|
In [[mathematics]], the '''[[Closed-form expression|explicit formulae]] for [[L-function]]s''' are relations between sums over the [[complex number]] zeroes of an [[L-function]] and sums over [[Prime number|prime powers]], introduced by {{harvtxt|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 [[Hans Carl Friedrich von Mangoldt|von Mangoldt]], see below) for the normalized prime-counting function {{math|π<sub>0</sub>(''x'')}} which is related to the [[prime-counting function]] {{math|π(''x'')}} by<ref>{{Cite web |title=Explicit Formulae (L-function) |url=https://encyclopedia.pub/entry/32287 |access-date=2023-06-14 |website=encyclopedia.pub |language=en}}</ref>
:<math>\pi_0(x) = \frac{1}{2} \lim_{h\to 0} \left[\,\pi(x+h) + \pi(x-h)\,\right]\,,</math>
which takes the [[arithmetic mean]] of the limit from the left and the limit from the right at discontinuities.{{efn|The original prime counting function can easily be recovered via <math>~\pi(x) = \pi_0(x+1)~</math> for all <math>~x \ge 3~.</math>}} His formula was given in terms of the related function
:<math>f(x) = \pi_0(x) + \frac{1}{2}\,\pi_0(x^{1/2}) + \frac{1}{3}\,\pi_0(x^{1/3}) + \cdots</math>
in which a prime power {{math|''p''<sup>''n''</sup>}} counts as {{frac|1|{{mvar|n}}}} of a prime. The normalized [[prime-counting function]] can be recovered from this function by
:<ref>{{Cite journal |last=Li |first=Xian-Jin |date=April 2004 |title=Explicit formulas for Dirichlet and Hecke $L$-functions |journal=Illinois Journal of Mathematics |volume=48 |issue=2 |pages=491–503 |doi=10.1215/ijm/1258138394 |issn=0019-2082|doi-access=free }}</ref><math>\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,</math>
where {{math|''μ''(''n'')}} is the [[Möbius function]]. Riemann's formula is then
:<math>f(x) = \operatorname{li}(x) - \sum_\rho \operatorname{li}(x^\rho) - \log(2) + \int_x^\infty \frac{dt}{~t\,(t^2-1)~\log(t)~}</math>
involving a sum over the non-trivial zeros {{mvar|ρ}} of the Riemann zeta function. The sum is not [[Absolute convergence|absolutely convergent]], but may be evaluated by taking the zeros in order of the absolute value of their imaginary part. The function {{math|li}} occurring in the first term is the (unoffset) [[logarithmic integral function]] given by the [[Cauchy principal value]] of the divergent integral
:<math>\operatorname{li}(x) = \int_0^x \frac{dt}{\,\log(t)\,}\,.</math>
The terms {{math|li(''x''<sup>''ρ''</sup>)}} involving the zeros of the zeta function need some care in their definition as {{math|li}} has [[branch point]]s at 0 and 1, and are defined by [[analytic continuation]] in the complex variable {{mvar|ρ}} in the region {{math|''x'' > 1}} and {{math|Re(''ρ'') > 0}}. The other terms also correspond to zeros: The dominant term {{math|li(''x'')}} comes from the pole at {{math|''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 {{harvnb|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]] {{mvar|ψ}} <ref>Weisstein, Eric W. [http://mathworld.wolfram.com/ExplicitFormula.html Explicit Formula] on MathWorld.</ref>
:<math>\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})</math>
where the LHS is an inverse Mellin transform with
:<math>\sigma > 1\,, \quad \psi(x) = \sum_{p^k \le x} \log p\,,
\quad \text{and} \quad \psi_0(x) = \frac{1}{2} \lim_{h\to 0} (\psi(x+h) + \psi(x-h))</math>
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:<ref name=Ing77>Ingham (1990) p.77</ref>
:<math>\sum_\rho\frac{x^\rho}{\rho} = \lim_{T \to \infty} S(x,T) </math> {{pad|2em}} where {{pad|2em}} <math> S(x,T) = \sum_{\rho:\left|\Im \rho\right| \le T} \frac{x^\rho}{\rho}\,.</math>
The error involved in truncating the sum to {{math|''S''(''x'',''T'')}} is always smaller than {{math|ln(''x'')}} in absolute value, and when divided by the [[natural logarithm]] of {{mvar|x}}, has absolute value smaller than {{math|{{frac|''x''|''T''}}}} divided by the distance from {{mvar|x}} to the nearest prime power.<ref>[https://math.stackexchange.com/q/497949 Confused about the explicit formula for ψ0(x)]</ref>
==Weil's explicit formula ==
| 