The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agrees with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak proved that the behavior of zeros near the central point of holomorphic cusp forms agrees with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions φ . We prove similar results for families of cuspidal Maass forms, the other natural family of GL 2 /Q L-functions. For suitable weight functions on the space of Maass forms, the limiting behavior agrees with the expected orthogonal group. We prove this for supp( φ ) ⊆ (−3/2, 3/2) when the level N tends to infinity through the square-free numbers; if the level is fixed the support decreases to being contained in (−1, 1), though we still uniquely specify the symmetry type by computing the 2-level density.

In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x/2, x]. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that π(x) − π(x/2) ≥ n for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n ∼ p 2n (where p m is the mth prime). In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c,n such that for all x ≥ R c,n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c,n exists for all n and all c, that R c,n ∼ p n 1−c as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes, and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping; this was first observed by Sondow, Nicholson, and Noe in the case c = 1/2. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.

In 1845, Bertrand conjectured that for all integers x ≥ 2, there exists at least one prime in (x/2, x]. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any n ≥ 1, there is a (smallest) prime R n such that π(x) − π(x/2) ≥ n for all x ≥ R n . In 2009 Sondow called R n the nth Ramanujan prime and proved the asymptotic behavior R n ∼ p 2n (where p m is the mth prime). He and Laishram proved the bounds p 2n < R n < p 3n , respectively, for n > 1. In the present paper, we generalize the interval of interest by introducing a parameter c ∈ (0, 1) and defining the nth c-Ramanujan prime as the smallest integer R c,n such that for all x ≥ R c,n , there are at least n primes in (cx, x]. Using consequences of strengthened versions of the Prime Number Theorem, we prove that R c,n exists for all n and all c, that R c,n ∼ p n 1−c as n → ∞, and that the fraction of primes which are c-Ramanujan converges to 1 − c. We then study finer questions related to their distribution among the primes, and see that the c-Ramanujan primes display striking behavior, deviating significantly from a probabilistic model based on biased coin flipping. This model is related to the Cramer model, which correctly predicts many properties of primes on large scales, but has been shown to fail in some instances on smaller scales.

Abstract. In this paper we obtain a weighted average formula for special values of L-functions attached to normalized elliptic modular forms of weight k and full level. These results are obtained by studying the pullback of a Siegel Eisenstein series and working out an explicit spectral decomposition.

The Katz-Sarnak Density Conjecture states that the behavior of zeros of a family of L-functions near the central point (as the conductors tend to zero) agree with the behavior of eigenvalues near 1 of a classical compact group (as the matrix size tends to infinity). Using the Petersson formula, Iwaniec, Luo and Sarnak [ILS] proved that the behavior of zeros near the central point of holomorphic cusp forms agree with the behavior of eigenvalues of orthogonal matrices for suitably restricted test functions. We prove a similar result for level 1 cuspidal Maass forms, the other natural family of GL 2 L-functions. We use the explicit formula to relate sums of our test function at scaled zeros to sums of the Fourier transform at the primes weighted by the L-function coefficients, and then use the Kuznetsov trace formula to average the Fourier coefficients over the family. There are numerous technical obstructions in handling the terms in the trace formula, which are surmounted through the use of smooth weight functions for the Maass eigenvalues and results on Kloosterman sums and Bessel and hyperbolic functions.