2019/02/14 by René Gy, Gy, René
Mathematics · #11A07 (Primary) #11B68 (Secondary) #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A07 #msc:11B68
paper · pdf · doi:10.48550/arxiv.1902.05258
32 pages, 0 figure
arxiv created 2019/02/14 · openalex publication_date 2019/02/14 · arxiv updated 2019/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We derive p-adic expansions for the generalized Harmonic numbers H(j)p-1 and H(j)(p-1)/(2) involving the Bernoulli numbers Bj and the the base-2 Fermat quotient qp. While most of our results are not new, we obtain them elementarily, without resorting to the theory of p-adic L-functions as was the case previously. Moreover, we show that ∑j=0n-1(\frac(2j+1-1)(j+1)\frac(2j+2-1)(j+2)\fracBj+22jH(j+1)(p-1)/(2)+2(-1)j\fracqpj+1j+1)pj≡ 0 \pmod pn holds under the condition that p >(n+1)/(2). This is another generalization, modulo any prime power, of the old p-congruence H(p-1)/(2)+2qp ≡ 0 \bmod p attributed to Eisenstein, which is stronger than the one which has been published recently.