Skewes's number and Neuenstein, Hesse: Difference between pages

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{{Infobox German Location
In [[number theory]], [[Skewes’s number]] can refer to several extremely large numbers used by the [[South Africa]]n mathematician [[Stanley Skewes]] as [[upper bound]]s for
|Art = Gemeinde
the smallest [[natural number]] {{math|<VAR >x</VAR >}} for which
|Name = Neuenstein
:{{math|&pi;(<VAR >x</VAR >) > li(<VAR >x</VAR >)}}
|Wappen = Neuenstein (Hessen).png
where {{math|&pi;(<VAR >x</VAR >)}} is the [[prime-counting function]] and {{math|li(<VAR >x</VAR >)}} is the [[logarithmic integral function]].
|lat_deg = 50 |lat_min = 56
The numbers found by Skewes are now only of historical interest, because computer calculations have produced much smaller estimates. {{as of|2007}}, these calculations suggest that the smallest such {{math|<VAR >x</VAR >}} is close to {{math|1.397&times;10<sup>316</sup>}}.
|lon_deg = 09 |lon_min = 33
|Lageplan =
|Bundesland = Hessen
|Regierungsbezirk = Kassel
|Landkreis = Hersfeld-Rotenburg
|Höhe = 307
|Fläche = 64.84
|Einwohner = 3154
|Stand = 2006-06-30
|PLZ = 36286
|Vorwahl = 06677<br>06621 (Untergeis/Gittersdorf)
|Kfz = HEF
|Gemeindeschlüssel = 06 6 32 014
|Gliederung = 8 districts
|Adresse = Freiherr-v.-Stein-Straße 5<br />36286 Neuenstein
|Website = [http://www.neuenstein.net/ www.neuenstein.net]
|Bürgermeister = Walter Glänzer
|Partei = CDU
}}
'''Neuenstein''' is a municipality in the [[Hersfeld-Rotenburg]] district, in [[Hesse]], [[Germany]].


{{Towns and municipalities in Hersfeld-Rotenburg district}}
==Skewes's numbers==
[[John Edensor Littlewood]], Skewes' teacher, proved in {{harv|Littlewood|1914}} that there is such a number (and so, a first such number); and indeed found that the sign of the difference &pi;(''x'')&nbsp;&minus;&nbsp;li(''x'') changes infinitely often. All numerical evidence then available seemed to suggest that &pi;(''x'') is always less than li(''x''), though mathematicians familiar with Riemann's work on the Riemann zeta function would probably have realized that occasional exceptions were likely by the argument [[#Riemann's formula|given below]] (and the claim sometimes made that Littlewood's result was a big surprise to experts seems doubtful). Littlewood's proof did not, however exhibit a concrete such number ''x''; it was not an [[Effective results in number theory|effective result]].


{{Hesse-geo-stub}}
{{harvtxt|Skewes|1933}} proved that, assuming that the [[Riemann hypothesis]] is true, there exists a number ''x'' violating &pi;(''x'') < li(''x'') below
:<math>e^{e^{e^{79}}}</math>


[[de:Neuenstein (Hessen)]]
(now sometimes called '''first Skewes' number'''), which is approximately equal to
[[eo:Neuenstein (Hesio)]]
:<math>10^{10^{8.85 \times 10^{33}}}</math>.
[[it:Neuenstein (Assia)]]

[[nl:Neuenstein (Hessen)]]
In {{harv|Skewes|1955}}, without assuming the Riemann hypothesis, he managed to prove that there must exist a value of ''x'' below
[[pl:Neuenstein (Hesja)]]
:<math>10^{10^{10^{963}}}</math>
[[ro:Neuenstein (Hessen)]]

[[ru:Нойенштайн (Гессен)]]
(sometimes called '''second Skewes' number''').
[[vo:Neuenstein (Hessen)]]
Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to [[George Kreisel]], this was at the time not considered obvious even in principle. The approach called ''[[unwinding (proof theory)|unwinding]]'' in [[proof theory]] looks directly at proofs and their structure to produce bounds. The other way, more often seen in practice in number theory, changes proof structure enough so that absolute constants can be made more explicit.

Skewes's result was celebrated partly because the proof structure used [[excluded middle]], which is not ''a priori'' a constructive argument (it divides into two cases, and it is not computable in which case one is working).
Although both Skewes numbers are big compared to most numbers encountered in mathematical proofs, neither is anywhere near as big as [[Graham's number]].
==More recent estimates==
These (enormous) upper bounds have since been reduced considerably by using large scale computer calculations of zeros of the Riemann zeta function. The first estimate for the actual value of a crossover point was given by {{harvtxt|Lehman|1966}}, who
showed that somewhere between 1.53&times; 10<sup>1165</sup> and 1.65&times; 10<sup>1165</sup> there are more than 10<sup>500</sup> consecutive integers ''x'' with &pi;(''x'') > li(''x'').
Without assuming the Riemann hypothesis, {{harvs|txt=yes|authorlink=H. J. J. te Riele|first=H. J. J.|last= te Riele |year= 1987}} proved an upper bound of 7{{e|370}}. A better estimation was 1.39822{{e|316}} discovered by {{harvtxt|Bays|Hudson|2000}}, who showed there are at least 10<sup>153</sup> consecutive integers somewhere near this value where &pi;(''x'') > li(''x''), and suggested that there are probably at least 10<sup>311</sup>. {{harvtxt|Chao|Plymen|2005}} gave a small improvement and correction to the result of Bays and Hudson. {{harvtxt|Demichel|2005}} suggested that the first crossover point is near the slightly smaller value 1.397162914&times;10<sup>316</sup>, though {{as of|2007|lc=on}} his work has not been published or independently checked.
Bays and Hudson found a few much smaller values of ''x'' where &pi;(x) gets close to li(x);
the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though
computer calculations suggest they are unlikely to exist. Rigorously, {{harvtxt|Rosser|Schoenfeld|1962}} proved that there are no crossover points below ''x'' = 10<sup>8</sup>, and this lower bound was subsequently improved by {{harvtxt|Brent|1975}} to 8{{e|10}}, and by {{harvtxt|Kotnik|2008}} to 10<sup>14</sup>.
There is no explicit value ''x'' known for certain to have the property &pi;(x) > li(x), though computer calculations suggest some explicit numbers that are quite likely to satisfy this.

{{harvtxt|Wintner|1941}} showed that the proportion of integers for which &pi;(''x'')>li(''x'') is positive, and {{harvtxt|Rubinstein|Sarnak|1994}} showed that this proportion is about .00000026, which is surprisingly large given how far one has to go to find the first example.

==Riemann's formula==
Riemann gave an explicit formula for &pi;(x), whose leading terms are (ignoring some subtle convergence questions)
:<math>\pi(x) = li(x) - li(\sqrt{x})/2 - \sum_\rho li(x^\rho)</math> + smaller terms
where the sum is over zeros &rho; of the Riemann zeta function. The largest error term in the approximation &pi;(x) = li(x) (if the [[Riemann hypothesis]] is true) is li(x<sup>1/2</sup>)/2, showing that li(''x'') is usually larger than &pi;(x). The other terms above are somewhat smaller, and moreover tend to have different complex arguments so mostly cancel out. Occasionally however, many of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term li(x<sup>1/2</sup>)/2. The reason why the Skewes number is so large is that these smaller terms are quite a lot smaller than the leading error term, mainly because the first complex zero of the zeta function has quite a large imaginary part, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of ''N'' random complex numbers having roughly the same argument is about 1 in 2<sup>''N''</sup>. This explains why &pi;(''x'') is sometimes larger than li(''x''), and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the Riemann zeta function. The argument above is not a proof, as it assume the zeros of the Riemann zeta function are random which is not true. Roughly speaking, Littlewood's proof consists of showing that in some sense the zeros are "sufficiently random" for this argument to work.

In the unlikely event that the Riemann hypothesis is false, the argument is much simpler, essentially because the terms li(''x''<sup>&rho;</sup>) for zeros violating the Riemann hypothesis (with real part greater than 1/2) are eventually larger than li(''x''<sup>1/2</sup>).

The reason for the term li(''x''<sup>1/2</sup>)/2 is that, roughly speaking, li(''x'') is counting not primes, but primes powers ''p''<sup>''n''</sup> weighted by 1/''n'', and li(''x''<sup>1/2</sup>)/2 is a sort of correction term coming from squares of primes.

==References==
*{{citation|id={{MR|1752093}}|first=C.|last= Bays |first2=R. H.|last2= Hudson |url=http://www.ams.org/mcom/2000-69-231/S0025-5718-99-01104-7/S0025-5718-99-01104-7.pdf
|title=A new bound for the smallest x with π(x)>li(x) |journal=Mathematics of Computation|volume=69|year=2000
|issue= 231|pages= 1285–1296}}
*{{citation|id={{MR|0369287}}|first=R. P.|last= Brent |title=Irregularities in the distribution of primes and twin primes
|journal=Mathematics of Computation|volume=29|year=1975 |pages= 43–56}}
*{{citation|title=A new bound for the smallest x with &pi;(x) > li(x)
|first=Kuok Fai|last= Chao|first2= Roger|last2= Plymen|year=2005|url=http://arXiv.org/abs/math/0509312}}
*{{citation|first= T.|last= Kotnik |url=http://dx.doi.org/10.1007/s10444-007-9039-2
|title=The prime-counting function and its analytic approximations |journal=Advances in Computational Mathematics|volume=29|year=2008|pages= 55-70}}
*{{citation|first= R. Sherman |last=Lehman|title= On the difference π(x) − li(x)|journal= Acta Arith. |volume=11 |year=1966|pages= 397–410
|id={{MR|0202686}}}}
* {{citation|first=J. E.|last= Littlewood|title=Sur la distribution des nombres premiers|journal=Comptes Rendus|volume= 158 |year=1914|pages= 1869-1872}}
*{{citation|first= S.|last= Skewes|title=On the difference π(''x'')&nbsp;&minus;&nbsp;Li(''x'')|journal=Journal of the London Mathematical Society|volume=8|year=1933|pages= 277-283}}
*{{citation|id={{MR|0067145}}| first= S.|last= Skewes|title=On the difference π(''x'')&nbsp;&minus;&nbsp;Li(''x'') (II)|journal=Proceedings of the London Mathematical Society|volume= 5 |year=1955|pages= 48-70}}
*{{citation|id={{MR|0866118}}|first= H. J. J. |last=te Riele|title=On the sign of the difference π(''x'')&nbsp;&minus;&nbsp;Li(''x'')|journal=Mathematics of Computation|volume=48|year=1987|pages= 323-328
|url=http://links.jstor.org/sici?sici=0025-5718%28198701%2948%3A177%3C323%3AOTSOTD%3E2.0.CO%3B2-N }}
*{{citation|id={{MR|0137689}}|first= J. B.|last= Rosser |first2= L.|last2= Schoenfeld
|title=Approximate formulas for some functions of prime numbers |journal=Illinois Journal of Mathematics|volume=6|year=1962|pages= 64-94}}
*{{citation|id={{MR|1329368}}
|last=Rubinstein|first= M.|last2= Sarnak|first2= P.
|title=Chebyshev's bias
|journal=Experiment. Math. |volume=3 |year=1994|issue= 3|pages= 173-197
|url= http://projecteuclid.org/euclid.em/1048515870 }}
*{{citation|id={{MR|0004255}}|last= Wintner|first= A. |title=On the distribution function of the remainder term of the prime number theorem|journal= Amer. J. Math.|volume= 63|year=1941|pages= 233-248
|url=http://links.jstor.org/sici?sici=0002-9327%28194104%2963%3A2%3C233%3AOTDFOT%3E2.0.CO%3B2-N }}

==External links==
*{{citation|url=http://www.mybloop.com/dmlpat/maths/li_crossover_pi.pdf|title= The prime counting function and related subjects|version= 3.1415
|accessdate= 2007 November 4|year=2005
|first= Patrick|last= Demichel}} Contains many graphs of the difference &pi;(x) &minus; li(x) near crossover points.

{{Large numbers}}

[[Category:Large numbers]]
[[Category:Number theory]]

[[ca:Nombre de Skewes]]
[[cs:Skewesovo číslo]]
[[de:Skewes' Zahl]]
[[fr:Nombre de Skewes]]
[[ko:스큐스 수]]
[[it:Numero di Skewes]]
[[ja:スキューズ数]]
[[pt:Número de Skewes]]
[[ru:Число Скьюза]]
[[sk:Skewesovo číslo]]
[[fi:Skewesin luku]]
[[sv:Skewes tal]]
[[th:จำนวนสกีว]]

Revision as of 04:35, 11 October 2008

Neuenstein
Municipality
Coat of arms of Neuenstein
Coat of arms
Location of Neuenstein
Map
CountryGermany
StateHesse
Admin. regionKassel
DistrictHersfeld-Rotenburg
Subdivisions8 districts
Government
 • MayorWalter Glänzer (CDU)
Area
 • Total64.84 km2 (25.03 sq mi)
Elevation
307 m (1,007 ft)
Population
 (2022-12-31)[1]
 • Total3,170
 • Density49/km2 (130/sq mi)
Time zoneUTC+01:00 (CET)
 • Summer (DST)UTC+02:00 (CEST)
Postal codes
36286
Dialling codes06677
06621 (Untergeis/Gittersdorf)
Vehicle registrationHEF
Websitewww.neuenstein.net

Neuenstein is a municipality in the Hersfeld-Rotenburg district, in Hesse, Germany.

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This Hesse location article is a stub. You can help Wikipedia by expanding it.

  1. ^ "Bevölkerung in Hessen am 31.12.2022 nach Gemeinden" (XLS) (in German). Hessisches Statistisches Landesamt. June 2023.
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