International Journal of Number Theory Vol. 9, No. 5 (2013) 1289–1299 c  World Scientific Publishing Company DOI: 10.1142/S1793042113500279 ANOTHER LOOK AT IWASAWA λ-INVARIANTS OF p-ADIC MEASURES ON Z n p AND Γ-TRANSFORMS RUPAM BARMAN Department of Mathematical Sciences, Tezpur University Napaam-784028, Sonitpur, Assam, India rupamb@tezu.ernet.in Received 21 August 2012 Accepted 5 February 2013 Published 9 May 2013 In [Iwasawa λ-invariants of p-adic measures on Z n p and their Γ-transforms, J. Number Theory 132(10) (2012) 2258–2266; Iwasawa λ-invariants and Γ-transforms of p-adic mea- sure on Z n p , Int. J. Number Theory 6(8) (2010) 1819–1829], the author and Saikia defined Iwasawa λ-invariants for multi-variable power series and proved a relation between the Iwasawa λ-invariant of a p-adic measure on Z n p and its Γ-transform. In this paper, we give two new definitions of λ-invariants for multi-variable power series which generalize the λ-invariant of single variable power series and prove that the new λ-invariants also satisfy the same relation. We also give algebraic interpretations of the new λ-invariants and discuss an application of our main results. Keywords : p-Adic measure; Γ-transform; Iwasawa invariants; Mahler coefficients. Mathematics Subject Classification 2010: 11F85, 11S80 1. Introduction The theory of Γ-transform plays an important role in the study of Iwasawa invariants of abelian number fields. For example, Sinnott [7] proved that to compute the μ-invariant of a function that can be expressed as Γ-transform of a power series, it is enough to know the μ-invariant of the series. Using this he gave an elegant new proof of the theorem of Ferrero and Washington that the Iwasawa μ-invariant is zero for the cyclotomic Z p -extension of any abelian number field. Following his approach, Rosenberg also found a new proof of the same theorem. In addition, he gave an upper bound on the corresponding λ-invariant (see [6]). In [4], Katz showed that the p-adic L-functions of a totally real number field K do indeed arise from roughly rational function measures on Z d p , where d =[K : Q]. Motivated by the work of Katz, we studied Iwasawa invariants of p-adic measures on Z n p and their Γ-transforms for any n ≥ 1 (see [1, 3]). We now recall some preliminaries from [5, Chap. 4]. Let p be a fixed odd prime. Let Z p denote the ring of p-adic integers, Q p the field of p-adic numbers, and Z × p the 1289