Zeros On The Critical Line
Hardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, by considering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small) positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating the zeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths.
Most zeros lie close to the critical line. More precisely, Bohr & Landau (1914) showed that for any positive ε, all but an infinitely small proportion of zeros lie within a distance ε of the critical line. Ivić (1985) gives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least 1/2+ε.
Read more about this topic: Riemann Hypothesis
Famous quotes containing the words critical and/or line:
“The critical spirit never knows when to stop meddling.”
—Mason Cooley (b. 1927)
“What is line? It is life. A line must live at each point along its course in such a way that the artist’s presence makes itself felt above that of the model.... With the writer, line takes precedence over form and content. It runs through the words he assembles. It strikes a continuous note unperceived by ear or eye. It is, in a way, the soul’s style, and if the line ceases to have a life of its own, if it only describes an arabesque, the soul is missing and the writing dies.”
—Jean Cocteau (1889–1963)