We consider Dirichlet L-functions L(s, χ) where χ is a real, non-principal character modulo q. Using Pintz's refinement of Page's theorem, we prove that for q ≥ 3 the function L(s, χ) has at most one real zero β with 1 − 1.011/ log q < β < 1.

The Stanley-Elder theorem asserts that the number of j's in the partitions of n is equal to the number of parts that appear at least j times in a given partition of n, summed over all partitions of n. In this paper, we prove that the number of partitions of n with crank > j equals to half the total number of j's in the Frobenius symbols for n.

We generalize the generating series of the Dyson ranks and M 2ranks of overpartitions to obtain k-fold variants, and give a combinatorial interpretation of each. The k-fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy's first and second Frobenius representations of overpartitions, respectively.1 This convention ensures that mirroring the diagram across its main diagonal will produce the Young tableau of another overpartition, more commonly known as conjugating the overpartition.