An explicit transformation for the series $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$, or equivalently, $\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re$(y)>0$, which takes y to $1/y$, is obtained for the first time. This series transforms into a series containing the derivative of $R(z)$, a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of $\psi _1(z)$ (the derivative of $R(z)$) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function $E_{2, b}(z)$ evaluated at $b=1$, all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of $\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as $y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of $ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as $\delta \to 0$.

It is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) and ζ(2Nm + 1). A result complementary to the aforementioned generalization is obtained for any even N and m ∈ ℤ. It generalizes a transformation of Wigert and can be regarded as a formula for ζ(2m + 1 − 1/N). Applications of these transformations include a generalization of the transformation for the logarithm of Dedekind eta-function η(z), Zudilin- and Rivoal-type results on transcendence of certain values, and a transcendence criterion for Euler's constant γ.

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Generating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.

We present two infinite families of generalized Lambert series identities, and deduce several known identities from them. They include an identity due to M. Jackson, a corollary of Ramanujan's $_1\psi_1$-summation formula, and a recent identity of G. E. Andrews, R. P. Lewis and Z.-G. Liu.