A p-ADIC VIEW OF MULTIPLE SUMS OF POWERS

Abstract

Washington [p-Adic L-functions and sums of powers, J. Number Theory 69 (1998) 50-61] gave an explicit p-adic expansion of Sigma(mp)(j=1p(sic)) 1/j(r) as a power series in m. The coefficients are values of p-adic L-functions. Let q = 4 if p = 2 and q = p otherwise. In this paper, we prove an explicit p-adic expansion of the multiple sums of powers Sigma(j1, ..., jn= 0) (mp-1)(p(sic)(j1+ ... +jn)) 1/(qt + j1 + ... + jn) (r+ n-1) as a p-adic power series in m. The coefficients are values of multiple two-variable p-adic L-functions. Washington's formula is a special case of the formula given in this paper when n = 1 and t = 0.