\(p\)-adic \(L\)-functions and sums of powers
Let \(p\) be a prime. The author studies the sums \(\sum_j j^{-r}\), where \(r\) is an integer and \(j\) runs over the range \(1,\dots,np-1\), with \(n \geq 1\), excluding the multiples of \(p\). He expands these sums explicitly as \(p\)-adic power series in \(n\). The coefficients of the power series are values of \(p\)-adic \(L\)-functions \(L_p(s,\chi)\), where \(\chi\) is a power of the Teichmüller character. The case of actual interest is that with \(r \geq 1\); otherwise the result is a simple variation of the classical formula for sums of powers in terms of Bernoulli polynomials. Several congruences, both classical and novel, for \(\sum_{j=1}^{p-1} j^{-r}\) follow as corollaries. The argument is a nice application of the ``explicit'' expression for \(L_p(s,\chi)\) due to the author. In fact, this argument allows the author to generalize the results to sums of the type \(\sum_j \chi (j)j^{-r}\), where \(\chi\) is any Dirichlet character and \(np\) is replaced by \(nF\), with \(F\) a multiple of \(p\) and the conductor of \(\chi\). This work was inspired by a result of \textit{D. Boyd} [Exp. Math. 3, No. 4, 287-302 (1994; Zbl 0838.11015)] dealing with the question of when \(p\) divides the partial sums of the harmonic series. The present paper concludes with a heuristic discussion about the corresponding question concerning \(\sum_{j=1}^\infty j^{-r}\) with \(r \geq 2\).
partial sums of generalized harmonic series
scientific article; zbMATH DE number 1207952
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