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On the largest square divisor of shifted primes

Volume 196 / 2020

Jori Merikoski Acta Arithmetica 196 (2020), 349-386 MSC: Primary 11N13; Secondary 11N36. DOI: 10.4064/aa190725-16-1 Published online: 6 August 2020


We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^\theta $ for $\theta =1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $\theta =1/2-\varepsilon $ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $\theta =4/9-\varepsilon .$ As in the work of Matomäki, we apply Harman’s sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $\theta =1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument to the one Zhang (2014) used to obtain equidistribution beyond the Bombieri–Vinogradov range for primes with respect to smooth moduli. To optimize the argument we incorporate technical refinements from the Polymath project (2014). Since the moduli are squares, the method produces complete exponential sums modulo squares of primes which are estimated using the results of Cochrane and Zheng (2000).


  • Jori MerikoskiDepartment of Mathematics and Statistics
    University of Turku
    FI-20014 University of Turku, Finland

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