PreprintPDF Available

More crop circles drawn by Riemann's zeta function

Authors:
Yuri Matiyasevich at Steklov Institute of Mathematics at St.Petersburg
Yuri Matiyasevich
  • Steklov Institute of Mathematics at St.Petersburg
Preprints and early-stage research may not have been peer reviewed yet.
Download file PDF
Read file
Download citation
Content uploaded by Yuri Matiyasevich
Author content
All content in this area was uploaded by Yuri Matiyasevich on Oct 25, 2020
Content may be subject to copyright.
υM(α, τ, s)
υM(α, τ, s) =
M
X
m=1
bM,m(α, τ )m−s
α τ bM,m(α, τ )
α τ s
∞
X
M=1
υM(α, τ, s) = η(s) (∗)
η(s) =
∞
X
n=1
(−1)n+1n−s= (1 −2×2−s)ζ(s)
(∗)
h(1 −σ+ it)υM(α, τ, 1−σ+ it)
h(σ+ it)υM(α, τ, σ + it)→e2α
h(s) = π−s
2(s−1)Γ(1 + s
2)
1−2×2−s.
ΥM(α, τ, σ, t) = υM(α, τ, σ + it)
η(σ+ it)
M α τ σ
t σ
η(s) =
∞
X
n=1
(−1)n+1n−s= (1 −2×2−s)ζ(s).
ξ(1 −s) = ξ(s),
ξ(s) = h(s)η(s),
h(s) = π−s
2(s−1)Γ(1 + s
2)
1−2×2−s.
η(s)
ηN(α, τ, s) =
N
X
n=1
aN,n(α, τ )n−s
N
α τ
aN,n(α, τ ) = aR
N,n(α, τ )+iaI
N,n(α, τ ), n = 1, . . . , N,
aR
N,1(α, τ ), . . . , aR
N,N (α, τ ), aI
N,1(α, τ ), . . . , aI
N,N (α, τ )
aN,1(α, τ )=1,
aR
N,1(α, τ )=1, aI
α,N,1(τ) = 0.
ξN(α, τ, s) = h(s)ηN(α, τ, s),
k= 0,...,2N−3
Im eiαdk
dtkξN(α, τ, 1
2+ it)t=τ= Im eiαdk
dtkξ(1
2+ it)t=τ.
α= 0
Im eiαdk
dtkξN(0, τ, 1
2+ it)t=τ= 0.
ηN(0, τ, s)
ξ(s)
θ(t)
η(s)
α= 0
α τ s N → ∞
ηN(α, τ, s)→η(s).
υM(α, τ, s) = (η1(α, τ, s), M = 1,
ηM(α, τ, s)−ηM−1(α, τ, s), M > 1.
ηN(α, τ, s) =
N
X
M=1
υM(α, τ, s).
η(s) =
∞
X
M=1
υM(α, τ, s),
1 = ξ(1 −σ+ it)
ξ(σ+ it)
=h(1 −σ+ it)η(1 −σ+ it)
h(σ+ it)η(σ+ it)
=P∞
M=1 h(1 −σ+ it)υM(α, τ, 1−σ+ it)
P∞
M=1 h(σ+ it)υM(α, τ, σ + it).
α= 0
M→ ∞
h(1 −σ+ it)υM(0, τ, 1−σ+ it)
h(σ+ it)υM(0, τ, σ + it)→1.
α= 0
ΥM(α, τ, σ, t) = υM(α, τ , σ + it)
η(σ+ it)
M= 350 α= 0 τ= 30 σ= 1/3t6.53
101.76
t
α
ΥM(α, τ, σ, t)α0143
144 π1
144 π
ΥM(τ, σ, t) ΥM(0, τ , σ, t)
ηM(τ, s)
α658
1728 π783
1728 π
1
1728 π
ΥM(α, τ, σ, t)M α τ σ
ΥM(α, τ, σ, t)
M α τ t σ
M= 350 α= 0 σ0
1t
σ= 1/2
CII
N(τ, t) RII
N(τ, t)
ΥN(0, τ , σ, t)−CII
N(τ, t)
RII
N(τ, t)
σ[0,1]
α6= 0
h(1 −σ+ it)υM(α, τ, 1−σ+ it)
h(σ+ it)υM(α, τ, σ + it)→e2iα.
1
α
h(1−σ+it)υM(0,τ,1−σ+it)
h(σ+it)υM(0,τ,σ+it)−1
M σ =−2, t = 20 σ= 1/3, t = 30 σ=−1, t = 35
25 4.516 . . . ·10−51.143 . . . ·10−86 3.002 . . . ·10−18
50 1.091 . . . ·10−24 1.377 . . . ·10−196 1.330 . . . ·10−53
75 8.447 . . . ·10−50 1.602 . . . ·10−311 8.233 . . . ·10−94
100 1.512 . . . ·10−79 1.300 . . . ·10−431 1.715 . . . ·10−139
125 5.115 . . . ·10−116 3.154 . . . ·10−558 1.412 . . . ·10−191
150 6.115 . . . ·10−155 3.303 . . . ·10−687 5.330 . . . ·10−246
200 1.537 . . . ·10−241 1.837 . . . ·10−953 7.298 . . . ·10−363
250 3.126 . . . ·10−339 1.274 . . . ·10−1230 1.437 . . . ·10−490
300 1.452 . . . ·10−444 3.221 . . . ·10−1515 1.156 . . . ·10−625
350 3.818 . . . ·10−555 1.491 . . . ·10−1805 1.236 . . . ·10−766
τ= 30 M σ t
h(1−σ+it)υM(0,τ,1−σ+it)
h(σ+it)υM(0,τ,σ+it)−1
t σ = 1/5σ= 1/3σ= 2/5
0 9.691 . . . ·10−230 9.300 . . . ·10−230 9.191 . . . ·10−230
3 5.491 . . . ·10−261 5.319 . . . ·10−261 5.271 . . . ·10−261
6 2.195 . . . ·10−296 2.125 . . . ·10−296 2.106 . . . ·10−296
9 1.908 . . . ·10−335 1.348 . . . ·10−335 1.236 . . . ·10−335
12 3.707 . . . ·10−384 3.496 . . . ·10−384 3.438 . . . ·10−384
15 5.190 . . . ·10−440 4.752 . . . ·10−440 4.634 . . . ·10−440
18 9.545 . . . ·10−507 6.539 . . . ·10−507 5.898 . . . ·10−507
21 4.140 . . . ·10−596 3.198 . . . ·10−596 2.971 . . . ·10−596
22 3.859 . . . ·10−632 2.771 . . . ·10−632 2.521 . . . ·10−632
23 8.170 . . . ·10−673 5.289 . . . ·10−673 4.671 . . . ·10−673
24 1.357 . . . ·10−719 7.506 . . . ·10−720 6.336 . . . ·10−720
25 1.042 . . . ·10−774 4.450 . . . ·10−775 3.488 . . . ·10−775
26 7.770 . . . ·10−842 2.074 . . . ·10−842 1.421 . . . ·10−842
27 1.096 . . . ·10−927 8.619 . . . ·10−929 4.148 . . . ·10−929
28 2.090 . . . ·10−1049 1.043 . . . ·10−1051 2.259 . . . ·10−1052
29 3.180 . . . ·10−1250 4.230 . . . ·10−1259 9.803 . . . ·10−1262
30 6.906 . . . ·10−1628 1.491 . . . ·10−1805 5.862 . . . ·10−1960
31 1.603 . . . ·10−1251 2.111 . . . ·10−1260 4.878 . . . ·10−1263
32 3.754 . . . ·10−1052 1.821 . . . ·10−1054 3.914 . . . ·10−1055
33 7.589 . . . ·10−932 7.006 . . . ·10−933 3.536 . . . ·10−933
34 7.893 . . . ·10−846 2.075 . . . ·10−846 1.415 . . . ·10−846
35 8.634 . . . ·10−779 3.735 . . . ·10−779 2.939 . . . ·10−779
36 1.519 . . . ·10−723 7.609 . . . ·10−724 6.215 . . . ·10−724
37 4.753 . . . ·10−678 3.164 . . . ·10−678 2.814 . . . ·10−678
38 7.534 . . . ·10−639 5.469 . . . ·10−639 4.990 . . . ·10−639
39 3.947 . . . ·10−604 3.051 . . . ·10−604 2.834 . . . ·10−604
40 5.121 . . . ·10−573 4.152 . . . ·10−573 3.911 . . . ·10−573
42 4.530 . . . ·10−519 3.928 . . . ·10−519 3.771 . . . ·10−519
45 3.699 . . . ·10−452 3.208 . . . ·10−452 3.065 . . . ·10−452
48 2.899 . . . ·10−400 2.738 . . . ·10−400 2.694 . . . ·10−400
51 6.993 . . . ·10−356 6.708 . . . ·10−356 6.628 . . . ·10−356
54 1.871 . . . ·10−316 1.775 . . . ·10−316 1.741 . . . ·10−316
M= 350 τ= 30 σ t
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ= 1/3t11.7 18 18
22
ΥM(α, τ, σ, t)M= 350
α= 0 τ= 30 σ0 1 t
ΥM(α, τ, σ, t)M= 350
α= 0 τ= 30 σ0 1 t
ΥM(α, τ, σ, t)M= 350
α= 0 τ= 30 σ0 1 t
tmin max
14.10 0.99977 1.00000
14.12 0.99990 1.00000
14.16 1.00000 1.00018
14.18 1.00000 1.00033
20.98 0.99904 1.00000
21.00 0.99947 1.00000
21.04 1.00000 1.00047
21.06 1.00000 1.00102
24.98 0.99964 1.00000
25.00 0.99986 1.00000
25.04 1.00000 1.00039
25.06 1.00000 1.00067
30.38 0.99687 1.00000
30.40 0.99815 1.00000
30.44 1.00000 1.00126
30.46 1.00000 1.00309
32.90 0.99981 1.00000
32.92 0.99990 1.00000
32.96 1.00000 1.00022
32.98 1.00000 1.00044
37.54 0.99752 1.00000
37.56 0.99851 1.00000
37.60 1.00000 1.00088
37.62 1.00000 1.00224
40.88 0.99687 1.00000
40.90 0.99838 1.00000
40.94 1.00000 1.00208
40.96 1.00000 1.00429
43.28 0.99940 1.00000
43.30 0.99961 1.00000
43.34 1.00000 1.00023
43.36 1.00000 1.00064
tmin max
47.96 0.99046 1.00000
47.98 0.99423 1.00000
48.02 1.00000 1.00402
48.04 1.00000 1.01024
49.74 0.99939 1.00000
49.76 0.99969 1.00000
49.80 1.00000 1.00080
49.82 1.00000 1.00159
52.94 0.99815 1.00000
52.96 0.99933 1.00000
53.00 1.00000 1.00214
53.02 1.00000 1.00378
56.40 0.99455 1.00000
56.42 0.99669 1.00000
56.46 1.00000 1.00199
56.48 1.00000 1.00519
59.30 0.98698 1.00000
59.32 0.99179 1.00000
59.36 1.00000 1.00473
59.38 1.00000 1.01320
60.80 1.00000 1.00067
60.82 1.00000 1.00018
60.86 0.99989 1.00000
60.88 0.99999 1.00009
65.08 0.99101 1.00000
65.10 0.99622 1.00000
65.14 1.00000 1.00982
65.16 1.00000 1.01850
67.04 0.99505 1.00000
67.06 0.99733 1.00000
67.10 1.00000 1.00318
67.12 1.00000 1.00682
M= 350 α= 0 τ= 30
σ0 1
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)M= 25
α τ = 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)
α M = 25 τ= 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
ΥM(α, τ, σ, t)
α M = 25 τ= 17 σ0 1 t= 13.8t= 14.0
t= 14.4t= 14.6
Охота за нулями рядов Дирихле средствами линейной алгебры. III (Hunting zeros of Dirichlet series by linear algebra. III)
Preprint
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.
View
Crop circles drawn by Riemann's zeta function and some other its nearby properties
Preprint
Full-text available
  • Jul 2019
Under nearby properties of the Riemann's zeta function we mean properties of approximations to this function, or, more generally, properties of functions which are similar to the zeta function in certain respects. Of these properties the most interesting are those that cannot be formulated in terms of the zeta function alone. In the paper we consider a particular approximations to the alternating zeta function η(s)=∑n=1∞(−1)n+1n−s\eta(s)=\sum_{n=1}^\infty (-1)^{n+1} n^{-s} by finite Dirichlet series ηN(τ,s)=∑n=1NaN,n(τ)n−s\eta_N(\tau,s)=\sum_{n=1}^N a_{N, n}(\tau)n^{-s} with coefficients depending on a real parameter~τ\tau (these coefficients are defined via the values of the Riemann–Siegel theta function and its derivatives at point 1/2+\myi \tau). The paper presents numerical evidence that the difference ηN(τ,s)−ηM(τ,s)\eta_N(\tau,s)-\eta_M(\tau,s) nearly (with high accuracy) satisfies the functional equation for the alternating zeta function. The paper also contains a large number of plots of the ratios \eta_N(\tau,\sigma+\myi t)/\eta_m(\tau,\sigma+\myi t) as functions of t for fixed M, N, and σ\sigma. These plots have very interesting structure: each consists of a tower of almost circular arcs (``crop circles'') each containing one point corresponding to the value of the ratio for t equal to the imaginary part of a nontrivial zeta zero.
View