In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

Riemann's explicit formula

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]}

${\displaystyle \pi _{0}(x)={\frac {1}{2}}\lim _{h\to 0}\left\,,}$

which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities.^[a] His formula was given in terms of the related function

${\displaystyle f(x)=\pi _{0}(x)+{\frac {1}{2}}\,\pi _{0}(x^{1/2})+{\frac {1}{3}}\,\pi _{0}(x^{1/3})+\cdots }$

in which a prime power pⁿ counts as 1⁄n of a prime. The normalized prime-counting function can be recovered from this function by

^[1]${\displaystyle \pi _{0}(x)=\sum _{n}{\frac {1}{n}}\,\mu (n)\,f(x^{1/n})=f(x)-{\frac {1}{2}}\,f(x^{1/2})-{\frac {1}{3}}\,f(x^{1/3})-{\frac {1}{5}}\,f(x^{1/5})+{\frac {1}{6}}\,f(x^{1/6})-\cdots ,}$

where μ(n) is the Möbius function. Riemann's formula is then

${\displaystyle f(x)=\operatorname {li} (x)-\sum _{\rho }\operatorname {li} (x^{\rho })-\log(2)+\int _{x}^{\infty }{\frac {dt}{~t\,(t^{2}-1)~\log(t)~}}}$

involving a sum over the non-trivial zeros ρ of the Riemann zeta function. The sum is not absolutely convergent, but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function li occurring in the first term is the (unoffset) logarithmic integral function given by the Cauchy principal value of the divergent integral

${\displaystyle \operatorname {li} (x)=\int _{0}^{x}{\frac {dt}{\,\log(t)\,}}\,.}$

The terms li(x^ρ) 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. The other terms also correspond to zeros: The dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros. This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their "expected" positions. (For graphs of the sums of the first few terms of this series see Zagier 1977.)

The first rigorous proof of the aforementioned formula was given by von Mangoldt in 1895: it started with a proof of the following formula for the Chebyshev's function ψ ^[2]

${\displaystyle \psi _{0}(x)={\dfrac {1}{2\pi i}}\int _{\sigma -i\infty }^{\sigma +i\infty }\left(-{\dfrac {\zeta '(s)}{\zeta (s)}}\right){\dfrac {x^{s}}{s}}\,ds=x-\sum _{\rho }{\frac {~x^{\rho }\,}{\rho }}-\log(2\pi )-{\dfrac {1}{2}}\log(1-x^{-2})}$

where the LHS is an inverse Mellin transform with

${\displaystyle \sigma >1\,,\quad \psi (x)=\sum _{p^{k}\leq x}\log p\,,\quad {\text{and}}\quad \psi _{0}(x)={\frac {1}{2}}\lim _{h\to 0}(\psi (x+h)+\psi (x-h))}$

and the RHS is obtained from the residue theorem, and then converting it into the formula that Riemann himself actually sketched.

This series is also conditionally convergent and the sum over zeroes should again be taken in increasing order of imaginary part:^[3]

${\displaystyle \sum _{\rho }{\frac {x^{\rho }}{\rho }}=\lim _{T\to \infty }S(x,T)}$   where   ${\displaystyle S(x,T)=\sum _{\rho :\left|\Im \rho \right|\leq T}{\frac {x^{\rho }}{\rho }}\,.}$

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.^[4]

Weil's explicit formula

There are several slightly different ways to state the explicit formula.^[5] André Weil's form of the explicit formula states

Zdroj:https://en.wikipedia.org?pojem=Weil's_explicit_formula
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