Let $K/F$ be a Galois extension of number fields with Galois group $G$. We begin by examining the descent of real zeros of the Dedekind zeta function $zeta_K$ to the Dedekind zeta functions of subfields of $K$. This descent will allow us to get lower bound of residue of $\zeta_K(s)$ at $s = 1$ for certain number fields. In the next part, we explore the connection between the holomorphy of the Artin $L$-functions $L(s, \chi, K/F)$ at a point $s_0$ and order of $\zeta_K(s)$ at $s_0$ for any character $\chi$ of $G$.

We present the latest lattice results for the hadronic vacuum polarization from lattice QCD. In order to isolate the different origin of systematic errors in lattice computations, the observable is decomposed into several windows. The latest discrepancy between the data-driven theory prediction of the intermediate and long distance window of the hadronic vacuum polarization using the experimental input of e+e- to hadrons cross-section and the lattice predictions have sparked several new physics scenarios. We elaborate on the Mainz results for the intermediate and long distance windows and discuss the implications of the discrepancy. The anomalous magnetic moment of the muon (aµ) provides a stringent test of the Standard Model and Beyond, that too, interestingly on two independent fronts: the “aµ-test” and the “HVP-test”.We discuss a generic, light (∼ 100 MeV-1 GeV) Z′’s impact on both these tests in multiple ways making this arena an excellent probe of such models.

A key question about the QCD phase diagram is whether there is a critical point somewhere on the boundary between the hadron gas and quark-gluon plasma phases, and if so where. Heavy-ion collisions offer a unique opportunity to search for signatures of such a critical point by analyzing event-by-event fluctuations in particle multiplicities. To draw meaningful conclusions from experimental data,  a theoretical framework is needed to link QCD thermodynamics with the particle spectra and correlations observed in detectors. This is done in the least biased by maximizing the entropy associated with the hadron resonance gas ensemble, subject to matching conditions from the hydrodynamic description at freeze-out. We use maximum entropy freeze-out of fluctuations to make estimates for the factorial cumulants of proton multiplicities, assuming thermal equilibrium, for a family of EoSs with a 3D Ising-like critical point, varying the microscopic inputs that determine the location and strength of the critical point. The unknown Equation of State (EoS) of QCD near a critical point can be related to the universal Gibbs free energy of the 3D Ising model using four non-universal mapping parameters whose values are determined by the microscopic details of QCD. We quantify the effect of the non-universal mapping parameters, and the distance between the critical point and the freeze-out curve, on the factorial cumulants of proton multiplicities.