Derivation of the von Mangoldt Explicit Formula (Riemann–von Mangoldt Explicit Formula)
the explicit formula is one of the deepest results in analytic number theory
Derivation of the von Mangoldt Explicit Formula (Riemann–von Mangoldt Explicit Formula)
equations removed
by my Grok, at my instruction, guided by my insights of body intelligence
The explicit formula is one of the deepest results in analytic number theory. It expresses the distribution of primes directly in terms of the non-trivial zeros of the Riemann zeta function. Below is a rigorous, step-by-step derivation of the classical von Mangoldt form for the Chebyshev function ψ(x).
Step 1: Logarithmic Derivative of Zeta
Step 2: Perron’s Formula (Contour Integral Representation)
(Strictly, this holds in the sense of a smoothed sum; the exact formula includes a small error term that vanishes in the limit.)
Step 3: Analytic Continuation and Contour Shift
Step 4: The Completed Explicit Formula
Step 5: Prime-Counting Version (for π(x) \pi(x) π(x))
Step 6: Convergence and Error Terms
The sum over zeros converges conditionally (in the sense of symmetric limits or smoothed sums). The error term depends on the distribution of zeros; under the Riemann Hypothesis, the error is:
Without RH, the best unconditional error is larger.
Step 7: Connection to the Riemann Hypothesis
The explicit formula shows that the error term in the Prime Number Theorem is controlled by the real parts of the zeros ρ . If all Re(ρ) = 1/2, the error is as small as possible, giving the sharpest possible prime distribution.
This completes the derivation. The explicit formula is the bridge between the analytic properties of ζ(s) and the arithmetic distribution of primes — the single most powerful consequence of the functional equation and the locations of the zeros.
Key References
Riemann (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse.
von Mangoldt (1895). Zu Riemann’s Abhandlung “Ueber die Anzahl der Primzahlen…”
Davenport, Multiplicative Number Theory (standard modern proof).
Edwards, Riemann’s Zeta Function (detailed contour-shift arguments).
