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Quantitative bounds for sets lacking polynomial progressions with shifted prime difference
Ben Krause, Hamed Mousavi, Terence Tao +1
We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes . The types of…
Products of consecutive integers with unusual anatomy
Terence Tao
Call an interval of consecutive natural numbers \emph{bad} if the product is divisible by the square of its largest prime factor; \emph{very…
Quantitative correlations and some problems on prime factors of consecutive integers
Terence Tao, Joni Teräväinen
We consider several old problems involving the number of prime divisors function , as well as the related functions and . Firstly, we show that there are infinit…
Rough numbers between consecutive primes
Ayla Gafni, Terence Tao
Using a sieve-theoretic argument, we show that almost all gaps between consecutive primes contain a natural number whose least prime factor $p(m…
On the number of exceptional intervals to the prime number theorem in short intervals
Ayla Gafni, Terence Tao
For a fixed exponent , it is expected that we have the prime number theorem in short intervals as . From the rece…
Decomposing a factorial into large factors
Boris Alexeev, Evan Conway, Matthieu Rosenfeld +4
Let denote the largest number such that can be expressed as the product of integers greater than or equal to . The bound was apparently es…
Higher uniformity of arithmetic functions in short intervals II. Almost all intervals
Kaisa Matomäki, Maksym Radziwiłł, Xuancheng Shao +2
We study higher uniformity properties of the von Mangoldt function , the Möbius function , and the divisor functions on short intervals for almost all $x \in…