Long Mollifiers

Let be a real polynomial with , let for some , and let

The Dirichlet polynomial is called a mollifier'' of the Riemann -function because it is an approximation to , and so should be much better behaved than near the -line.

The mean value of near the -line is a fundamental tools for studying zeros of the -function. The most general version currently used is the following formula of Conrey, Ghosh, and Gonek [ MR 90h:11077]

where , uniformly for . This formula is used in Levinson's method [ MR 58 #27837], and is known to be valid [ MR 90g:11120] for . Showing that the formula is valid for large is key to having good results.

Farmer [ MR 95a:11076] conjectured that the above formula should remain valid for all . This leads to a conjecture for an integral involving ratios of zeta-functions, which implies the pair-correlation conjecture. See the original paper [ MR 95a:11076] for some additional consequences.

It would be a significant result to prove that the above formula holds for some . Establishing it for would prove that more than half of the zeros of the -function are on the -line. Proving that it holds for all is more-or-less equivalent to the GUE conjecture, because it can be deduced from the formulas for ratios of zeta-functions.

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