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In [[mathematics]], the '''Bombieri–Vinogradov theorem''' (sometimes simply called '''Bombieri's theorem''') is a major result of [[analytic number theory]], obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by [[Mark Barban|Barban]] in 1961<ref>{{cite journal |first=M. B. |last=Barban |title=New applications of the 'large sieve' of Yu. V. Linnik |journal=Akad. Nauk. UzSSR Trudy. Inst. Mat. |volume=22 |year=1961 |issue= |pages=1–20 |doi= }}</ref> and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after [[Enrico Bombieri]]<ref>{{cite book |authorlink=Enrico Bombieri |first=E. |last=Bombieri |title=Le Grand Crible dans la Théorie Analytique des Nombres |edition=Seconde |series=Astérisque |volume=18 |location=Paris |year=1987 | zbl=0618.10042 }}</ref> and [[A. I. Vinogradov]],<ref>{{cite journal |first=A. I. |last=Vinogradov |title=The density hypothesis for Dirichlet L-series |language=Russian |journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=29 |issue=4 |year=1965 |pages=903–934 |mr=197414 }} Corrigendum. ibid. 30 (1966), pages 719-720. (Russian)</ref> who published on a related topic, the [[density hypothesis]], in 1965. Perhaps somewhat unjustly, Barban's name is often forgotten. | |||
This result is a major application of the [[large sieve method]], which developed rapidly in the early 1960s, from its beginnings in work of [[Yuri Linnik]] two decades earlier. Besides Bombieri, [[Klaus Roth]] was working in this area. | |||
==Statement of the Bombieri–Vinogradov theorem== | |||
Let ''A'' be any positive real number. Then | |||
:<math>\sum_{q\leq Q}\max_{y<x}\max_{1\le a\le q\atop (a,q)=1}\left|\psi(y;q,a)-{y\over\varphi(q)}\right|=O\left(x^{1/2}Q(\log x)^5\right),</math> | |||
if | |||
:<math>x^{1/2}\log^{-A}x\leq Q\leq x^{1/2}.</math> | |||
Here <math>\varphi(q)</math> is the [[Euler totient function]], which is the number of summands for the modulus ''q'', and | |||
:<math>\psi(x;q,a)=\sum_{n\le x\atop n\equiv a\bmod q}\Lambda(n),</math> | |||
where <math>\Lambda</math> denotes the [[von Mangoldt function]]. | |||
A verbal description of this result is that it addresses the error term in the [[Prime number theorem|prime number theorem for arithmetic progressions]], averaged over the moduli ''q'' up to ''Q''. For a certain range of ''Q'', which are around <math>\sqrt x</math> if we neglect logarithmic factors, the error averaged is nearly as small as <math>\sqrt x</math>. This is quite unobvious, and without the averaging is about of the strength of the [[Generalized Riemann Hypothesis]] (GRH). | |||
==See also== | |||
*[[Elliott–Halberstam conjecture]] (a generalization of Bombieri–Vinogradov) | |||
*[[Vinogradov's theorem]] (named after [[Ivan Matveyevich Vinogradov]]) | |||
==Notes== | |||
{{reflist}} | |||
==External links== | |||
*{{MathWorld|urlname=BombierisTheorem|title=Bombieri's Theorem}} | |||
{{DEFAULTSORT:Bombieri-Vinogradov theorem}} | |||
[[Category:Sieve theory]] | |||
[[Category:Theorems in analytic number theory]] | |||
Latest revision as of 18:07, 13 December 2013
In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by Barban in 1961[1] and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri[2] and A. I. Vinogradov,[3] who published on a related topic, the density hypothesis, in 1965. Perhaps somewhat unjustly, Barban's name is often forgotten.
This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of Yuri Linnik two decades earlier. Besides Bombieri, Klaus Roth was working in this area.
Statement of the Bombieri–Vinogradov theorem
Let A be any positive real number. Then
if
Here is the Euler totient function, which is the number of summands for the modulus q, and
where denotes the von Mangoldt function.
A verbal description of this result is that it addresses the error term in the prime number theorem for arithmetic progressions, averaged over the moduli q up to Q. For a certain range of Q, which are around if we neglect logarithmic factors, the error averaged is nearly as small as . This is quite unobvious, and without the averaging is about of the strength of the Generalized Riemann Hypothesis (GRH).
See also
- Elliott–Halberstam conjecture (a generalization of Bombieri–Vinogradov)
- Vinogradov's theorem (named after Ivan Matveyevich Vinogradov)
Notes
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External links
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Corrigendum. ibid. 30 (1966), pages 719-720. (Russian)