Convergence properties of a twisted Dirichlet series

We investigate the convergence behavior of a modified Dirichlet series in which the complex exponent has a fixed real part σ = 0.5 and an imaginary part that increases linearly with the summation index. This induces progressively faster phase oscillations while maintaining slow decay in magnitude. Despite the slow decay rate, numerical computations and analytic arguments indicate that the series remains bounded and convergent. This contrasts with the divergence of the classical Riemann zeta function at σ = 1 and highlights the stabilizing role of structured phase variation. A justification is provided using oscillatory integral approximations and summation techniques related to the Euler–Maclaurin formula. The convergence mechanism arises from enhanced cancellation due to controlled phase rotation. This construction provides a simple example of conditional convergence in the critical strip and suggests connections to phase-based interpretations of Dirichlet series.