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In this paper, we study a free-probabilistic model on the algebra of arithmetic functions by considering their asymptotic behavior. As an application, we concentrate on arithmetic functions arising from certain representations attached to the general linear group G L n . We then study conditions under which a Dirichlet series may be factored into a product of automorphic L-functions using asymptotic freeness.

This present collaborative research, undertaken in two different hemispheres, in an effort to address the challenge of early structural and sub-surface assessment of heritage marble architectures, like the Taj Mahal, using two complementary non-contact, non-invasive imaging techniques in the THz spectral range. In our previous work, it was already demonstrated that the complementary techniques of broadband Terahertz Time Domain Imaging (THz-TDI) and micro-Raman spectroscopy are successful in probing volume and surface damage in marble with Pietra-dura work. In the present work, the unique combination of THz-TDI and highly sensitive THz-Laser Feedback Interferometry (THz-LFI) have been explored to study sub-surface damage and irregularities of marble structures with Pietra-dura motif. These optical techniques hold immense possibility in large-scale architectural restoration projects as they collectively provide accurate structural depth profile up to several inches into the volume of the marble including the strain generated within the structure leading to potential cracks.