of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place. An equivalent relationship had been conjectured by Euler over a hundred years earlier, in 1749, for the Dirichlet eta function (the alternating zeta function): Incidentally, this relation gives an equation for calculating in the region 0 < < 1, i.e. where the η-series is convergent (albeit non-absolutely) in the larger half-plane (for a more detailed survey on the history of the functional equation, see e.g. Blagouchine). Riemann also found a symmetric version of the functional equation applying to the xi-function: which satisfies: (Riemann's original was slightly different.) The factor was not well-understood at the time of Riemann, until John Tate's (1950) thesis, in which it was shown that this so-called "Gamma factor" is in fact the local L-factor corresponding to the Archimedean place, the other factors in the Euler product expansion being the local L-factors of the non-Archimedean places. Zeros, the critical line, and the Riemann hypothesis The functional equation shows that the Riemann zeta function has zeros at . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The set is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line. For the Riemann zeta function on the critical line, see -function. Number of zeros in the critical strip Let be the number of zeros of in the critical strip , whose imaginary parts are in the interval . Trudgian proved that, if , then . The Hardy–Littlewood conjectures In 1914, Godfrey Harold Hardy proved that has infinitely many real zeros. Hardy and John Edensor Littlewood formulated two conjectures on the density and distance between the zeros of on intervals of large positive real numbers. In the following, is the