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Petersson Conjecture

Petersson considered the absolutely converging Dirichlet L-series

phi(s)=product_(p)1/(1-c(p)p^(-s)+p^(2k-1)p^(-2s)).
(1)

Writing the denominator as

1-c(p)x+p^(2k-1)x^2=(1-r_1x)(1-r_2x),
(2)

where

r_1+r_2=c(p)
(3)

and

r_1r_2=p^(2k-1),
(4)

Petersson conjectured that r_1 and r_2 are always complex conjugate, which implies

|r_1|=|r_2|=p^(k-1/2)
(5)

and

|c(p)|<=2p^(k-1/2).
(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.

See also

Dirichlet L-Series, Tau Conjecture

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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.

Referenced on Wolfram|Alpha

Petersson Conjecture

Cite this as:

Weisstein, Eric W. "Petersson Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PeterssonConjecture.html

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