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:

    It would be easy ... to regard the whole of world 3 as timeless, as Plato suggested of his world of Forms or Ideas.... I propose a different view—one which, I have found, is surprisingly fruitful. I regard world 3 as being essentially the product of the human mind.... More precisely, I regard the world 3 of problems, theories, and critical arguments as one of the results of the evolution of human language, and as acting back on this evolution.
    Karl Popper (1902–1994)

    I had crossed de line of which I had so long been dreaming. I was free; but dere was no one to welcome me to de land of freedom. I was a stranger in a strange land, and my home after all was down in de old cabin quarter, wid de ole folks, and my brudders and sisters. But to dis solemn resolution I came; I was free, and dey should be free also; I would make a home for dem in de North, and de Lord helping me, I would bring dem all dere.
    Harriet Tubman (c. 1820–1913)