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.
Explicit formula for the Riemann zeta function
There are several slightly different ways to state the explicit formula. Weil's form of the explicit formula states
where
- ρ runs over the non-trivial zeros of the zeta function
- p runs over positive primes
- m runs over positive integers
- F is a smooth function all of whose derivatives are rapidly decreasing
- φ is a Fourier transform of F:
- Φ(1/2 + it) = φ(t)
- Ψ(t) = -log(π) + Re(ψ(1/4 + it/2)), where ψ is the digamma function Γ′/Γ.
Roughly speaking, the explicit formula says the Fourier transform of the zeros of the zeta function is the set of prime powers plus some elementary factors.
The terms in the formula aries in the following way.
- The terms on the right hand side come from the logarithmic derivative of
- with the terms corresponding to the prime p coming from the Euler factor of p, and the term at the end involving Ψ coming from the gamma factor (the Euler factor at infinity).
- The left hand side is a sum over all zeros of ζ* counted with multiplicities, so thepoles at 0 and 1 are counted as zeros of order −1.
Generalizations
The Riemann zeta function can be replaced by a Dirichlet L-function of a Dirichlet character χ. The sum over prime powers then gets extra factors of χ(pm), and the terms Φ(0) and Φ(0) disappear because the L-series has no poles.
More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke L-series. The sum over primes then gets replaced by a sum over prime ideals.
Applications
Riemann's original use of the explicit formula way to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y1/2/log(y) for 0≤y≤x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x. The main term on the left if Φ(1); which turns out to be the dominant terms of the prime number theorem, and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function F does not satisfy the smoothness condition.)
Hilbert-Pólya conjecture
According 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.
See also
References
- Ingham, A.E. (1990), The Distribution of Prime Numbers, Cambridge University Press, ISBN 978-0-521-39789-6, MR 1074573 (Reissued with a foreword by R. C. Vaughan in 1990)
- Algebraic Number Theory, Serge Lang
- Riemann, Bernhard (1859), "Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse", Monatsberichte der Berliner Akademie
- Weil, André (1952), "Sur les "formules explicites" de la théorie des nombres premiers", Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 1952: 252–265, MR 0053152