The Σπ Operator: A Unified Framework for Divergent Series Summation

In this paper, I introduce a new summation operator Σπ that unifies three classical methods for handling divergent series: zeta regularization, Ramanujan summation, and Cesàro summation. I define the operator using a multi-indexed divergence framework capable of classifying polynomial, logarithmic, and iterated logarithmic growth. Key results:- A unified summation operator that retrieves zeta regularization, Ramanujan summation, and Cesàro summation as natural algebraic quotients- A multi-indexed ring structure that classifies polynomial, logarithmic, and iterated logarithmic growth- Shift stability and exponential boundary theorems- A concrete application to quantum field theory renormalization via hard cutoff regularization This work builds directly on Ramanujan's legacy in summability theory and provides a rigorous algebraic foundation for divergent series summation.