TITLE:
Ramanujan-Inspired Prime Sums through Prime Gaps
AUTHORS:
Payam Danesh, Raoul Bianchetti
KEYWORDS:
Partial Prime Sums, Prime Gaps, Ramanujan-Inspired Summation, Geometric Encoding
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
14,
2026
ABSTRACT: In this study, we develop a finite framework for studying sums of prime numbers through consecutive prime gaps. This work is motivated by Ramanujan’s influence on arithmetic decompositions, partition methods, and summation ideas, but the argument itself remains within ordinary finite number theory. The main result is an exact finite rearrangement of the partial prime sum: the sum of the first
n
primes is written as the contribution of the initial prime together with a weighted accumulation of consecutive prime gaps. Each gap receives a weight equal to the number of later primes affected by that gap. The same identity is also expressed geometrically, where the weighted gap contribution appears as the slope relation in a right triangle. Exact examples verify the formula for small prime sums, and the asymptotic discussion places the identity beside the classical leading scale predicted by the prime number theorem. This paper also separates finite identities from regularized infinite summations, while divergent infinite prime sums require a separate summation theory and cannot be treated as ordinary convergent series.