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Riemann zeta function

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Bernhard Riemann, in his famous 1859 paper,[1][2] analytically continued Euler's zeta function over the whole complex plane (except for a single pole of order 1 at \scriptstyle s \,=\, 1 \,, which corresponds to the diverging harmonic series). It is thus known as the Riemann zeta function \scriptstyle \zeta(s) \,, where \scriptstyle s \,=\,\mathfrak{R}(s) + i \mathfrak{I}(s) \,=\, \sigma + i t \, (this notation was introduced in his paper).

Contents

Analytic continuation

Analytic continuation to the right of the critical strip

The following infinite series converges for all complex numbers \scriptstyle s \, with real part greater than 1, and defines \scriptstyle \zeta(s) \, in the complex plane to the right of the critical strip, i.e. for \scriptstyle \sigma \,>\, 1 \,,

\zeta(s) := \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{\stackrel{p}{p \, {\rm prime}}} \frac{p^s}{p^s - 1} = \prod_{\stackrel{p}{p \, {\rm prime}}} \frac{1}{1 - \frac{1}{p^s}},\quad \mathfrak{R}(s) > 1, \,

where, in Euler's product (for the Riemann zeta function), the product is taken over all primes \scriptstyle p \,.

Analytic continuation within the critical strip

Within the critical strip, i.e. for \scriptstyle 0 \,<\, \sigma \,<\, 1 \,, we may use[3]

\zeta(s) = \left( \frac{2^{s-1}}{2^{s-1} - 1} \right) \, \phi(s) = \left( \frac{1}{1 - 2^{1-s}} \right) \, \phi(s) = \left( \frac{1}{1 - 2^{1-s}} \right) \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s},\quad 0 < \mathfrak{R}(s) < 1, \,

where \scriptstyle \phi(s) \, is Euler's alternating zeta function (which converges for \scriptstyle \mathfrak{R}(s) \,>\, 0 \,)

\phi(s) := \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s},\quad \mathfrak{R}(s) > 0. \,

Analytic continuation to the left of the critical strip

To the left of the critical strip, i.e. for \scriptstyle \sigma \,<\, 0 \,, we may use the functional equation

\zeta(s) = \left( 2^s \pi^{s-1} \sin\left( \frac{\pi s}{2} \right) \Gamma(1-s) \right) \zeta(1-s),\quad \mathfrak{R}(s) < 0, \,

which reveals the trivial zeros for negative even integers, since \scriptstyle \sin\left( \frac{\pi s}{2} \right) \,=\, 0 \, for even \scriptstyle s \,.

Integral formula

The Riemann zeta function, for all complex numbers \scriptstyle s \, with real part \scriptstyle \sigma \,>\, 1 \,, is given by the integral

\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{t^{s-1}}{e^t - 1} \, dt,\quad \Re(s) > 1, \,

where \scriptstyle \Gamma(n) \, is the Gamma function.

Laurent expansion of the Riemann zeta function

The Laurent expansion of the Riemann zeta function about \scriptstyle s \,=\, 1 \, gives

\zeta(s) = \frac{1}{s-1} + \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \, \gamma_n \, (s-1)^n = \frac{1}{s-1} + \gamma + \sum_{n=1}^{\infty} \frac{(-1)^n}{n!}\, \gamma_n \, (s-1)^n, \,

where \scriptstyle \gamma \,=\, \gamma_0 \, is the Euler-Mascheroni constant and \scriptstyle \gamma_n \, are the Stieltjes constants, sometimes referred to as generalized Euler constants.

Since

\lim_{s \to 1} \bigg[ \zeta(s) - \frac{1}{s-1}\bigg] = \gamma \,

it implies that \scriptstyle \gamma_0 \,=\, \gamma \,.

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 that \scriptstyle \zeta(0) \,=\, - \frac{1}{2} \, is the only rational number for nonnegative even integers)

\zeta(2n) = (-1)^{n+1} \, \frac{2^{2n-1} \, B_{2n} \, \pi^{2n}}{(2n)!},\quad n \ge 0, \,

where \scriptstyle B_n \, are the Bernoulli numbers. The values for \scriptstyle \zeta(2n),\, n \,\ge\, 1, \, are transcendental numbers (a rational number times \scriptstyle \pi^{2n} \,).

Riemann zeta function for even integers
\scriptstyle 2n \, \scriptstyle \zeta(2n) \,

\scriptstyle n \ge 0 \,
Decimal expansion
(Sequence of decimal digits)
A-number
0 - \tfrac{1}{2} \, −0.5
{5}
 
2 \tfrac{\pi^2}{6} \, 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 \tfrac{\pi^4}{90} \, 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 \tfrac{\pi^6}{945} \, 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 \tfrac{\pi^8}{9450} \, 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 \tfrac{\pi^{10}}{93555} \, 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 \tfrac{691 \pi^{12}}{638512875}  \, 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 \tfrac{2 \pi^{14}}{18243225} \, 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 \tfrac{3617 \pi^{16}}{325641566250} \, 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 \tfrac{43867 \pi^{18}}{38979295480125} \, 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 \tfrac{174611 \pi^{20}}{1531329465290625} \, 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, \scriptstyle \zeta(2n),\, n \,\ge\, 0 \,, have the generating function

G_{\{ \zeta(2n) \}}(x) := \sum_{n=0}^{\infty} \zeta(2n) \, x^{2n} = - \frac{\pi x}{2} \cot(\pi x) = -\frac{1}{2} + \frac{\pi^2}{6} x^2 + \frac{\pi^4}{90} x^4 + \frac{\pi^6}{945} x^6 + \cdots, \,

where \scriptstyle \cot(\pi x) \, is the cotangent function.

\scriptstyle \frac{\zeta(2n)}{\pi^{2n}},\, n \,\ge\, 0 \,, have the generating function

G_{\{ \frac{\zeta(2n)}{\pi^{2n}} \}}(x) := \sum_{n=0}^{\infty} \frac{\zeta(2n)}{\pi^{2n}} \, x^{2n} = - \frac{x}{2} \cot(x) = -\frac{1}{2} + \frac{1}{6} x^2 + \frac{1}{90} x^4 + \frac{1}{945} x^6 + \cdots, \,

where \scriptstyle \cot(x) \, is the cotangent function.

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 constant \scriptstyle \zeta(3) \,, proved irrational by Roger Apéry), let alone transcendental.

The Riemann zeta function for nonnegative odd integers (except for 1, where we have a pole of order 1) is given by the integral

\zeta(2n+1) = \frac{1}{(2n)!} \int_{0}^{\infty} \frac{t^{2n}}{e^t - 1} dt,\quad n \ge 1. \,
Riemann zeta function for odd integers
\scriptstyle 2n+1 \, \scriptstyle \zeta(2n+1) \,

\scriptstyle n \ge 0 \,
Decimal expansion
(Sequence of decimal digits)
A-number
1 Pole
(of order 1)
\scriptstyle \infty \,
(this is the unique pole, of order 1, of the Riemann zeta function) (\scriptstyle \zeta(1) \, gives the harmonic series)
 
3 \scriptstyle \frac{1}{2!} \int_{0}^{\infty} \frac{t^2}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{4!} \int_{0}^{\infty} \frac{t^4}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{6!} \int_{0}^{\infty} \frac{t^6}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{8!} \int_{0}^{\infty} \frac{t^8}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{10!} \int_{0}^{\infty} \frac{t^{10}}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{12!} \int_{0}^{\infty} \frac{t^{12}}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{14!} \int_{0}^{\infty} \frac{t^{14}}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{16!} \int_{0}^{\infty} \frac{t^{16}}{e^t - 1} dt \, 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 \scriptstyle \frac{1}{18!} \int_{0}^{\infty} \frac{t^{18}}{e^t - 1} dt \, 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

Apart from the trivial zeros (even negative integers), the Riemann zeta function doesn't have any zero outside the critical strip 0 < σ < 1 delimited by the lines σ = 0 and σ = 1 (neither can the zeros lie "too close" to those lines). Furthermore, the non-trivial zeros are symmetric about the real axis and the critical line σ = 1/2 and according to the Riemann Hypothesis, they all lie on the line σ = 1/2.

The nontrivial zeros of the Riemann zeta function[2] appear in the critical strip \scriptstyle 0 \,<\, \sigma \,<\, 1 \,

s = \sigma + i t,\quad 0 < \sigma < 1, \,

where the zeros appear in pairs (if \scriptstyle \epsilon \,\ne\, 0 \,) reflected through the critical line (corresponding to real part \scriptstyle \frac{1}{2} \,)

s = (\tfrac{1}{2} \pm \epsilon) + i t,\quad 0 \le \epsilon < \tfrac{1}{2}, \,

together with their conjugates

\overline{s} = (\tfrac{1}{2} \pm \epsilon) - i t,\quad 0 \le \epsilon < \tfrac{1}{2}. \,

All the known nontrivial zeros of the Riemann zeta function have real part \scriptstyle \frac{1}{2}. \,

The Riemann zeta function may be expressed in terms of the nontrivial zeros as (note the pole of order 1 at \scriptstyle s \,=\, 1 \,)

\zeta(s) = \frac{ \pi^{s/2} }{ s \, (s-1) \, \Gamma(\frac{s}{2}) } \prod_{\rho} \left( 1 - \frac{s}{\rho} \right), \,

where \scriptstyle \rho \, is a nontrivial zero (note that for each nontrivial zero \scriptstyle \rho \, in the upper half of the complex plane, we have a corresponding zero \scriptstyle \overline{\rho} \, , its complex conjugate, in the lower half of the complex plane.

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)

All the nontrivial zeros [of the Riemann zeta function] have real part \scriptstyle \frac{1}{2} \,, i.e.
s = (\tfrac{1}{2} \pm \epsilon) + i t,\quad \epsilon = 0. \,

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 \scriptstyle \frac{1}{2} \,, the more regular the distribution of the primes.

Table of nontrivial zeros

The first 100 (nontrivial) zeros of the Riemann zeta function, accurate to over 1000 decimal places.[4]

Table of nontrivial zeros[5]
n \, Imaginary part (base 10)

of \scriptstyle n \,th nontrivial zero (above the real axis)

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...

Sequences related to the nontrivial zeros

A002410 Nearest integer to imaginary part of \scriptstyle n \,-th zero of Riemann zeta function.

{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, ...}

A135297 Number of Riemann zeta function zeros on the critical line, less than \scriptstyle 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, ...}

A161914 Gaps between the nontrivial zeros of Riemann zeta function, rounded to nearest integers, with \scriptstyle 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, ...}

A100060 Consider the nontrivial zeros of the Riemann zeta function on the critical line, \scriptstyle \frac{1}{2} + i t \,. \scriptstyle a(n) \, tells where the second difference of the imaginary part is positive (denoted by 1) or negative (denoted by 0).

{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, ...}

A117537 These are the locations of the midpoints of consecutive zeros of the Riemann zeta function on the critical line with increasingly large normalized spacing; equivalently, of consecutive real zeros of the \scriptstyle Z \, function (see Z function). If \scriptstyle t \, and \scriptstyle s \, are consecutive zeros of the \scriptstyle Z \, function, we define their normalized spacing as \scriptstyle \frac{s-t}{2 \pi \, \ln \left( \frac{s+t}{4 \pi} \right)} \,. The sequence above is found by taking \scriptstyle r \,=\, \frac{\ln 2}{2 \pi} \,\cdot\, \frac{s+t}{2} \, and rounding to the nearest integer. These values \scriptstyle r \, have a marked tendency to be close to integer values and all of the terms of the above sequence are actually contained in the intervals \scriptstyle \frac{\ln 2}{2 \pi} \,\cdot\, [s, t] \,.

{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 the \scriptstyle Z \, function for increasing real \scriptstyle t \,. If \scriptstyle Z'(s) \,=\, 0 \, is a positive zero of the derivative of \scriptstyle Z \,, then \scriptstyle |Z(s)| \, is the peak value. We renormalize \scriptstyle s \, by \scriptstyle r \,=\, s \,\cdot\, \frac{\ln 2}{2 \pi} \, and round to the nearest integer to get the terms of the sequence. The fractional parts of these values are not randomly distributed; \scriptstyle r \, shows a very strong tendency to be near an integer.

{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, ...}

A117538 Locations of the increasing peak values of the integral of the absolute value of the Riemann zeta function between successive zeros on the critical line. This can also be defined in terms of the \scriptstyle Z \, function; if \scriptstyle s \, and \scriptstyle t \, are successive zeros of a renormalized \scriptstyle Z \, function, \scriptstyle z(x) \,=\, \frac{2 \pi}{\ln 2} \,\cdot\, x \, Z(x) \,, then take the integral between \scriptstyle s \, and \scriptstyle t \, of \scriptstyle |z(x)| \,. For each successively higher value of this integral, the corresponding term of the integer sequence is \scriptstyle r \,=\, \frac{s+t}{2} \, rounded to the nearest integer.

{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



Notes

  1. 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. 2.0 2.1 Riemann's 1859 Manuscript, Clay Mathematics Institute 2010.
  3. Since
    \sum_{n=1}^\infty \frac{(-1)^{n}}{n^s} + \sum_{n=1}^\infty \frac{1}{n^s} = 2 \, \sum_{k=1}^\infty \frac{1}{(2k)^s} = 2^{1-s} \, \sum_{n=1}^\infty \frac{1}{n^s} \,
    and thus
    \zeta(s) = \left( \frac{1}{1 - 2^{1-s}} \right) \, \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s},\quad \mathfrak{R}(s) > 0. \,
    Note that analytic continuation is always unique.
  4. 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.
  5. 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.

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