We estimate the density of integers which have more than one divisor in an interval (y, z] with z ≈ y + y/(log y) log 4−1. As a consequence, we determine the precise range of z such that most integers which have at least one divisor in (y, z] have exactly one such divisor.
The probabilistic method is one of the most significant contributions of Paul Erd˝ os. Indeed, Paul himself said, during his eightieth birthday conference in Keszthely, Hungary, that he believes the method will live long after him. This was the only time I heard him making any comment about the significance and impact of his work. He was always more… (More)
This is Part I of a two-part feature on Paul Erd˝ os following his centennial. There are eleven articles by leading experts who have reflected on the remarkable life, contributions, and influence of this towering figure of twentieth century mathematics. Here in Part I we have contributions from Krishnaswami Alladi and The 100th birth anniversary of the… (More)