 Smoothed explicit formula for the Chebyshev function

ID: smoothed-explicit-formula-for-the-chebyshev-function

For

x > 1

, moving the Mellin contour past the pole at one, the Nontrivial zeros of the Riemann zeta function, and the remaining poles gives

Ψ_{1} (x) = \frac{x ^{2}}{2} - \sum_{ρ} \frac{x ^{ρ + 1}}{ρ ( ρ + 1 )} - x lo g (2 π) + \frac{ζ ^{'} ( - 1 )}{ζ ( - 1 )} - \sum_{k \geq 1} \frac{x ^{1 - 2 k}}{2 k ( 2 k - 1 )} .

Zeros are included with multiplicity. The Riemann–von Mangoldt formula makes the zero sum absolutely convergent. If every nontrivial zero has real part below one, division by

x^{2}

Ψ_{1} (x) \sim x^{2} /2

. Monotonicity of

ψ

then recovers

ψ (x) \sim x

by upper and lower difference quotients.

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