Can one use random matrix theory to predict the horizontal distribution of the real parts of the zeros of ? It is known that the Riemann Hypothesis is equivalent to the assertion that each non-real zero of has real part greater than or equal to 1/2. Moreover, if such a zero of has real part 1/2, then it is also a zero of (and so a multiple zero of ). These assertions are the point of departure for Levinson's work on zeros of the Riemann zeta-function on the critical line [MR 54 #5135]. It would be interesting to know the horizontal distribution of these zeros; in particular what proportion of them with ordinates between and are within of the 1/2-line? See the paper [ MR 98k:11119] of Soundararajan for the best theoretical result in this direction.
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