Վիքիպեդիա

«Մասնակից:MHamlet/Սևագրություն/1»–ի խմբագրումների տարբերություն

Դիտել պատմությունը
← Նախորդ տարբՀաջորդ տարբ →
Content deleted Content added
ՎիզուալԵլատեքստ
No edit summary
No edit summary
Տող 1.
[[Թվերի տեսություն|Թվերի տեսության]] մեջ '''Կրամերի վարկածը''', որն ձևակերպվել է Շվեդ մաթեմատիկոս [[Հարալդ Կրամեր]]ի կողմից [[1936]]թ․–ին,<ref name="Cramér1936">{{Citation |last=Կրամեր |first=Հարալդ |title=On the order of magnitude of the difference between consecutive prime numbers |url=http://matwbn.icm.edu.pl/ksiazki/aa/aa2/aa212.pdf |journal=Acta Arithmetica |volume=2 |year=1936 |pages=23–46 }} {{ref-en}}</ref> [[պարզ թվերի միջև միջակայքեր|հաջորդական պարզ թվերի միջև միջակայքերի]] չափսի գնահատումն է․ հաջորդական պարզ թվերի միջև միջակայքերն միշտ փոքր են, ուստի [[վարկած]]ը քանակապես ցույց է տալիս, թե որքան փոքր կարող են լինել դրանք։ Այն պնդում է, որ
{{Ֆուրիեյի ձևափոխություններ}}
:<math>p_{n+1}-p_n=O((\log p_n)^2),\ </math>
[[Պատկեր:Fourier Series.svg|thumb|right|180px|[[Քառակուսի ալիք]]ի համար Ֆուրիեյի շարքի առաջին չորս մասնակի գումարները]]
որտեղ ''p''<sub>''n''</sub>–ը ''n''–րդ [[պարզ թիվ]]ն է, ''O''–ն՝ [[մեծ O նշանակում]]ը, "log"–ը՝ [[բնական լոգարիթմ]]ը։ While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement
'''Ֆուրիեյի շարքեր''', [[պարբերական ֆունկցիա]]յի կամ պարբերական ազդակների կազմալուծումը պարզ (հնարավոր է՝ անվերջ թվով) տատանողական ֆունկցիաների, այսինքն՝ [[սինուսոիդ|սինուսների և կոսինուսների]] (կամ [[բարդ ցուցչային ֆունկցիա]]ների) գումարի։ Ֆուրիեյի շարքերի ուսումնասիրումը [[Ֆուրիեյի անալիզ]]ի մի մասն է կազմում։
:<math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1,</math>
and this formulation is often called Cramér's conjecture in the literature.
 
Neither form of Cramér's conjecture has yet been proven or disproven.
The Fourier series is named in honour of [[Jean-Baptiste Joseph Fourier]] (1768–1830), who made important contributions to the study of [[trigonometric series]], after preliminary investigations by [[Leonhard Euler]], [[Jean le Rond d'Alembert]], and [[Daniel Bernoulli]].<ref group="nb">These three did some [[wave equation#Notes|important early work on the wave equation]], especially D'Alembert. Euler's work in this area was mostly [[Euler-Bernoulli beam equation|comtemporaneous/ in collaboration with Bernoulli]], although the latter made some independent contributions to the theory of waves and vibrations ([http://books.google.co.uk/books?id=olMpStYOlnoC&pg=PA214&lpg=PA214&dq=bernoulli+solution+wave+equation&source=bl&ots=h8eN69CWRm&sig=lRq2-8FZvcXIjToXQI4k6AVfRqA&hl=en&sa=X&ei=RqOhUIHOIOa00QWZuIHgCw&ved=0CCEQ6AEwATg8#v=onepage&q=bernoulli%20solution%20wave%20equation&f=false see here, pg.s 209 & 210, ]).</ref> Fourier introduced the series for the purpose of solving the [[heat equation]] in a metal plate, publishing his initial results in his 1807 ''[[Mémoire sur la propagation de la chaleur dans les corps solides]]'' (''Treatise on the propagation of heat in solid bodies''), and publishing his ''Théorie analytique de la chaleur'' in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on [[Deferent and epicycle|deferents and epicycles]].
 
==Heuristic justification==
The heat equation is a [[partial differential equation]]. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a [[sine]] or [[cosine]] wave. These simple solutions are now sometimes called [[Eigenvalue, eigenvector and eigenspace|eigensolutions]]. Fourier's idea was to model a complicated heat source as a superposition (or [[linear combination]]) of simple sine and cosine waves, and to write the [[superposition principle|solution as a superposition]] of the corresponding [[eigenfunction|eigensolutions]]. This superposition or linear combination is called the Fourier series.
Cramér's conjecture is based on a [[Probabilistic number theory|probabilistic]] model (essentially a [[heuristic]]) of the primes, in which one assumes that the probability of a [[natural number]] of size ''x'' being prime is 1/log ''x''. This is known as the '''Cramér model''' of the primes.
Cramér proved that in this model, the above conjecture holds true with [[Almost sure convergence|probability one]].<ref name="Cramér1936" />
 
==Proven results on prime gaps==
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of [[function (mathematics)|function]] and [[integral]] in the early nineteenth century. Later, [[Peter Gustav Lejeune Dirichlet]]<ref>Lejeune-Dirichlet, P. "[[List of important publications in mathematics#Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données|Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données]]". (In French), transl. "On the convergence of trigonometric series which serve to represent an arbitrary function between two given limits". Journal f¨ur die reine und angewandte Mathematik, Vol. 4 (1829) p. 157–169.</ref> and [[Bernhard Riemann]]<ref>{{cite web|url = http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/| title=Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe |language=German|work=[[Habilitationschrift]], [[Göttingen]]; 1854. Abhandlungen der [[Göttingen Academy of Sciences|Königlichen Gesellschaft der Wissenschaften zu Göttingen]], vol. 13, 1867''. Published posthumously for Riemann by [[Richard Dedekind]]|trans_title=About the representability of a function by a trigonometric series|accessdate= 19 May 2008 <!--DASHBot-->| archiveurl= http://web.archive.org/web/20080520085248/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Trig/ | archivedate= 20 May 2008| deadurl= no}}</ref><ref>D. Mascre, Bernhard Riemann: Posthumous Thesis on the Representation of Functions by Triginometric Series (1867). [http://books.google.co.uk/books?id=UdGBy8iLpocC&printsec=frontcover#v=onepage&q&f=false Landmark Writings in Western Mathematics 1640–1940], Ivor Grattan-Guinness (ed.); pg. 492. Elsevier, 20 May 2005.Accessed 7 Dec 2012.</</ref><ref>[http://books.google.co.uk/books?id=uP8SF4jf7GEC&printsec=frontcover#v=onepage&q&f=false Theory of Complex Functions: Readings in Mathematics], by Reinhold Remmert; pg 29. Springer, 1991. Accessed 7 Dec 2012.</ref> expressed Fourier's results with greater precision and formality.
Cramér also gave a [[conditional proof]] of the much [[List of mathematical jargon#weak|weaker]] statement that
:<math>p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n)</math>
on the assumption of the [[Riemann hypothesis]].<ref name="Cramér1936" />
 
In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,<ref>{{Citation |last=Westzynthius |first=E. |title={{lang|de|Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind}} |journal=Commentationes Physico-Mathematicae Helingsfors |volume=5 |issue= |year=1931 |pages=1–37 | zbl=0003.24601 | jfm=57.0186.02 }}.</ref>
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are [[Sine wave|sinusoid]]s. The Fourier series has many such applications in [[electrical engineering]], [[oscillation|vibration]] analysis, [[acoustics]], [[optics]], [[signal processing]], [[image processing]], [[quantum mechanics]], [[econometrics]],<ref>{{cite book |first=Marc |last=Nerlove |first2=David M. |last2=Grether |first3=Jose L. |last3=Carvalho |year=1995 |title=Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics |location= |publisher=Elsevier |isbn=0-12-515751-7 }}</ref> [[Thin-shell structure|thin-walled shell]] theory,<ref>{{cite book |first=Wilhelm |last=Flugge |year=1957 |title=Statik und Dynamik der Schalen |publisher=Springer-Verlag |location=Berlin }}</ref> etc.
:<math>\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.</math>
 
==Heuristics<span id="Cramér–Granville conjecture" />==
[[Daniel Shanks]] conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture.<ref>{{Citation |first=Daniel |last=Shanks |title=On Maximal Gaps between Successive Primes |journal=Mathematics of Computation |volume=18 |issue=88 |year=1964 |pages=646–651 |doi=10.2307/2002951 |publisher=American Mathematical Society |jstor=2002951 }}.</ref>
 
In the random model,
:<math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = c,</math> with <math>c = 1.</math>
But this constant, <math>c</math>, may not apply to all the primes, by [[Maier's theorem]]. As pointed out by [[Andrew Granville]],<ref>{{Citation |last=Granville |first=A. |title=Harald Cramér and the distribution of prime numbers |journal=Scandinavian Actuarial Journal |volume=1 |issue= |year=1995 |pages=12–28 |url=http://www.dartmouth.edu/~chance/chance_news/for_chance_news/Riemann/cramer.pdf }}.</ref> a refinement of Cramér's model taking into account divisibility by small primes suggests that <math>c \ge 2e^{-\gamma}\approx1.1229\ldots</math>, where <math>\gamma</math> is the [[Euler–Mascheroni constant]].
 
[[Thomas Nicely]] has calculated many large prime gaps.<ref>{{Citation |last=Nicely |first=Thomas R. |doi=10.1090/S0025-5718-99-01065-0 |mr=1627813 |issue=227 |journal=Mathematics of Computation |pages=1311–1315 |title=New maximal prime gaps and first occurrences |url=http://www.trnicely.net/gaps/gaps.html |volume=68 |year=1999 }}.</ref> He measures the quality of fit to Cramér's conjecture by measuring the ratio <math>R</math> of the logarithm of a prime to the square root of the gap; he writes, “For the largest known maximal gaps, <math>R</math> has remained near 1.13.” However, <math>1/R^2</math> is still less than 1, and it does not provide support to Granville's refinement that c should be greater than 1.
 
==See also==
*[[Prime number theorem]]
*[[Legendre's conjecture]] and [[Andrica's conjecture]], much weaker but still unproven upper bounds on prime gaps
* [[Firoozbakht’s conjecture]]
* [[Maier's theorem]] on the numbers of primes in short intervals for which the model predicts an incorrect answer
 
==References==
{{Reflist}}
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 | isbn=978-0-387-20860-2 | zbl=1058.11001 | at=A8 }}
* {{cite journal | authorlink=János Pintz | last1=Pintz | first1=János | title=Cramér vs. Cramér. On Cramér's probabilistic model for primes | url=http://projecteuclid.org/euclid.facm/1229619660 | mr=2363833 | year=2007 | journal= Functiones et Approximatio Commentarii Mathematici | volume=37 | pages=361–376 | zbl=1226.11096 | issn=0208-6573 }}
* {{cite book | last=Soundararajan | first=K. | authorlink=Kannan Soundararajan | chapter=The distribution of prime numbers | editor1-last=Granville | editor1-first=Andrew | editor1-link=Andrew Granville | editor2-last=Rudnick | editor2-first=Zeév | title=Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005 | location=Dordrecht |publisher=[[Springer-Verlag]] | series=NATO Science Series II: Mathematics, Physics and Chemistry | volume=237 | pages=59-83 | year=2007 | isbn=978-1-4020-5403-7 | zbl=1141.11043 }}
 
==External links==
*{{mathworld|title=Cramér Conjecture|urlname=CramerConjecture}}
*{{mathworld|title=Cramér-Granville Conjecture|urlname=Cramer-GranvilleConjecture}}
 
{{DEFAULTSORT:Cramer's Conjecture}}
[[Category:Analytic number theory]]
[[Category:Conjectures about prime numbers]]
Ստացված է «https://hy.wikipedia.org/wiki/Մասնակից:MHamlet/Սևագրություն/1» էջից