Identities for number series and their reciprocals: Dirac delta function approach

TL;DR

This paper introduces a Dirac delta function approach to derive identities for number series and their reciprocals, providing a simple proof connecting the prime counting function and the logarithmic integral.

Contribution

The paper presents a novel application of the Dirac delta function to derive identities in number theory, including a new proof relating pi-function and Li-function.

Findings

Derived identities for number series using delta function

Provided a simple proof linking pi-function and Li-function

Demonstrated efficiency of the delta function approach

Abstract

Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function (pi-function) and logarithmic integral (Li-function) can be obtained.