The Smoothing of Prime Penalties: How Chaotic Modular Remainders Converge to a Basel Constant

If you take a massive number, divide it by every prime number smaller than itself, and add up all the remainders, what do you get? Because prime numbers are notoriously unpredictable, you might expect the result to be completely random. However, this paper reveals a beautiful hidden pattern. As your starting number gets larger and larger, the sum of these remainders perfectly settles at exactly 35.5% of the sum of the primes themselves. Even more surprisingly, the exact mathematical constant that dictates this ratio is in terms of π, connecting the chaotic world of prime remainders directly to πππ and the geometry of circles. We subsequently found out that this ratio is applicable to not only prime numbers, but other number types as well. For details, please refer to the paper.