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تسجيل مستخدم جديدLarge prime factors on short intervals
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تأليف
Jori Merikoski
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We show that for all large enough $x$ the interval $[x,x+x^{1/2}log^{1.39}x]$ contains numbers with a prime factor $p > x^{18/19}.$ Our work builds on the previous works of Heath-Brown and Jia (1998) and Jia and Liu (2000) concerning the same problem for the longer intervals $[x,x+x^{1/2+epsilon}].$ We also incorporate some ideas from Harmans book `Prime-detecting sieves (2007). The main new ingredient that we use is the iterative argument of Matomaki and Radziwi{l}{l}(2016) for bounding Dirichlet polynomial mean values, which is applied to obtain Type II information. This allows us to take shorter intervals than in the above-mentioned previous works. We have also had to develop ideas to avoid losing any powers of $log x$ when applying Harmans sieve method.
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