Zeros of Ramanujan-type Polynomials

Ramanujan's notebooks contain many elegant identities and one of the celebrated identities is a formula for $\zeta(2k+1)$. In 1972, Grosswald gave an extension of the Ramanujan's formula for $\zeta(2k+1)$, which contains a polynomial of degree $2k+2$. This polynomial is now well-known as the Ramanujan polynomial$R_{2k+1}(z)$, first studied by Gun, Murty, and Rath. Around the same time, Murty, Smith and Wang proved that all the non-real zeros of $R_{2k+1}(z)$ lie on the unit circle. Recently, Chourasiya, Jamal, and the first author found a new polynomial while obtaining a Ramanujan-type formula for Dirichlet $L$-functions and named it as Ramanujan-type polynomial $R_{2k+1,p}(z)$. In the same paper, they conjectured that all the non-real zeros of $R_{2k+1,p}(z)$ lie on the circle $|z|=1/p$. The main goal of this paper is to present a proof of this conjecture.Comment: 14 pages, comments are welcome

Magnetic and electrical properties of RCo2Mn (R=Ho, Er) compounds

The RCo2Mn (R= Ho and Er) alloys, crystallizing in the cubic MgCu2-type structure, are isostructural to RCo2 compounds. The excess Mn occupies both the R and the Co atomic positions. Magnetic, electrical and heat capacity measurements have been done in these comounds. The Curie temperature is found to be 248 K and 222 K for HoCo2Mn and ErCo2Mn respectively, which are considerably higher than that of the corresponding RCo2 compounds. Saturation magnetization values calculated in these samples are less compared to that of the corresponding RCo2 compounds. Heat capacity data have been fitted with the nonmagnetic contribution with Debye temperature =250 K and electronic coefficient=26 mJ mol^-1K^-2.Comment: 13 pages, 5 figures, 2 table

Bressoud-Subbarao type weighted partition identities for a generalized divisor function

In 1984, Bressoud and Subbarao obtained an interesting weighted partition identity for a generalized divisor function, by means of combinatorial arguments. Recently, the last three named authors found an analytic proof of the aforementioned identity of Bressoud and Subbarao starting from a $q$-series identity of Ramanujan. In the present paper, we revisit the combinatorial arguments of Bressoud and Subbarao, and derive a more general weighted partition identity. Furthermore, with the help of a fractional differential operator, we establish a few more Bressoud-Subbarao type weighted partition identities beginning from an identity of Andrews, Garvan and Liang. We also found a one-variable generalization of an identity of Uchimura related to Bell polynomials.Comment: 18 pages, Comments are welcome

A Dirichlet character analogue of Ramanujan's formula for odd zeta values

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series, $$\sum_{n=1}^{\infty} \frac{n^{N-2h} }{\exp(n^N x)-1},$$ for $N \in \mathbb{N}$ and $h\in \mathbb{Z}$ with some restriction on $h$. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for $\zeta(2m+1)$ while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely, $$\sum_{r=1}^{q}\sum_{n=1}^{\infty} \frac{\chi(r)n^{N-2h}{\exp\left(-\frac{r}{q}n^N x\right)}}{1-\exp({-n^N x})},$$ where $\chi$ denotes a Dirichlet character modulo $q$, $N\in 2\mathbb{N}$ and with some restriction on the variable $h$. In the current paper, we investigate the above series for {\it any} $N \in \mathbb{N}$ and $h \in \mathbb{Z}$. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for $\zeta(2m+1)$. Moreover, we establish a new identity for $L(1/3, \chi)$ analogous to Ramanujan's famous identity for $\zeta(1/2)$.Comment: 24 pages, comments are welcome