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.

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
• 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