An Integrated Proof of the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture using Digit Sum Patterns

The paper proposes a model linking the arithmetic information of elliptic curves with prime number digit sum patterns, aiming to resolve the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture using the L-function. The methodology combines the error term a_p of elliptic curves with the digit sum rule of natural numbers S(N)(mod3), filtering the Dirichlet coefficients a_n of the L-function L(E,s) based on the asymmetry where digit sums of primes (excluding 3) avoid multiples of 3. This L-function aligns all non-trivial zeros on the critical line Re(s)=1/2, offering a basis for proving the Riemann Hypothesis, and by demonstrating that the order of vanishing at s=1 matches the algebraic rank of the elliptic curve, the paper addresses the core proposition of the BSD conjecture.