Proof of the Riemann Hypothesis Using Modified Trigonometric Series

The Riemann hypothesis, first proposed by Bernhard Riemann in a famous 1859 paper, surmised that all nontrivial zeros of the analytically continued Riemann Zeta function, ζ(s), have real part equal to ½. Due to the intimate connection between the zeta zeros and the distribution of prime numbers (enumerated by Riemann in the same paper), verification of the Riemann hypothesis is widely regarded as being the most important unsolved problem in pure mathematics. In this contribution, the Riemann hypothesis is proven by assuming a generalized zeta zero within the critical strip of the form ρ = σ + it (where 0 < σ < 1, and σ,t are real), and applying the result that ρ must also be a zero (i.e., ζ(ρ) = ζ(ρ) = 0) utilizing an alternative Dirichlet series representation of ζ(s) that is absolutely convergent within the critical strip. These two equations (combined and rewritten using Euler’s formula) are used to derive two modified trigonometric series of real variables (σ, t) whose sums must converge to finite values. It is then demonstrated that the series converge if and only if the coefficients vanish, requiring that σ = ½.