On P-adic Properties Of Generalized Harmonic Series

Let i and r be positive integers,the nth partial sum of the generalized har-monic series is defined byHn（r）=（?）.Let p be a prime number and v_pbe a p-adic exponential valuation,we define the p-divisible set J_p（r）={n>0|v_p（H_n（r））>0}.The goal of this article is to discuss some basic properties of the general-ized harmonic series,including the exponential valuations,congruence relation-s,p-divisible sets and some related conjectures.The Wolstenholme’s theorem asserts that（?）≡1 mod p³.for p>3.If this congruence holds modulo p⁴,then p is called a Wolstenholme prime.Further,there is a close link between the generalized harmonic series and Wolstenholme type congruences.So the Wolstenholme theorem,Wolstenholme prime and the converse of Wolstenholme theorem will be introduced in detail.In this article,first we study the relation between v_p(H_npk（r）)and v_p（H_n（r））.Second,we obtain a congruence equation modulo p¹²involving H_p-1（r）.Third,for p-divisible set J₁₁（r）,we can assure it is finite for all 1≤r≤3×10⁸,moreover,ifr（?）1 or 3 mod 10,J₁₁（r）is also finite.Fourth,we get a congruence for_?modulo p¹²,and we provide a family of integers,for which the converse of Wol-stenholme theorem holds.