Explicit formulae for L-functions

In mathematics, the explicit formulae for L-functions relate sums taken over the complex number zeroes of a given L-function to sums over prime powers. The first case was found by Riemann for the Riemann zeta function, where sums over its complex zeroes are identified with other sums over prime numbers. 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

${\displaystyle \Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )=\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}(F(\log(p^{m})+F(-\log(p^{m}))-{\frac {1}{2\pi }}\int _{-\infty }^{\infty }\phi (t)\Psi (t)dt}$ 

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
• ψ is a Fourier transform of F: ${\displaystyle \psi (t)=\int _{-\infty }^{\infty }F(x)e^{itx}dx}$ 
• 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

${\displaystyle \zeta ^{*}(s)=\Gamma (s/2)\pi ^{-s/2}\prod _{p}{\frac {1}{1-p^{-s}}}}$ 

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.

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 y^1/2/log(y) for 0≤y≤x. 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

In terms suggested by the Hilbert-Pólya conjecture, one of the major heuristics underlying the Riemann hypothesis and its supposed connection with functional analysis, the complex zeroes ρ should be closely linked to the eigenvalues of some linear operator T. A sum

${\displaystyle \sum =\sum _{\rho }F(\rho ),\!}$ 

would then have this interpretation: use the functional calculus of operators, supposed to apply to T, to form

${\displaystyle F({\widehat {T}}),\!}$ 

and then take its trace. In a formal sense, ignoring all the difficult points of mathematical analysis involved, this will be Σ. Therefore the existence of such 'trace formulae' for T means that the explicit formulae essentially encode the nature of T, from the point of view of spectral theory, at least as far as its eigenvalues (spectrum) is concerned.

For the case the Spectrum is just the one belonging to a Hamiltonian H , the semiclassical approach can give a definition of the sum by means of an integral of the form:

${\displaystyle \Sigma =\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-aH(x,p)}\ dx\ dp,\!}$ 

taking our operator ${\displaystyle \scriptstyle F({\widehat {T}})\!}$  to be ${\displaystyle e^{-\scriptstyle {\widehat {H}}}\!}$  valid when a is small and positive or pure imaginary.

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

• Selberg trace formula

References

• Ingham, A.E. (1990), The Distribution of Prime Numbers, Cambridge University Press, ISBN 978-0-521-39789-6, MR 1074573
• 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