In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication… (More)

T he Sudoku puzzle has become a very popular puzzle that many newspapers carry as a daily feature. The puzzle consists of a 9×9 grid in which some of the entries of the grid have a number from 1 to… (More)

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of… (More)

Article history: Received 13 October 2007 Revised 9 May 2008 Available online 17 December 2008 Communicated by David Goss In 1997, Serre proved an equidistribution theorem for eigenvalues of Hecke… (More)

Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler’s -function. Let q be a positive integer greater than 1 and γ denote Euler’s constant. We show that all the numbers… (More)

The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that… (More)

For a fixed hyperelliptic curve C given by the equation y 1⁄4 f ðxÞ with f A Z1⁄2x having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm… (More)

Published 15 We obtain a new proof of an asymptotic formula for the coefficients of the j-invariant 16 of elliptic curves. Our proof does not use the circle method. We use Laplace's method 17 of… (More)

1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which… (More)

a(p) = 2p~ @ ) cos 0(p). Since we know the truth of the Ramanujan-Petersson conjecture, it follows that the 0(p)'s are real. Inspired by the Sato-Tate conjecture for elliptic curves, Serre [14]… (More)