Riemann Hypothesis - Zeros On The Critical Line

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:

    Somewhere it is written that parents who are critical of other people’s children and publicly admit they can do better are asking for it.
    Erma Bombeck (20th century)

    There’s a line between love and fascination that’s hard to see on an evening such as this.
    Ned Washington (1901–1976)