In this paper we shall make complete structural elucidation of the explicit formula for the (discrete) mean square of Dirichlet $L$-function at integral arguments, save for the case $s=1$, this being completely settled in [1] recently. We shall treat the cases of negative and positive integers arguments separately, the former case being a preliminary and inclusive in the second. It will turn out that in respective cases the characteristic difference properties of Bernoulli polynomials and of the Hurwitz zeta-function are essential and telescoping the resulting difference equations, we obtain the results, revealing the underlying simple structure (known before 1905 at least).

The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a sketch of a proof of this conjecture.

We give a description of the dual of W. Singer’s algebraic transfer in the May spectral sequence and use this description to prove new results on the image of the algebraic transfer in higher homological degrees.