84 citations · 187 across the 10 of their papers we have counts for
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On indefinite integral ternary quadratic forms
Alexander Gamburd, Amit Ghosh, Peter Sarnak +1
We resolve two problems pertaining to indefinite integral ternary quadratic forms, one highlighted by Margulis and the other initiated by Serre, both from 1990. To do so we develop…
Markoff Surfaces and Strong Approximation: 1
Jean Bourgain, Alexander Gamburd, Peter Sarnak
We prove results pertaining to strong approximation for Markoff triples in the case of prime moduli.
An asymptotic formula for integer points on Markoff-Hurwitz varieties
Alex Gamburd, Michael Magee, Ryan Ronan
We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2}=ax_{1}x_{2}\ldots x_{n}+k. \] When $n…
Markoff Triples and Strong Approximation
Jean Bourgain, Alex Gamburd, Peter Sarnak
We investigate the transitivity properties of the group of morphisms generated by Vieta involutions on the solutions in congruences to the Markoff equation as well as to other Mark…
Generalization of Selberg's 3/16 Theorem and Affine Sieve
Jean Bourgain, Alex Gamburd, Peter Sarnak
A celebrated theorem of Selberg states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We prove a generalization of Selberg's theorem for…
Pseudomoments of the Riemann zeta-function and pseudomagic squares
Brian Conrey, Alex Gamburd
We compute integral moments of partial sums of the Riemann zeta function on the critical line and obtain an expression for the leading coefficient as a product of the standard arit…