Theorumber

Each module in the application allows real-time parameter tuning and real-time viewing of calculation results and curve changes. The Dirichlet module can select templates such as Zeta (s), M ö bius, and divisor function to adjust Re (s) and coefficient growth rate, view convergence domain determination and partial sum curves; Perron module demonstrates the contour integral approximation of ψ (x), θ (x), and π (x), supporting custom x and truncation height T. Sieve provides screening methods such as Eratosthenes and Sundaram, which can gradually or automatically play the elimination process; Exp. Sums calculates the classical bound (van der Corput, Weyl, etc.) for the exponent and S (N, α) and compares it with the true modulus length; Continuation shows the analytical extension of Zeta, η, and λ within the critical band through functional equations; PNT uses a seven step proof framework to concatenate the Prime Number Theory, and performs numerical verification using π (x) and li (x).