Higher Mahler measure for cyclotomic polynomials and Lehmer’s question

Abstract

The k-higher Mahler measure of a non-zero polynomial P is the integral of log k |P | on the unit circle. In this note, we consider Lehmer's question (which is a long-standing open problem for k = 1) for k > 1 and find some interesting formulas for 2-and 3-higher Mahler measure of cyclotomic polynomials.

Log-Sine Evaluations of Mahler Measures

“…As described in [17], it is a long-standing question due to Lehmer as to whether µ(P) can be arbitrarily close to zero for single-variable integer polynomials P. For higher Mahler measures, [17,Theorem 7] shows that for k = 1, 2, . .…”

“…It was shown in ( Can one find any natural polynomial sequences so that µ(P n , Q n ) tends to zero with n, and so generalize [17,Theorem 7]?…”

“…To evaluate µ(1 + x + y + z), we follow Boyd [11,Appendix 1] and observe, on applying Jensen's formula, that for complex constants a and b, [3] Log-sine evaluations of Mahler measures 17 Writing w = y/z, we see that µ(1 + x + y + z) = µ(1 + x + z(1 + w)) = µ(log |1 + w| ∨ log |1 + x|) (1.6)…”

The Zeta Mahler measure of $(z^n − 1)/(z − 1)$

“…In [LaSi11] Lalin and Sinha used polynomials of type P n (z) := (z n − 1)/(z − 1) with n ∈ N, n > 1, while answering analogous Lehmer's question [EvWa09] for odd indexed higher Mahler measure. Although the main purpose of their paper was not formulating a practically useful form of m 2k+1 (P n ), the expression that the authors achieved is significantly complicated.…”

Limiting value of higher Mahler measure

“…In [5] Lalin and Sinha answered Lehmer's question [3] for higher Mahler measure by finding non-trivial lower bounds for m k on Z[z] for k ≥ 2.…”