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Riemann ζ function
s = 1 |
ζ (s) |
s = ℜ(s) + i ℑ(s) = σ + i t |
Contents
Analytic continuation
Analytic continuation to the right of the critical strip
The following infinite series converges for all complex numberss |
ζ (s) |
σ > 1 |
-
ζ (s) := ∞∑ n = 1
=1 n s ∏ p
p prime
=p s p s − 1 ∏ p
p prime
, ℜ(s) > 1,1 1 − 1 p s
p |
Analytic continuation within the critical strip
Within the critical strip, i.e. for0 < σ < 1 |
-
ζ (s) =
η (s) =2 s − 1 2 s − 1 − 1
η (s) =1 1 − 2 1 − s 1 1 − 2 1 − s ∞∑ n = 1
, 0 < ℜ(s) < 1,(−1) n +1 n s
η (s) |
ℜ (s) > 0 |
-
η (s) := ∞∑ n = 1
, ℜ(s) > 0.(−1) n +1 n s
Analytic continuation to the left of the critical strip
To the left of the critical strip, i.e. forσ < 0 |
-
ζ (s) = 2 s π s − 1 sin
Γ(1 − s) ζ (1 − s), ℜ(s) < 0,π s 2
sin (
|
s |
Integral formula
The Riemann zeta function, for all complex numberss |
σ > 1 |
-
ζ (s) = 1 Γ(s) ∫ ∞ 0
d t, ℜ(s) > 1,t s − 1 e t − 1
Γ(n) |
Laurent expansion of the Riemann zeta function
The Laurent expansion of the Riemann zeta function abouts = 1 |
-
ζ (s) =
+1 s − 1 ∞∑ n = 0
γn (s − 1) n =(−1) n n!
+ γ +1 s − 1 ∞∑ n = 1
γn (s − 1) n,(−1) n n!
γ = γ0 |
γn |
Since
-
ζ (s) −
= γ,1 s − 1
γ0 = γ |
Riemann zeta function for nonnegative integers
Riemann zeta function for nonnegative even integers
The Riemann zeta function for negative even integers is 0 (those are the trivial zeros of the Riemann zeta function).
The Riemann zeta function for nonnegative even integers is given by (note thatζ (0) = −
|
-
ζ (2 n) = (−1) n +1
, n ≥ 0,2 2 n − 1 B2 n π 2 n (2 n)!
Bn |
ζ (2 n), n ≥ 1, |
π 2n |
|
|
Decimal expansion (Sequence of decimal digits) |
A-number | |||
---|---|---|---|---|---|---|
0 |
|
− 0.5 {5} |
||||
2 |
|
1.644934066848226436472415166646... {1, 6, 4, 4, 9, 3, 4, 0, 6, 6, 8, 4, 8, 2, 2, 6, 4, 3, 6, 4, 7, 2, 4, 1, 5, 1, 6, 6, 6, 4, 6, 0, 2, 5, 1, 8, 9, 2, 1, 8, 9, 4, 9, 9, 0, 1, 2, 0, 6, 7, 9, 8, 4, 3, 7, 7, 3, 5, 5, 5, ...}
|
A013661 | |||
4 |
|
1.082323233711138191516003696541... {1, 0, 8, 2, 3, 2, 3, 2, 3, 3, 7, 1, 1, 1, 3, 8, 1, 9, 1, 5, 1, 6, 0, 0, 3, 6, 9, 6, 5, 4, 1, 1, 6, 7, 9, 0, 2, 7, 7, 4, 7, 5, 0, 9, 5, 1, 9, 1, 8, 7, 2, 6, 9, 0, 7, 6, 8, 2, 9, 7, ...}
|
A013662 | |||
6 |
|
1.0173430619844491397145179297909... {1, 0, 1, 7, 3, 4, 3, 0, 6, 1, 9, 8, 4, 4, 4, 9, 1, 3, 9, 7, 1, 4, 5, 1, 7, 9, 2, 9, 7, 9, 0, 9, 2, 0, 5, 2, 7, 9, 0, 1, 8, 1, 7, 4, 9, 0, 0, 3, 2, 8, 5, 3, 5, 6, 1, 8, 4, 2, 4, 0, ...}
|
A013664 | |||
8 |
|
1.004077356197944339378685238508... {1, 0, 0, 4, 0, 7, 7, 3, 5, 6, 1, 9, 7, 9, 4, 4, 3, 3, 9, 3, 7, 8, 6, 8, 5, 2, 3, 8, 5, 0, 8, 6, 5, 2, 4, 6, 5, 2, 5, 8, 9, 6, 0, 7, 9, 0, 6, 4, 9, 8, 5, 0, 0, 2, 0, 3, 2, 9, 1, 1, ...}
|
A013666 | |||
10 |
|
1.0009945751278180853371459589003... {1, 0, 0, 0, 9, 9, 4, 5, 7, 5, 1, 2, 7, 8, 1, 8, 0, 8, 5, 3, 3, 7, 1, 4, 5, 9, 5, 8, 9, 0, 0, 3, 1, 9, 0, 1, 7, 0, 0, 6, 0, 1, 9, 5, 3, 1, 5, 6, 4, 4, 7, 7, 5, 1, 7, 2, 5, 7, 7, 8, ...}
|
A013668 | |||
12 |
|
1.0002460865533080482986379980477... {1, 0, 0, 0, 2, 4, 6, 0, 8, 6, 5, 5, 3, 3, 0, 8, 0, 4, 8, 2, 9, 8, 6, 3, 7, 9, 9, 8, 0, 4, 7, 7, 3, 9, 6, 7, 0, 9, 6, 0, 4, 1, 6, 0, 8, 8, 4, 5, 8, 0, 0, 3, 4, 0, 4, 5, 3, 3, 0, 4, ...}
|
A013670 | |||
14 |
|
1.00006124813505870482925854510513... {1, 0, 0, 0, 0, 6, 1, 2, 4, 8, 1, 3, 5, 0, 5, 8, 7, 0, 4, 8, 2, 9, 2, 5, 8, 5, 4, 5, 1, 0, 5, 1, 3, 5, 3, 3, 3, 7, 4, 7, 4, 8, 1, 6, 9, 6, 1, 6, 9, 1, 5, 4, 5, 4, 9, 4, 8, 2, 7, 5, ...}
|
A013672 | |||
16 |
|
1.0000152822594086518717325714876367... {1, 0, 0, 0, 0, 1, 5, 2, 8, 2, 2, 5, 9, 4, 0, 8, 6, 5, 1, 8, 7, 1, 7, 3, 2, 5, 7, 1, 4, 8, 7, 6, 3, 6, 7, 2, 2, 0, 2, 3, 2, 3, 7, 3, 8, 8, 9, 9, 0, 4, 7, 1, 5, 3, 1, 1, 5, 3, 1, 0, ...}
|
A013674 | |||
18 |
|
1.0000038172932649998398564616446219... {1, 0, 0, 0, 0, 0, 3, 8, 1, 7, 2, 9, 3, 2, 6, 4, 9, 9, 9, 8, 3, 9, 8, 5, 6, 4, 6, 1, 6, 4, 4, 6, 2, 1, 9, 3, 9, 7, 3, 0, 4, 5, 4, 6, 9, 7, 2, 1, 8, 9, 5, 3, 3, 3, 1, 1, 4, 3, 1, 7, ...}
|
A013676 | |||
20 |
|
1.0000009539620338727961131520386834... {1, 0, 0, 0, 0, 0, 0, 9, 5, 3, 9, 6, 2, 0, 3, 3, 8, 7, 2, 7, 9, 6, 1, 1, 3, 1, 5, 2, 0, 3, 8, 6, 8, 3, 4, 4, 9, 3, 4, 5, 9, 4, 3, 7, 9, 4, 1, 8, 7, 4, 1, 0, 5, 9, 5, 7, 5, 0, 0, 5, ...}
|
A013678 |
Generating function of the Riemann zeta function for nonnegative even integers
The even zeta constants,ζ (2 n), n ≥ 0 |
-
G{ζ (2 n)}(x) := ∞∑ n = 0
cot (π x) = −π x 2
+1 2
x 2 +π 2 6
x 4 +π 4 90
x 6 + ⋯,π 6 945
cot (π x) |
For
|
-
G{
}(x) :=ζ (2 n) π 2 n ∞∑ n = 0
x 2 n = −ζ (2 n) π 2 n
cot x = −x 2
+1 2
x 2 +1 6
x 4 +1 90
x 6 + ⋯,1 945
cot x |
Riemann zeta function for nonnegative odd integers
The Riemann zeta function for odd integers has no known closed-form formula. It is not known whether those values are irrational (except for Apéry’s constantζ (3) |
The Riemann zeta function for nonnegative odd integers (except for 1, where we have a pole of order 1) is given by the integral
-
ζ (2 n + 1) = 1 (2 n)! ∫ ∞ 0
d t, n ≥ 1.t 2 n e t − 1
|
|
Decimal expansion (Sequence of decimal digits) |
A-number | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | Pole (of order 1) |
(this is the unique pole, of order 1, of the Riemann zeta function) (
|
||||||||||
3 |
|
1.2020569031595942853997381615114... {1, 2, 0, 2, 0, 5, 6, 9, 0, 3, 1, 5, 9, 5, 9, 4, 2, 8, 5, 3, 9, 9, 7, 3, 8, 1, 6, 1, 5, 1, 1, 4, 4, 9, 9, 9, 0, 7, 6, 4, 9, 8, 6, 2, 9, 2, 3, 4, 0, 4, 9, 8, 8, 8, 1, 7, 9, 2, 2, 7, ...}
|
A002117 | |||||||||
5 |
|
1.036927755143369926331365486457... {1, 0, 3, 6, 9, 2, 7, 7, 5, 5, 1, 4, 3, 3, 6, 9, 9, 2, 6, 3, 3, 1, 3, 6, 5, 4, 8, 6, 4, 5, 7, 0, 3, 4, 1, 6, 8, 0, 5, 7, 0, 8, 0, 9, 1, 9, 5, 0, 1, 9, 1, 2, 8, 1, 1, 9, 7, 4, 1, 9, ...}
|
A013663 | |||||||||
7 |
|
1.008349277381922826839797549849... {1, 0, 0, 8, 3, 4, 9, 2, 7, 7, 3, 8, 1, 9, 2, 2, 8, 2, 6, 8, 3, 9, 7, 9, 7, 5, 4, 9, 8, 4, 9, 7, 9, 6, 7, 5, 9, 5, 9, 9, 8, 6, 3, 5, 6, 0, 5, 6, 5, 2, 3, 8, 7, 0, 6, 4, 1, 7, 2, 8, ...}
|
A013665 | |||||||||
9 |
|
1.002008392826082214417852769232... {1, 0, 0, 2, 0, 0, 8, 3, 9, 2, 8, 2, 6, 0, 8, 2, 2, 1, 4, 4, 1, 7, 8, 5, 2, 7, 6, 9, 2, 3, 2, 4, 1, 2, 0, 6, 0, 4, 8, 5, 6, 0, 5, 8, 5, 1, 3, 9, 4, 8, 8, 8, 7, 5, 6, 5, 4, 8, 5, 9, ...}
|
A013667 | |||||||||
11 |
|
1.000494188604119464558702282526... {1, 0, 0, 0, 4, 9, 4, 1, 8, 8, 6, 0, 4, 1, 1, 9, 4, 6, 4, 5, 5, 8, 7, 0, 2, 2, 8, 2, 5, 2, 6, 4, 6, 9, 9, 3, 6, 4, 6, 8, 6, 0, 6, 4, 3, 5, 7, 5, 8, 2, 0, 8, 6, 1, 7, 1, 1, 9, 1, 4, ...}
|
A013669 | |||||||||
13 |
|
1.000122713347578489146751836526... {1, 0, 0, 0, 1, 2, 2, 7, 1, 3, 3, 4, 7, 5, 7, 8, 4, 8, 9, 1, 4, 6, 7, 5, 1, 8, 3, 6, 5, 2, 6, 3, 5, 7, 3, 9, 5, 7, 1, 4, 2, 7, 5, 1, 0, 5, 8, 9, 5, 5, 0, 9, 8, 4, 5, 1, 3, 6, 7, 0, ...}
|
A013671 | |||||||||
15 |
|
1.000030588236307020493551728510... {1, 0, 0, 0, 0, 3, 0, 5, 8, 8, 2, 3, 6, 3, 0, 7, 0, 2, 0, 4, 9, 3, 5, 5, 1, 7, 2, 8, 5, 1, 0, 6, 4, 5, 0, 6, 2, 5, 8, 7, 6, 2, 7, 9, 4, 8, 7, 0, 6, 8, 5, 8, 1, 7, 7, 5, 0, 6, 5, 6, ...}
|
A013673 | |||||||||
17 |
|
1.000007637197637899762273600293... {1, 0, 0, 0, 0, 0, 7, 6, 3, 7, 1, 9, 7, 6, 3, 7, 8, 9, 9, 7, 6, 2, 2, 7, 3, 6, 0, 0, 2, 9, 3, 5, 6, 3, 0, 2, 9, 2, 1, 3, 0, 8, 8, 2, 4, 9, 0, 9, 0, 2, 6, 2, 6, 7, 9, 0, 9, 5, 3, 7, ...}
|
A013675 | |||||||||
19 |
|
1.00000190821271655393892565695779... {1, 0, 0, 0, 0, 0, 1, 9, 0, 8, 2, 1, 2, 7, 1, 6, 5, 5, 3, 9, 3, 8, 9, 2, 5, 6, 5, 6, 9, 5, 7, 7, 9, 5, 1, 0, 1, 3, 5, 3, 2, 5, 8, 5, 7, 1, 1, 4, 4, 8, 3, 8, 6, 3, 0, 2, 3, 5, 9, 3, ...}
|
A013677 |
Zeros
Trivial zeros
The trivial zeros of the Riemann zeta function are the complex numbers with real part corresponding to the negative even integers:
- {–2, –4, –6, –8, –10, –12, –14, –16, –18, –20, –22, –24, –26, –28, –30, –32, –34, –36, –38, –40, –42, –44, –46, –48, –50, –52, –54, –56, –58, –60, –62, –64, –66, –68, –70, –72, –74, –76, –78, –80, ...}
Nontrivial zeros
The nontrivial zeros of the Riemann zeta function[2] appear in the critical strip0 < σ < 1 |
-
s = σ + i t, 0 < σ < 1,
ϵ ≠ 0 |
|
-
s = (
± ϵ ) + i t, 0 ≤ ϵ <1 2
,1 2
together with their conjugates
-
s̅ = (
± ϵ ) − i t, 0 ≤ ϵ <1 2
.1 2
|
s = 1 |
-
ζ (s) = π s / 2 s (s − 1) Γ(
)s 2 ∏ ρ
=s ρ π s / 2 s (s − 1) Γ(
)s 2 ∏ ρ: ℑ( ρ) > 0
,| ρ |2 − 2 s ℜ( ρ) + s 2| ρ |2
Γ(s) |
ρ |
ρ |
x̅ρ |
Riemann Hypothesis
In his famous 1859 paper,[1] Bernhard Riemann proposed the conjecture (used as a hypothesis for numerous conditional proofs in number theory)
Hypothesis (Riemann Hypothesis, 1859). (Riemann)Since many “conditional proofs” assume the truth of the conjecture, it became known as the Riemann Hypothesis. The nontrivial zeros reveal information about the distribution of the primes: the closer the real part of the nontrivial zeros lies to
All the nontrivial zeros [of the Riemann zeta function] have real part, i.e.
1 2
s = ( ± ϵ ) + i t, ϵ = 0.
1 2
|
Table of nontrivial zeros
The first 100 (nontrivial) zeros of the Riemann zeta function, accurate to over 1000 decimal places.[4]
|
Imaginary part (base 10) of
|
OEIS | ||
---|---|---|---|---|
1 | 14.134725141734693790457251983562470270784257115699243175685567460149... | A058303 | ||
2 | 21.022039638771554992628479593896902777334340524902781754629520403587... | A065434 | ||
3 | 25.010857580145688763213790992562821818659549672557996672496542006745... | A065452 | ||
4 | 30.424876125859513210311897530584091320181560023715440180962146036993... | A065453 | ||
5 | 32.935061587739189690662368964074903488812715603517039009280003440784... | A192492 | ||
6 | 37.586178158825671257217763480705332821405597350830793218333001113622... | |||
7 | 40.918719012147495187398126914633254395726165962777279536161303667253... | |||
8 | 43.327073280914999519496122165406805782645668371836871446878893685521... | |||
9 | 48.005150881167159727942472749427516041686844001144425117775312519814... | |||
10 | 49.773832477672302181916784678563724057723178299676662100781955750433... | |||
11 | 52.970321477714460644147296608880990063825017888821224779900748140317... | |||
12 | 56.446247697063394804367759476706127552782264471716631845450969843958... | |||
13 | 59.347044002602353079653648674992219031098772806466669698122451754746... | |||
14 | 60.831778524609809844259901824524003802910090451219178257101348824808... | |||
15 | 65.112544048081606660875054253183705029348149295166722405966501086675... | |||
16 | 67.079810529494173714478828896522216770107144951745558874196669551694... | |||
17 | 69.546401711173979252926857526554738443012474209602510157324539999663... | |||
18 | 72.067157674481907582522107969826168390480906621456697086683306151488... | |||
19 | 75.704690699083933168326916762030345922811903530697400301647775301574... | |||
20 | 77.144840068874805372682664856304637015796032449234461041765231453151... | |||
21 | 79.337375020249367922763592877116228190613246743120030878438720497101... | |||
22 | 82.910380854086030183164837494770609497508880593782149146571306283235... | |||
23 | 84.735492980517050105735311206827741417106627934240818702735529689045... | |||
24 | 87.425274613125229406531667850919213252171886401269028186455557938439... | |||
25 | 88.809111207634465423682348079509378395444893409818675042199871618814... | |||
26 | 92.491899270558484296259725241810684878721794027730646175096750489181... | |||
27 | 94.651344040519886966597925815208153937728027015654852019592474274513... | |||
28 | 95.870634228245309758741029219246781695256461224987998420529281651651... | |||
29 | 98.831194218193692233324420138622327820658039063428196102819321727565... | |||
30 | 101.31785100573139122878544794029230890633286638430089479992831871523... | |||
31 | 103.72553804047833941639840810869528083448117306949576451988516579403... | |||
32 | 105.44662305232609449367083241411180899728275392853513848056944711418... | |||
33 | 107.16861118427640751512335196308619121347670788140476527926471042155... | |||
34 | 111.02953554316967452465645030994435041534596839007305684619079476550... | |||
35 | 111.87465917699263708561207871677059496031174987338587381661941961969... | |||
36 | 114.32022091545271276589093727619107980991765772382989228772843104130... | |||
37 | 116.22668032085755438216080431206475512732985123238322028386264231147... | |||
38 | 118.79078286597621732297913970269982434730621059280938278419371651419... | |||
39 | 121.37012500242064591894553297049992272300131063172874230257513263573... | |||
40 | 122.94682929355258820081746033077001649621438987386351721195003491528... | |||
41 | 124.25681855434576718473200796612992444157353877469356114035507691395... | |||
42 | 127.51668387959649512427932376690607626808830988155498248279977930068... | |||
43 | 129.57870419995605098576803390617997360864095326465943103047083999886... | |||
44 | 131.08768853093265672356637246150134905920354750297504538313992440777... | |||
45 | 133.49773720299758645013049204264060766497417494390467501510225885516... | |||
46 | 134.75650975337387133132606415716973617839606861364716441697609317354... | |||
47 | 138.11604205453344320019155519028244785983527462414623568534482856865... | |||
48 | 139.73620895212138895045004652338246084679005256538260308137013541090... | |||
49 | 141.12370740402112376194035381847535509030066087974762003210466509596... | |||
50 | 143.11184580762063273940512386891392996623310243035463254859852295728... |
A002410 Nearest integer to imaginary part of
n |
- {14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, ...}
A013629 Floor of imaginary parts of zeros of Riemann zeta function.
- {14, 21, 25, 30, 32, 37, 40, 43, 48, 49, 52, 56, 59, 60, 65, 67, 69, 72, 75, 77, 79, 82, 84, 87, 88, 92, 94, 95, 98, 101, 103, 105, 107, 111, 111, 114, 116, 118, 121, 122, 124, 127, 129, 131, 133, 134, 138, ...}
A092783 Ceiling of imaginary parts of zeros of Riemann zeta function.
- {15, 22, 26, 31, 33, 38, 41, 44, 49, 50, 53, 57, 60, 61, 66, 68, 70, 73, 76, 78, 80, 83, 85, 88, 89, 93, 95, 96, 99, 102, 104, 106, 108, 112, 112, 115, 117, 119, 122, 123, 125, 128, 130, 132, 134, 135, 139, ...}
n |
- {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 14, 14, 15, 15, 16, ...}
a (1) = 14 |
- {14, 7, 4, 5, 3, 5, 3, 2, 5, 2, 3, 3, 3, 1, 4, 2, 2, 3, 4, 1, 2, 4, 2, 3, 1, 4, 2, 1, 3, 2, 2, 2, 2, 4, 1, 2, 2, 3, 3, 2, 1, 3, 2, 2, 2, 1, 3, 2, 1, 2, 3, 1, 3, 1, 2, 3, 1, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, ...}
A124288 Indices of unstable zeros of the Riemann zeta function.
- {1, 3, 6, 9, 13, 17, 21, 26, 30, 33, 40, 44, 50, 54, 61, 67, 70, 78, 79, 90, 93, 101, 109, 112, 117, 124, 134, 139, 147, 149, 153, 165, 167, 175, 186, 189, 197, 201, 214, 218, 219, 234, 235, 240, 253, 255, ...}
A124289 Unstable twins: pairs of consecutive numbers in A124288 (indices of unstable zeros of the Riemann zeta function).
- {78, 79, 218, 219, 234, 235, 299, 300, 370, 371, 500, 501, ...}
|
a (n) |
- {1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...}
t |
s |
Z |
|
r =
|
r |
|
- {2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190, ...}
Absolute value of the Riemann zeta function
Peaks of the absolute value of the Riemann zeta function along the critical line
A117536 These are the locations of the increasingly larger peaks of the absolute value of the Riemann zeta function along the critical line. Equivalently, the locations of the increasingly large peaks of the absolute value of theZ |
t |
Z ′(s) = 0 |
Z |
| Z (s) | |
s |
r = s ⋅
|
r |
- {0, 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, ...}
Z |
s |
t |
Z |
z (x) =
|
s |
t |
| Z (s) | |
r =
|
- {2, 5, 7, 12, 19, 31, 41, 53, 72, 130, 171, 224, 270, 764, 954, 1178, 1395, 1578, 2684, 3395, 7033, 8269, 8539, 14348, 16808, 36269, 58973, ...}
See also
- Euler’s zeta function
- Riemann zeta function
- Euler products
- Euler’s alternating zeta function
- Prime zeta function
- Hurwitz zeta function
- Multivariate zeta function
Notes
- ↑ 1.0 1.1 Riemann, G.F.B., “Über die Anzahl der Primzahlen unter einer gegebenen Grösse” Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671–680, Nov. 1859.
- ↑ 2.0 2.1 Riemann's 1859 Manuscript, Clay Mathematics Institute 2010.
- ↑ Since
-
ζ (s) − η (s) = ∞∑ n = 1
+1 n s ∞∑ n = 1
=(−1) n n s 2 ∞∑ k = 1
= 2 1 − s1 (2 k ) s ∞∑ n = 1
= 2 1 − s ζ (s),1 n s
-
ζ (s) = 1 1 − 2 1 − s ∞∑ n = 1
, ℜ(s) > 0.(−1) n +1 n s
-
- ↑ Andrew M. Odlyzko, The first 100 zeros of the Riemann zeta function, accurate to over 1000 decimal places, were computed by Andrew M. Odlyzko of the University of Minnesota at his previous position at AT&T Labs - Research.
- ↑ Data extracted from above mentioned file with permission of Andrew M. Odlyzko.
References
- Bernhard Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of primes less than a given quantity), Monatsberichte der Berliner Akademie, November 1859.
External links
- Andrew Odlyzko, Tables of zeros of the Riemann zeta function.