Andrew Granville's Home Page

2012 Publications

Sharp transitions in making squares (with Ernie Croot, Robin Pemantle and Prasad Tetali)
Annals of Mathematics, 175 (2012), 1507-1550.

In the fastest-performing integer factoring algorithms, one creates a sequence of integers (in a pseudo-random way) and wishes to rapidly determine a subsequence whose product is a square. In 1994 Pomerance stated the following problem which encapsulates all of the key issues: Select integers \( a_1, a_2, \ldots \) at random from the interval \( [1, x]\) , until some (non-empty) subsequence has product equal to a square. Find a good estimate for the expected stopping time of this process. A good solution should allow one to determine the optimal choice of parameters in many factoring algorithms. Pomerance (1994), using an idea of Schroeppel (1985), showed that with probability \( 1 - o(1)\) the first subsequence whose product equals a square occurs after at least \( J_0^{1-o(1)}\) integers have been selected, but no more than \( J_0\), for an appropriate (explicitly determined) \( J_0 = J_0(x)\) . We tighten Pomerance?s interval to \( [(\pi/4)(e^{-\gamma}-o(1)) J_0, (e^{-\gamma}+o(1)) J_0]\) , where \( \gamma\) is the Euler-Mascheroni constant, and believe that the correct interval is \( [ (e^{-\gamma}-o(1)) J_0, (e^{-\gamma}+o(1)) J_0]\) , a "sharp threshold". In our proof we confirm the well-established belief that, typically, none of the integers in the square product have large prime factors. The heart of the proof of our upper bound lies in delicate calculations in probabilistic graph theory, supported by comparative estimates on smooth numbers using precise information on saddle points.


Zeta functions for ideal classes in real quadratic fields, at \(s=0\) (with Andras Biro)
Journal of Number Theory, 132 (2012), 1807-1829.

For any given fractional ideal \( I \) of a real quadratic field, Jozsef Beck introduced the twisted partial zeta function \( \zeta_I(s,\chi) := \sum_a \chi(Na)/(Na)^s\), where the sum is over all integral ideals that are equivalent to \( I \), and \( \chi\) is some Dirichlet character. We give a short, easily computable formula to evaluate \( \zeta_I(0,\chi)\) in terms of the cycle of reduced quadratic forms associated with \( I\). We generalize our formula to \( \zeta_I(1-k,\chi)\) for all integers \(k\geq 1\), and discuss connections between these formulae and small class numbers.

For a class of Lucas sequences \( \{ x_n\}_{n\geq 0}\) , we show that if \( n\) s a positive integer then \( x_n\) has a primitive prime factor which divides \( x_n\) to an odd power, except perhaps when \( n =\) 1, 2, 3 or 6. This has several desirable consequences.