Chowla–Selberg formula

In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the Gamma function at rational values. The name comes from a 1967 joint paper of Chowla and Selberg.^[1] The basic result was already in much earlier work of the Czech mathematician Mathias Lerch.^[2][3]

In logarithmic form, the formula shows that in certain cases the sum

$\sum \chi(r)\log \Gamma\left( \frac{r}{D} \right)$

can be evaluated (by modular form theory). Here χ is the quadratic residue symbol modulo D, where −D is the discriminant of an imaginary quadratic field. The sum is taken over 0 < r < D, with the usual convention χ(r) = 0 if r and D have a common factor.

The origin of such formulae is now seen to be in the theory of complex multiplication, and in particular in the theory of periods of an abelian variety of CM-type. This has led to much research and generalisation. In particular the analogue for p-adic numbers, involving a p-adic gamma function, was initiated by Benedict Gross and Neal Koblitz; and is important in the theory of p-adic periods.

See also

• Multiplication theorem

Notes

1. ^ Chowla & Selberg 1967
2. ^ Lerch 1897
3. ^ Schappacher 1988, §3.2

References

• Chowla, Sarvadaman; Selberg, Atle (1967), "On Epstein's Zeta-function", Journal für die reine und angewandte Mathematik 227 (227): 86–110, doi:10.1515/crll.1967.227.86, MR0215797 
• Lerch, Mathias (1897), "Sur quelques formules relatives au nombre des classes", Bulletin des sciences mathématiques 21: 290–304 
• Schappacher, Norbert (1988), Periods of Hecke characters, Lecture Notes in Mathematics, 1301, Berlin: Springer-Verlag, MR0935127 

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