Deduction of Semi-Optimal Mollifier for Obtaining Lower Bound for N(0)(T) for Riemann's Zeta-Function.
A mollifier played a key role in showing N(0)(T) > 1/3N(T) for large T in ref. 1 [Levinson, N. (1974) Advan. Math. 13, 383-436]. A basic problem in ref. 1 was that of obtaining an upper bound for a sum of two terms, one larger than the other. Here a deductive procedure is given for finding a mollifier that actually minimizes the larger term. An Euler-Lagrange equation is obtained. (Optimization of the sum of both the major and minor terms appears to be formidable.) The actual improvement effected by the optimized mollifier over the ad hoc mollifier of ref. 1 is unfortunately only 1.4%. To obtain a usable mollifier it is necessary to blur the optimization procedure by smoothing at several stages of the deduction. The procedure is of more interest than the particular application because of the small improvement in this case.
Levinson
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Department of Mathematics, Room 2-365, Massachusetts Institute of Technology, Cambridge, Mass. 02139.