.5
1.0
1.5
2.0
N= 50 m= 1 τ= 14 s0= 1/3 + 15i
s00 = 6/7 + 14i
z
-0.5 0.5 1.0 1.5
0.5
1.0
1.5
2.0
N= 50 m= 1 τ= 14
s0= 1/3 + 15i s00 = 6/7 + 14i
z
-0.5 0.5 1.0 1.5
0.5
1.0
1.5
2.0
N= 50 m= 2 τ= 14
s0= 1/3 + 15i s00 = 6/7 + 14i
z
-0.5 0.5 1.0 1.5
0.5
1.0
1.5
2.0
N= 50 m= 3 τ= 14
s0= 1/3 + 15i s00 = 6/7 + 14i
z
-0.5 0.5 1.0 1.5
0.5
1.0
1.5
2.0
N= 50 m= 4 τ= 14
s0= 1/3 + 15i s00 = 6/7 + 14i
n aN,n
1 1
2−1+2.2181 . . . ·10−18 + 2.1692 . . . ·10−18i
3 1 −7.4732 . . . ·10−17 + 3.4719 . . . ·10−17i
4−1−2.3549 . . . ·10−15 −8.5455 . . . ·10−16i
5 1 −6.2490 . . . ·10−14 −2.2430 . . . ·10−14i
6−1−1.4053 . . . ·10−12 + 2.7476 . . . ·10−13i
7 1 −1.0939 . . . ·10−11 + 2.2234 . . . ·10−11i
8−1+2.3595 . . . ·10−10 + 2.5368 . . . ·10−10i
9 1 + 3.2566 . . . ·10−9−2.2589 . . . ·10−9i
10 −1−2.2472 . . . ·10−8−3.0146 . . . ·10−8i
11 1 −2.0449 . . . ·10−7+ 2.1883 . . . ·10−7i
12 −1+1.8217 . . . ·10−6+ 8.8525 . . . ·10−7i
13 1 + 5.9806 . . . ·10−7−1.1725 . . . ·10−5i
14 −1−5.3270 . . . ·10−5+ 2.4969 . . . ·10−5i
15 1 + 2.1656 . . . ·10−4+ 1.3776 . . . ·10−4i
16 −1−6.5702 . . . ·10−5−9.7882 . . . ·10−4i
17 1 −2.3916 . . . ·10−3+ 2.2778 . . . ·10−3i
18 −1+9.8214 . . . ·10−3+ 6.0047 . . . ·10−4i
19 1 −1.7518 . . . ·10−2−1.9273 . . . ·10−2i
20 −1+7.5775 . . . ·10−4+ 6.1461 . . . ·10−2i
21 1 + 8.0453 . . . ·10−2−1.0175 . . . ·10−1i
22 −1−2.3703 . . . ·10−1+ 6.3810 . . . ·10−2i
23 1 + 3.9449 . . . ·10−1+ 1.3681 . . . ·10−1i
24 −1−4.0384 . . . ·10−1−4.9640 . . . ·10−1i
25 1 + 1.5331 . . . ·10−1+ 8.7256 . . . ·10−1i
26 −1+3.1348 . . . ·10−1−1.0663 . . . ·100i
27 1.9177 . . . ·10−1+ 9.7936 . . . ·10−1i
28 1.4072 . . . ·10−1−6.8703 . . . ·10−1i
29 −2.4974 . . . ·10−1+ 3.5976 . . . ·10−1i
30 2.0534 . . . ·10−1−1.2730 . . . ·10−1i
31 −1.1565 . . . ·10−1+ 1.6925 . . . ·10−2i
32 4.7456 . . . ·10−2+ 1.2747 . . . ·10−2i
33 −1.3946 . . . ·10−2−1.0968 . . . ·10−2i
34 2.6239 . . . ·10−3+ 4.7430 . . . ·10−3i
35 −1.5360 . . . ·10−4−1.3648 . . . ·10−3i
36 −7.6192 . . . ·10−5+ 2.7027 . . . ·10−4i
37 2.7103 . . . ·10−5−3.5293 . . . ·10−5i
38 −4.3908 . . . ·10−6+ 2.6483 . . . ·10−6i
39 3.7618 . . . ·10−7−6.7133 . . . ·10−8i
40 −1.3591 . . . ·10−8−2.5085 . . . ·10−9i
aN,n(τ)N= 40 τ= 15
ηN(τ,s)
η(s)−1
s τ = 15 τ= 20 τ= 25
−1 + 50i 8.481 . . . ·10−46 6.379 . . . ·10−49 4.947 . . . ·10−50
−1 + 30i 1.821 . . . ·10−56 6.835 . . . ·10−56 1.460 . . . ·10−54
−1 + 20i 5.988 . . . ·10−59 1.657 . . . ·10−57 1.641 . . . ·10−55
−1−10i 2.083 . . . ·10−56 1.023 . . . ·10−50 1.169 . . . ·10−43
−1−30i 1.105 . . . ·10−35 2.325 . . . ·10−31 1.878 . . . ·10−27
50i 3.395 . . . ·10−45 2.880 . . . ·10−48 2.194 . . . ·10−49
30i 5.563 . . . ·10−56 2.070 . . . ·10−55 4.372 . . . ·10−54
20i 1.086 . . . ·10−58 2.987 . . . ·10−57 2.950 . . . ·10−55
−10i 2.271 . . . ·10−56 1.046 . . . ·10−50 5.504 . . . ·10−44
−30i 6.743 . . . ·10−36 1.202 . . . ·10−31 8.521 . . . ·10−28
1/2 + 50i 7.979 . . . ·10−45 7.488 . . . ·10−48 5.694 . . . ·10−49
1/2 + 30i 8.882 . . . ·10−56 3.302 . . . ·10−55 6.963 . . . ·10−54
1/2 + 20i 1.229 . . . ·10−58 3.378 . . . ·10−57 3.334 . . . ·10−55
1/2−10i 2.292 . . . ·10−56 1.068 . . . ·10−50 3.783 . . . ·10−44
1/2−30i 4.745 . . . ·10−36 7.755 . . . ·10−32 5.133 . . . ·10−28
1 + 50i 2.707 . . . ·10−45 2.880 . . . ·10−48 2.194 . . . ·10−49
1 + 30i 5.563 . . . ·10−56 2.070 . . . ·10−55 4.372 . . . ·10−54
1 + 20i 1.086 . . . ·10−58 2.987 . . . ·10−57 2.950 . . . ·10−55
1−10i 2.271 . . . ·10−56 9.879 . . . ·10−51 2.251 . . . ·10−44
1−30i 1.284 . . . ·10−36 1.918 . . . ·10−32 1.183 . . . ·10−28
η(s)ηN(τ, s)N= 50 τ
s
pN,m (τ,s)
qN,m (τ)η(s)−1
m s = 1/7 + 14i s= 1/2 + 14i s= 2/3 + 14i
0 8.682 . . . ·10−24 2.473 . . . ·10−23 9.612 . . . ·10−24
1 4.874 . . . ·10−23 1.388 . . . ·10−22 5.401 . . . ·10−23
2 1.146 . . . ·10−23 2.998 . . . ·10−23 1.469 . . . ·10−23
3 6.415 . . . ·10−22 1.826 . . . ·10−21 7.041 . . . ·10−22
4 8.350 . . . ·10−22 −2.378 . . . ·10−21 −9.383 . . . ·10−22
5 8.720 . . . ·10−20 −2.483 . . . ·10−19 −9.729 . . . ·10−20
6 6.652 . . . ·10−18 −1.894 . . . ·10−17 −7.455 . . . ·10−18
7 2.868 . . . ·10−17 8.169 . . . ·10−17 3.216 . . . ·10−17
8 1.172 . . . ·10−16 3.339 . . . ·10−16 1.314 . . . ·10−16
9 2.703 . . . ·10−16 −7.697 . . . ·10−16 −3.001 . . . ·10−16
10 3.458 . . . ·10−15 9.848 . . . ·10−15 3.861 . . . ·10−15
11 8.516 . . . ·10−14 2.425 . . . ·10−13 9.510 . . . ·10−14
12 2.063 . . . ·10−12 5.876 . . . ·10−12 2.305 . . . ·10−12
13 1.078 . . . ·10−11 3.071 . . . ·10−11 1.205 . . . ·10−11
14 3.983 . . . ·10−11 1.134 . . . ·10−10 4.452 . . . ·10−11
15 2.261 . . . ·10−11 6.439 . . . ·10−11 2.519 . . . ·10−11
16 2.566 . . . ·10−10 7.315 . . . ·10−10 2.949 . . . ·10−10
17 1.940 . . . ·10−85.526 . . . ·10−82.165 . . . ·10−8
18 1.129 . . . ·10−73.217 . . . ·10−71.259 . . . ·10−7
19 2.996 . . . ·10−78.532 . . . ·10−73.333 . . . ·10−7
20 3.712 . . . ·10−7−1.056 . . . ·10−6−4.094 . . . ·10−7
21 2.637 . . . ·10−6−7.472 . . . ·10−6−2.802 . . . ·10−6
22 1.474 . . . ·10−5−4.187 . . . ·10−5−1.595 . . . ·10−5
23 2.554 . . . ·10−5−7.250 . . . ·10−5−2.752 . . . ·10−5
24 5.362 . . . ·10−51.528 . . . ·10−46.012 . . . ·10−5
25 1.083 . . . ·10−13.088 . . . ·10−11.243 . . . ·10−1
η(s)N= 50 τ= 18
s m
λN,m (τ,1−σ+it)
λN,m (τ,σ+it)−1
m τ = 14 τ= 16 τ= 18
1 1.04 . . . ·10−139 2.73 . . . ·10−108 2.66 . . . ·10−79
2 9.96 . . . ·10−137 1.13 . . . ·10−107 3.07 . . . ·10−77
3 1.14 . . . ·10−131 1.13 . . . ·10−104 5.78 . . . ·10−77
4 7.31 . . . ·10−128 2.47 . . . ·10−103 1.03 . . . ·10−75
5 5.74 . . . ·10−124 6.58 . . . ·10−102 3.52 . . . ·10−73
6 9.30 . . . ·10−122 3.53 . . . ·10−98 6.51 . . . ·10−71
7 1.39 . . . ·10−117 1.29 . . . ·10−97 9.06 . . . ·10−71
8 1.93 . . . ·10−113 1.30 . . . ·10−96 1.67 . . . ·10−68
9 1.14 . . . ·10−110 1.43 . . . ·10−95 5.67 . . . ·10−67
10 8.40 . . . ·10−106 2.49 . . . ·10−93 8.30 . . . ·10−66
11 5.23 . . . ·10−104 4.58 . . . ·10−91 2.82 . . . ·10−64
12 7.31 . . . ·10−101 6.53 . . . ·10−90 1.40 . . . ·10−61
13 2.81 . . . ·10−97 6.71 . . . ·10−87 9.62 . . . ·10−61
14 1.20 . . . ·10−92 3.13 . . . ·10−86 3.45 . . . ·10−58
15 3.99 . . . ·10−89 1.08 . . . ·10−81 1.55 . . . ·10−57
16 1.47 . . . ·10−88 2.28 . . . ·10−81 4.64 . . . ·10−55
17 4.31 . . . ·10−86 1.03 . . . ·10−78 4.83 . . . ·10−53
18 1.19 . . . ·10−80 1.82 . . . ·10−77 1.94 . . . ·10−52
19 9.88 . . . ·10−76 1.03 . . . ·10−75 4.37 . . . ·10−50
20 9.87 . . . ·10−76 4.11 . . . ·10−74 7.30 . . . ·10−49
21 2.60 . . . ·10−74 4.80 . . . ·10−72 7.83 . . . ·10−46
22 4.70 . . . ·10−70 1.45 . . . ·10−69 8.06 . . . ·10−46
23 1.08 . . . ·10−66 1.03 . . . ·10−66 8.45 . . . ·10−44
24 3.19 . . . ·10−63 4.50 . . . ·10−63 2.03 . . . ·10−42
25 1.39 . . . ·10−60 1.57 . . . ·10−61 1.48 . . . ·10−38
26 1.85 . . . ·10−58 9.40 . . . ·10−58 1.18 . . . ·10−38
27 1.28 . . . ·10−54 5.85 . . . ·10−56 1.19 . . . ·10−37
28 6.68 . . . ·10−52 4.57 . . . ·10−53 1.66 . . . ·10−35
29 1.69 . . . ·10−48 6.80 . . . ·10−52 5.49 . . . ·10−35
30 4.71 . . . ·10−46 2.19 . . . ·10−47 1.53 . . . ·10−31
N= 50 σ= 1/3t= 14
m τ
λN,m(τ, s)
m τ = 14 τ= 18
1−3.96 . . . ·10−134 −1.53 . . . ·10−272i 3.20 . . . ·10−137 −6.27 . . . ·10−216i
2 4.06 . . . ·10−132 −1.50 . . . ·10−267i−2.36 . . . ·10−133 + 5.32 . . . ·10−210i
3−1.87 . . . ·10−128 + 7.53 . . . ·10−259i 1.21 . . . ·10−131 + 5.13 . . . ·10−208i
4−3.17 . . . ·10−126 + 8.32 . . . ·10−253i 6.50 . . . ·10−128 −4.94 . . . ·10−203i
5 6.75 . . . ·10−123 + 1.34 . . . ·10−245i 4.45 . . . ·10−126 −1.15 . . . ·10−198i
6 5.49 . . . ·10−122 + 1.74 . . . ·10−242i 1.37 . . . ·10−122 −6.56 . . . ·10−193i
7 1.73 . . . ·10−118 + 8.37 . . . ·10−235i−1.49 . . . ·10−121 −9.90 . . . ·10−192i
8 2.97 . . . ·10−116 + 1.93 . . . ·10−228i−1.17 . . . ·10−118 −1.43 . . . ·10−186i
9−7.63 . . . ·10−114 −2.92 . . . ·10−223i−6.12 . . . ·10−116 + 2.54 . . . ·10−182i
10 −3.03 . . . ·10−111 −8.29 . . . ·10−216i 3.82 . . . ·10−114 + 2.33 . . . ·10−179i
11 5.30 . . . ·10−110 + 9.38 . . . ·10−213i 3.69 . . . ·10−112 −7.63 . . . ·10−176i
12 5.23 . . . ·10−108 + 1.24 . . . ·10−207i−2.84 . . . ·10−109 + 2.94 . . . ·10−170i
13 −1.94 . . . ·10−105 + 1.75 . . . ·10−201i−2.72 . . . ·10−108 + 1.91 . . . ·10−168i
14 −6.72 . . . ·10−103 −2.44 . . . ·10−194i−4.55 . . . ·10−105 + 1.15 . . . ·10−162i
15 −6.92 . . . ·10−101 + 8.41 . . . ·10−189i−4.86 . . . ·10−103 + 5.56 . . . ·10−160i
16 1.05 . . . ·10−99 −4.76 . . . ·10−187i 2.27 . . . ·10−101 −7.76 . . . ·10−156i
17 −1.20 . . . ·10−97 + 1.38 . . . ·10−182i−6.44 . . . ·10−99 + 2.29 . . . ·10−151i
18 −5.93 . . . ·10−95 + 2.04 . . . ·10−174i−1.58 . . . ·10−97 + 2.25 . . . ·10−149i
19 −8.03 . . . ·10−93 + 2.64 . . . ·10−92i 4.95 . . . ·10−95 + 1.59 . . . ·10−144i
20 −8.03 . . . ·10−93 −2.64 . . . ·10−92i 3.10 . . . ·10−93 −1.66 . . . ·10−141i
21 3.74 . . . ·10−90 + 3.23 . . . ·10−163i−2.17 . . . ·10−90 + 4.25 . . . ·10−90i
22 −4.91 . . . ·10−87 + 6.50 . . . ·10−156i−2.17 . . . ·10−90 −4.25 . . . ·10−90i
23 9.86 . . . ·10−86 −2.92 . . . ·10−151i−5.44 . . . ·10−87 −3.39 . . . ·10−130i
24 1.57 . . . ·10−83 −1.37 . . . ·10−145i 3.51 . . . ·10−85 −5.27 . . . ·10−127i
25 6.54 . . . ·10−82 + 2.42 . . . ·10−141i 8.04 . . . ·10−82 + 8.25 . . . ·10−120i
26 −4.02 . . . ·10−80 + 1.99 . . . ·10−137i−9.38 . . . ·10−82 −7.65 . . . ·10−120i
27 −9.45 . . . ·10−78 −3.13 . . . ·10−131i−6.29 . . . ·10−80 −6.13 . . . ·10−117i
28 2.34 . . . ·10−76 −3.95 . . . ·10−127i 1.27 . . . ·10−77 + 1.81 . . . ·10−112i
29 6.46 . . . ·10−74 −2.63 . . . ·10−121i 1.24 . . . ·10−75 −4.77 . . . ·10−110i
30 −3.48 . . . ·10−72 −7.33 . . . ·10−117i−7.35 . . . ·10−74 −5.04 . . . ·10−73i
λN,l(τ, s)N= 50 s= 1/2+14i
m τ
N τ t σ max
ωN,m,k (τ,1−σ+it)
ωN,m,k (τ,σ+it)−1
40 14 14 0 5.46246 . . . ·10−87
40 14 14 1/7 3.20371 . . . ·10−87
40 14 14 1/3 3.30046 . . . ·10−87
40 14 15 0 3.18719 . . . ·10−77
40 14 15 1/7 1.85486 . . . ·10−78
40 14 15 1/3 1.88463 . . . ·10−79
50 14 14 0 3.02829 . . . ·10−115
50 14 14 1/7 8.93416 . . . ·10−116
50 14 14 1/3 4.20312 . . . ·10−116
50 14 15 0 3.86955 . . . ·10−103
50 14 15 1/7 1.16495 . . . ·10−104
50 14 15 1/3 5.25409 . . . ·10−106
50 16 14 1/7 8.56135 . . . ·10−85
50 16 14 1/3 2.69217 . . . ·10−85
50 16 15 1/7 2.65126 . . . ·10−112
50 16 15 1/3 2.99890 . . . ·10−114
50 18 14 1/7 5.06519 . . . ·10−57
50 18 14 1/3 3.86475 . . . ·10−57
50 18 15 1/7 4.30181 . . . ·10−69
50 18 15 1/3 2.54209 . . . ·10−69
WN,m(τ, σ + it)
WN,m(τ, 1−σ+ it)m= 1 N τ σ t
n ωN,m,n(τ, +∞)
−39 4.2. . . ·10100
−38 −1.1. . . ·10102
−37 1.4. . . ·10103
−36 −1.1. . . ·10104
−35 6.8. . . ·10104
−34 −3.1. . . ·10105
−33 1.1. . . ·10106
−32 −3.4. . . ·10106
−31 8.8. . . ·10106
−30 −1.9. . . ·10107
−29 3.5. . . ·10107
−28 −5.7. . . ·10107
−27 8.1. . . ·10107
−26 −1.0. . . ·10108
−25 1.1. . . ·10108
−24 −1.0. . . ·10108
−23 9.3. . . ·10107
−22 −7.1. . . ·10107
−21 4.8. . . ·10107
−20 −2.9. . . ·10107
−19 1.5. . . ·10107
−18 −7.5. . . ·10106
−17 3.2. . . ·10106
−16 −1.1. . . ·10106
−15 3.9. . . ·10105
−14 −1.1. . . ·10105
−13 2.8. . . ·10104
n ωN,m,n(τ, +∞)
−12 −6.1. . . ·10103
−11 1.1. . . ·10103
−10 −1.8. . . ·10102
−9 2.4. . . ·10101
−8−2.7. . . ·10100
−7 2.5. . . ·1099
−6−1.9. . . ·1098
−5 1.1. . . ·1097
−4−5.3. . . ·1095
−3 1.8. . . ·1094
−2−4.1. . . ·1092
−1−1.3. . . ·1091
0 1
1−3.6. . . ·1091
2 9.4. . . ·1092
3−2.3. . . ·1094
4 5.4. . . ·1095
5−1.0. . . ·1097
6 1.6. . . ·1098
7−2.0. . . ·1099
8 2.2. . . ·10100
9−1.9. . . ·10101
10 1.4. . . ·10102
11 −8.8. . . ·10102
12 4.7. . . ·10103
13 −2.1. . . ·10104
14 8.8. . . ·10104
n ωN,m,n(τ, +∞)
15 −3.0. . . ·10105
16 9.4. . . ·10105
17 −2.5. . . ·10106
18 6.1. . . ·10106
19 −1.2. . . ·10107
20 2.4. . . ·10107
21 −4.0. . . ·10107
22 6.0. . . ·10107
23 −7.9. . . ·10107
24 9.2. . . ·10107
25 −9.6. . . ·10107
26 8.8. . . ·10107
27 −7.1. . . ·10107
28 5.0. . . ·10107
29 −3.1. . . ·10107
30 1.7. . . ·10107
31 −7.9. . . ·10106
32 3.1. . . ·10106
33 −1.0. . . ·10106
34 2.9. . . ·10105
35 −6.4. . . ·10104
36 1.0. . . ·10104
37 −1.3. . . ·10103
38 1.0. . . ·10102
39 −4.1. . . ·10100
WN,m(τ, +∞)N= 40 m= 1 τ= 14
Охота за нулями рядов Дирихле средствами линейной алгебры. III (Hunting zeros of Dirichlet series by linear algebra. III)
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Full-text available
  • Jun 2022
In the first part (\doi{10.13140/RG.2.2.29328.43528}) and in the second part (\doi{10.13140/RG.2.2.20434.22720}) the author presented numerical examples of calculation of approximate values of the zeros of the zeta-function, the alternating zeta function, and Davenport--Heilbronn function by by tools of linear algebra. This paper presents numerical examples of some non-evident ways of calculation of the values of the individual summands, n−s n^{-s} , from the Dirichlet series for the zeta-function when s is its trivial or non-trivial zero.
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