Unnormalized differences between zeros of L-functions

We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-functions. In particular, we show how the rank of an elliptic curve over Q is encoded in the sequences of zeros of other L-functions, not only the one associated to the curve.Comment: v3. Small corrections. Changed slightly one hypothesis in the definition of the set H of test functions, regarding the decay of the Fourier transform. To appear in Compositio Mat