  Multiple zeta values are real numbers defined by an infinite series
generalizing values of the Riemann zeta function at positive integers. Finite
truncations of this series are called multiple harmonic sums and are known to
have interesting arithmetic properties. When the truncation point is one less
than a prime $p$, the mod $p$ values of multiple harmonic sums are called
finite multiple zeta values. The present work introduces a new class of
congruence for multiple harmonic sums, which we call weighted congruences.
These congruences can hold modulo arbitrarily large powers of $p$. Unlike
results for finite multiple zeta values, weighted congruences typically involve
harmonic sums of multiple weights, which are multiplied by explicit powers of
$p$ depending on weight. We also introduce certain formal weighted congruences
inolving an infinite number of terms, which we call asymptotic relations. We
define a weighted analogue of the finite multiple zeta function, and give an
algebraic framework for classifying weighted congruences and asymptotic
relations.
