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Register a new userOn the largest square divisor of shifted primes
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Jori Merikoski
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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 Matomaki (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 $theta=4/9-varepsilon.$ Similarly as in the work of Matomaki, we apply Harmans 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 as 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).
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Let $ xgeq 1 $ be a large number, let $ [x]=x-{x} $ be the largest integer function, and let $ sigma(n)$ be the sum of divisors function. This note presents the first proof of the asymptotic formula for the average order $ sum_{pleq x}sigma([x/p])=c_0xlog log x+O(x) $ over the primes, where $c_0>0$ is a constant. More generally, $ sum_{pleq x}sigma([x/(p+a)])=c_0xlog log x+O(x) $ for any fixed integer $a$.
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