We give the expected asymptotic for the number of \( y\)-smooth integers up to \( x\), in the arithmetic progression \( a \pmod q\) provided \( q\leq y^\epsilon\).
If \( a_1,a_2,\ldots \) is a sequence of non-negative real numbers satisfying \( ma_m + \sum_{1\leq i\leq m-1} a_ia_{m-i}=2m+O(1)\), then either \( a_m=1+o(1)\) or \( a_m=1-(-1)^m+o(1)\).
Smoothing "smooth" numbers (with John Friedlander)
Philosophical Transactions of the Royal Society, 345 (1993), 339-348.
We show that the \( y\)-smooth integers are equidistributed in intervals of length \( \sqrt{x}y^{2+o(1)}\) around \( x\). In 2015 this has been improved by Matomaki and Radziwill to intervals of length \(A\sqrt{x}\) provided \( A\to \infty\) as \( x\to \infty\)
We give the expected asymptotic for the number of \( y\)-smooth integers up to \( x\), in the arithmetic progression \( a \pmod q\) provided \( q\leq y^{1-\epsilon}\) and \( q=x^{o(1)}\).
An upper bound in Goldbach's problem (with Jean-Marc Deshouillers, Wladyslaw Narkiewicz and Carl Pomerance)
Mathematics of Computation, 61 (1993), 209-213.
We show that 210 is the largest even integer \( 2n\) for which \( 2n-p\) is prime for every prime \( p\in [n/2,n-2] \).