TL;DR
This paper establishes lower bounds on the count of shifted primes with divisors in specific intervals, extending previous work and linking the distribution of such primes to primes in arithmetic progressions.
Contribution
It provides broad-range lower bounds for shifted primes with divisors in given intervals, complementing existing upper bounds and exploring their dependence on prime distribution.
Findings
Lower bounds depend heavily on primes in arithmetic progressions.
Bounds are applicable over a broad range of parameters y and z.
Application to counting shifted primes in multiplication tables.
Abstract
We bound from below the number of shifted primes p+s<x that have a divisor in a given interval (y,z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine the number of shifted primes that appear in a multiplication table up to multiplicative constants.
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