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Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study
Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005
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is greater than 1, and the limit inferior is less than 1. The Cramér model of primes predicts incorrectly that it has limit 1 when λ≥2 (using the
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fixed). He also used an equivalent of the number of primes in arithmetic progressions of sufficient length due to
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181:'s equivalent for the counting function of quasi-primes (set of numbers without prime factors lower to bound
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gave another proof, and also showed that most probabilistic models of primes incorrectly predict the
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366:{\displaystyle \int _{2}^{Y}\left(\sum _{2<p\leq x}\log p-\sum _{2<n\leq x}1\right)^{2}\,dx}
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519:, NATO Science Series II: Mathematics, Physics and Chemistry, vol. 237, Dordrecht:
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144:{\displaystyle {\frac {\pi (x+(\log x)^{\lambda })-\pi (x)}{(\log x)^{\lambda -1}}}}
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460:"Cramér vs. Cramér. On Cramér's probabilistic model for primes"
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511:(2007), "The distribution of prime numbers", in
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158:tends to infinity; more precisely the
32:in short intervals for which Cramér's
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39:The theorem states that if π is the
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411:The Michigan Mathematical Journal
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43:and λ is greater than 1 then
34:probabilistic model of primes
565:Theorems about prime numbers
407:"Primes in short intervals"
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215:{\displaystyle z=x^{1/u}}
177:proved his theorem using
154:does not have a limit as
515:; Rudnick, Zeév (eds.),
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41:prime-counting function
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376:of one version of the
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36:gives a wrong answer.
390:Maier's matrix method
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560:Probabilistic models
378:prime number theorem
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164:Borel–Cantelli lemma
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530:978-1-4020-5403-7
513:Granville, Andrew
509:Soundararajan, K.
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254:mean square error
235:{\displaystyle u}
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470:: 361–376,
549:Categories
539:1141.11043
502:1226.11096
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396:References
26:Maier 1985
486:0208-6573
433:0026-2285
335:≤
322:∑
318:−
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301:≤
288:∑
267:∫
244:Gallagher
131:−
128:λ
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97:π
94:−
86:λ
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57:π
458:(2007),
405:(1985),
384:See also
179:Buchstab
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170:Proofs
30:primes
175:Maier
525:ISBN
482:ISSN
429:ISSN
329:<
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535:Zbl
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16:In
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