Petersson Conjecture
المؤلف:
Apostol, T. M.
المصدر:
Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
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22-12-2019
1390
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.
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