Explicit formulae for L-functions
In his 1859 paper "On the Number of Primes Less Than a Given Magnitude" Riemann sketched an explicit formula (it was not fully proven until 1895 by von Mangoldt, see below) for the normalized prime-counting function π0(x) which is related to the prime-counting function π(x) by[citation needed] which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities.The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral The terms li(xρ) involving the zeros of the zeta function need some care in their definition as li has branch points at 0 and 1, and are defined by analytic continuation in the complex variable ρ in the region x > 1 and Re(ρ) > 0.This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions.This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part:[3] The error involved in truncating the sum to S(x,T) is always smaller than ln(x) in absolute value, and when divided by the natural logarithm of x, has absolute value smaller than x⁄T divided by the distance from x to the nearest prime power.So that the Fourier transform of the non trivial zeros is equal to the primes power symmetrized plus a minor term.Of course, the sum involved are not convergent, but the trick is to use the unitary property of Fourier transform which is that it preserves scalar product: whereAccording to the Hilbert–Pólya conjecture, the complex zeroes ρ should be the eigenvalues of some linear operator T. The sum over the zeros of the explicit formula is then (at least formally) given by a trace: Development of the explicit formulae for a wide class of L-functions was given by Weil (1952), who first extended the idea to local zeta-functions, and formulated a version of a generalized Riemann hypothesis in this setting, as a positivity statement for a generalized function on a topological group.More recent work by Alain Connes has gone much further into the functional-analytic background, providing a trace formula the validity of which is equivalent to such a generalized Riemann hypothesis.A slightly different point of view was given by Meyer (2005), who derived the explicit formula of Weil via harmonic analysis on adelic spaces.