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 principle goal of education in the schools should be creating men and women who are capable of doing new things, not simply repeating what other generations have done; men and women who are creative, inventive and discoverers, who can be critical and verify, and not accept, everything they are offered.”
—Jean Piaget (1896–1980)
“The line of separation was very distinct, and the Indian immediately remarked, “I guess you and I go there,—I guess there’s room for my canoe there.” This was his common expression instead of saying “we.” He never addressed us by our names, though curious to know how they were spelled and what they meant, while we called him Polis. He had already guessed very accurately at our ages, and said that he was forty-eight.”
—Henry David Thoreau (1817–1862)