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:
“Productive collaborations between family and school, therefore, will demand that parents and teachers recognize the critical importance of each other’s participation in the life of the child. This mutuality of knowledge, understanding, and empathy comes not only with a recognition of the child as the central purpose for the collaboration but also with a recognition of the need to maintain roles and relationships with children that are comprehensive, dynamic, and differentiated.”
—Sara Lawrence Lightfoot (20th century)
“This wild night, gathering the washing as if it were flowers
animal vines twisting over the line and
slapping my face lightly, soundless merriment
in the gesticulations of shirtsleeves ...”
—Denise Levertov (b. 1923)