Recently Montgomery and Odlyzko [1] showed, assuming the Riemann Hypothesis, that infinitely often consecutive zeros of the zeta function differ by at least 1.9799 times the average spacing and infinitely often they differ by at most 0.5179 times the average spacing. We improve their result by a better choice of a certain weight function which occurs in their proof and we give explicit bounds for what is possible by this method. In addition, we indicate a method of proof which is considerably simpler than that of Montgomery and Odlyzko.LetIn logy 271 where y ^ y' are consecutive ordinates of zeros of ((s). Note that the average of y' -y is 27r/log y.THEOREM. If the Riemann Hypothesis is true, then k > 2.337 and /i < 0.5172.We indicate the modifications in [1] that lead to our theorem. Let n. k nk $ K where 2sin(7rc(logn)/logr) = n log n so that |gf(n)| ^ --and K = V~A for some small 6 > 0. By (20) and (21) of [1], if logT h(c) < 1 for some choice of c, a ki and S > 0, then A ^ c, and if h(c) > 1 for some choice of c, a k , and S > 0, then ix ^ c.To obtain the bound for k we choose a k = d r (k)k~1 12 where d r is multiplicative and for a prime p,
ABSTRACT. This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the L 2 -restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex L ∞ -bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.
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