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:
“A third variety of drama ... begins as tragedy with scraps of fun in it ... and ends in comedy without mirth in it, the place of mirth being taken by a more or less bitter and critical irony.”
—George Bernard Shaw (1856–1950)
“I thank heaven for a man like Adolf Hitler, who built a front line of defense against the anti-Christ of Communism.”
—Frank Buchman (1878–1961)