On the Distribution of Imaginary Parts of Zeros of the Riemann Zeta Function

There is an intimate connection between the distribution of the nontrivial zeros of the Riemann zeta function ζ(s) and the distribution of prime numbers. Critical to many prime number problems is the horizontal distribution of zeros; here the Riemann Hypothesis (RH) asserts that the zeros all have real part 1 2 . There is also much interest in studying the distribution of the imaginary parts of the zeros (the vertical distribution). For example, one expects that their consecutive spacings follow the GUE distribution from random matrix theory. Originally discovered by Montgomery [12], who studied the pair correlation of zeros of the zeta function, this phenomenon has been investigated, for higher correlations and also for more general L−functions, by a number of authors, including Odlyzko [15], Hejhal [6], Rudnick and Sarnak [17], Katz and Sarnak[9], Murty and Perelli [13], and Murty and Zaharescu [14]. Let {y} denote the fractional part of y, which can be interpreted as the image of y in the torus T = R/Z. In this paper we look at the distribution of {αγ} where α is a fixed nonzero real number and γ runs over the imaginary parts of the zeros of ζ(s). The starting point is an old formula of Landau [11], which states that for each fixed x > 1,