61 citations · 445 across the 59 of their papers we have counts for
6 papers · 1 filter
Root optimization of polynomials in the number field sieve
Shi Bai, Richard P. Brent, Emmanuel Thomé
The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. T…
Lower bounds on maximal determinants of +-1 matrices via the probabilistic method
Richard P. Brent, Judy-anne H. Osborn, Warren D. Smith
We show that the maximal determinant D(n) for -matrices satisfies . Here is the Hadamard upper bound, and de…
Bounds on minors of binary matrices
Richard P. Brent, Judy-anne H. Osborn
We prove an upper bound on sums of squares of minors of {+1, -1} matrices. The bound is sharp for Hadamard matrices, a result due to de Launey and Levin (2009), but our proof is si…
On minors of maximal determinant matrices
Richard P. Brent, Judy-anne H. Osborn
By an old result of Cohn (1965), a Hadamard matrix of order n has no proper Hadamard submatrices of order m > n/2. We generalise this result to maximal determinant submatrices of H…
General lower bounds on maximal determinants of binary matrices
Richard P. Brent, Judy-anne H. Osborn
We give general lower bounds on the maximal determinant of n by n {+1,-1}-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier r…
On the sign of the real part of the Riemann zeta-function
Juan Arias de Reyna, Richard P. Brent, Jan van de Lune
We consider the distribution of on fixed lines , and in particular the density \[d(σ) = \lim_{T \rightarrow +\infty} \frac{1}{2T} |\{t \in [-T,+T]: |\argζ…