Petersson Conjecture

Petersson considered the absolutely converging Dirichlet L-series

(1)

Writing the denominator as

(2)

where

(3)

and

(4)

Petersson conjectured that  and  are always complex conjugate, which implies

(5)

and

(6)

This conjecture was proven by Deligne (1974), which also proved the tau conjecture as a special case. Deligne was awarded the Fields medal for his proof.

REFERENCES:

Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 140, 1997.

Deligne, P. "La conjecture de Weil. I." Inst. Hautes Études Sci. Publ. Math. 43, 273-307, 1974.

Deligne, P. "La conjecture de Weil. II." Inst. Hautes Études Sci. Publ. Math. 52, 137-252, 1980.