Mahler’s measure: proof of two conjectured formulae. (English) Zbl 1174.11086
Summary: In this note we prove the two formulae conjectured by D. W. Boyd [Exp. Math. 7, No. 1, 37–82 (1998; Zbl 0932.11069)],
\[
m(y^2 (x+1)^2 +y(x^2 +6x+1)+(x+1)^2 )=\frac83 L'({\chi}_{-4} ,-1),
\]
\[ m(y^2 (x+1)^2 +y(x^2 -10x+1)+(x+1)^2)=\frac{20}3 L'({\chi}_{-3} ,-1), \] where \(m\) denotes the logarithmic Mahler measure for two-variable polynomials.
\[ m(y^2 (x+1)^2 +y(x^2 -10x+1)+(x+1)^2)=\frac{20}3 L'({\chi}_{-3} ,-1), \] where \(m\) denotes the logarithmic Mahler measure for two-variable polynomials.
MSC:
| 11R06 | PV-numbers and generalizations; other special algebraic numbers; Mahler measure |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
