A Generalization of Wolstenholme’s Harmonic Series Congruence

Multiple Harmonic Sums I: Generalizations of Wolstenholme’s Theorem

for complex variables s1, . . . , sl satisfying Re(sj) + · · · + Re(sl) > l − j + 1 for all j = 1, . . . , l. We call |s| = s1 + · · · + sl the weight and denote the length by l(s). The special values of multiple zeta functions at positive integers have significant arithmetic and algebraic meanings, whose defining series (1) will be called MZV series, including the divergent ones like ζ(. . . ,...

A Binomial Sum Related to Wolstenholme’s Theorem

A certain alternating sum u(n) of n + 1 products of two binomial coefficients has a property similar to Wolstenholme’s theorem, namely u(p) ≡ −1 (mod p3) for all primes p ≥ 5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in qu...

Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme’s Theorem

We show that the set of composite positive integers n ≤ x satisfying the congruence (2n−1 n−1 ) ≡ 1 (mod n) is of cardinality at most x exp ( −(1/ √ 2 + o(1)) √ log x log log x ) as x →∞.

Mod p structure of alternating and non-alternating multiple harmonic sums

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the (p−1)-st partial sum of the harmonic series is congruent to 0 modulo p2. If one replaces the harmonic series by ∑ k≥1 1/n for k even, then the modulus has to be changed from p2 to just p. One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partia...

A q-Analogue of Wolstenholme's Harmonic Series Congruence