Shintani's celebrated invariants are conjectured to generate abelian extensions of real quadratic number fields, offering a potential solution to Hilbert's 12th problem in that setting. In this note, we derive new expressions for Shintani's invariants by generalizing an observation of Yamamoto, who showed that these invariants - originally formulated using the double sine function - can be expressed in terms of the q-Pochhammer symbol.

Let $\Omega$ be a nonempty set and let $\mathcal{A}$ be a $\sigma$-ring of subsets of $\Omega$ containing all singletons. Denote by $L(\mathcal{A})$ the Riesz space of real-valued $\mathcal{A}$-measurable functions on $\Omega$. This space is locally convex-solid when equipped with the topology $\tau_p$ of pointwise convergence on $\Omega$. We present several conditions equivalent to the nonexistence of other Hausdorff locally solid [respectively, locally convex-solid] topologies on $L(\mathcal{A})$. Some of them assert the nonexistence of $\sigma$-order continuous submeasures or two-valued measures on $\mathcal{A}$ vanishing on singletons.

Let $\overline{R_{\ell}^{*}}$ denote the number of overpartitions of $n$ where nonoverlined parts are $\ell$-regular and overlined parts are unrestricted. In this paper, we establish several Ramanujan like congruences and infinite families of congruences for $\overline{R_{\ell}^{*}}$ for $\ell=2k, 5, 6$ and $8k$. For example, for all $n\geq 0$ $$\overline{R_{184}^{*}}(16n+15)\equiv 0\pmod{1472}.$$

We prove an exact formula for the expected density of distribution of complex level crossings of random polynomials spanned by orthogonal polynomials on the unit circle and unit disk. We then inquire into the consequences of the asymptotical evaluations.

In 1977 Pohst conjectured a certain inequality for $n$ variables and gave a computer-assisted proof for $n\leq 10$. We give a proof for all $n$ using a combinatorial argument. This inequality yields a better bound for the regulator in terms of the discriminant for totally real number fields.

As is well known, we can average the eigenfunction y s of the hyperbolic Laplacian on the hyperbolic plane by Γ a lattice in SL(2, R) to obtain an automorphic form, the non-holomorphic Eisenstein series Ea(z, s). In this note, we choose a particular eigenfunction y s dx of the Hodge-Laplace operator for 1-forms on the hyperbolic plane. Then, we average by Γ to define a 1-form E 1 a (z, v), s . We see that E 1 a admits a Fourier expansion and calculates the corresponding coefficients. Also, we evaluate the integral γ E 1 a for when γ is a lifting of horocycles and closed geodesics in the unit tangent bundle. Finally, we will obtain an analog to the Rankin-Selberg method for E 1 a .