The truncation error of the Riemann–von Mangoldt explicit formula: effective exponent, summation schemes, and a numerical window study

This article studies the truncation error of the Riemann–von Mangoldt explicit formula from a numerical and methodological standpoint. It characterises the effective exponent governing the decay of the window-averaged error in the reconstruction of the Chebyshev function from the non-trivial zeros of the Riemann zeta function, for N between 1 and 500 zeros over the window x ∈ [500, 1500]. The measured effective exponent is α ≈ 0.26, well below the N−1/2 scale, and the functional form with a direct logarithmic correction is shown not to be supported by the data over the sampled interval. A separate analysis establishes that the exponent is not invariant under the choice of summation scheme, which is the methodological contribution of the work. Keywords for indexing and citation: Riemann–von Mangoldt explicit formula; truncation error; Chebyshev function; effective exponent; summation schemes; zeros of the zeta function; conditional convergence; numerical analysis in number theory.