Azizul Hoque

Pell-type equations and class number of the maximal real subfield of a cyclotomic field

We investigate the solvability of the Diophantine equation

An analogue of Wilton's formula and values of Dedekind zeta functions.

x^2-my^2=pm p

Exponents of class groups of certain imaginary quadratic fields

x2-my2=±p in integers for certain integer m and prime p. Then we apply these results to produce family of maximal real subfield of a cyclotomic field whose class number is strictly larger than 1.

On the solutions of a Lebesgue - Nagell type equation.

Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. In [28] (Proc Int Conf–Number Theory 1, 107–116, 2004), Murty and Pacelli gave an instructive proof of the quadratic reciprocity via the theta transformation formula and Gerver (Amer J Math 92, 33–55, 1970) [12] was the first to give a proof of differentiability/non-differentiability of Riemann’s function. The aim here is to survey some of the work done in these two directions and concentrates more onto a recent work of the first author along with Kanemitsu and Li (Res Number Theory 1, 14, 2015) [5]. In that work (Kanemitsu and Li, Res Number Theory 1, 14, 2015) [5], an integrated form of the theta function was utilised and the advantage of that is that while the theta function (Theta (tau )) is a dweller in the upper half-plane, its integrated form F(z) is a dweller in the extended upper half-plane including the real line, thus making it possible to consider the behaviour under the increment of the real variable, where the integration is along the horizontal line.