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Proceedings of the London Mathematical Society Advance Access originally published online on March 16, 2009
Proceedings of the London Mathematical Society 2009 99(2):484-520; doi:10.1112/plms/pdp009
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© 2009 London Mathematical Society

An orthogonal test of the L-functions Ratios conjecture

Steven J. Miller

Department of Mathematics and Statistics
Williams College
Williamstown, MA 01267
USA

Received 17 July 2008. Revision received 17 December 2008.

We test the predictions of (a weakened version of) the L-functions Ratios conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N -> {infty} through the primes or N = 1 and k -> {infty}. We study the main and lower-order terms in the 1-level density. We provide evidence for the Ratios conjecture by computing and confirming its predictions up to a power savings in the family's cardinality, at least for test functions whose Fourier transforms are supported in (–2, 2). We do this both for the weighted and unweighted 1-level density (where in the weighted case we use the Petersson weights), thus showing that either formulation may be used. These two 1-level densities differ by a term of size 1/log (k2 N). Finally, we show that there is another way of extending the sums arising in the Ratios conjecture, leading to a different answer (although the answer is such a lower-order term that it is hopeless to observe which is correct).

2000 Mathematics Subject Classification 11M26 (primary), 11M41, 15A52 (secondary).

This work was partly supported by NSF grant DMS0600848.


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