First integral of the Chebyshev function (${\Psi }_1(x)={\int }_0^x\psi (u)du$)

For the Second Chebyshev function, finite summation gives ${\Psi }_1(x)={\sum }_{n\le x}(x-n)\Lambda (n)$. For $c>1$, Mellin inversion of the kernel $1/(s(s+1))$ givesThe extra denominator makes this smoothing useful: the resulting sum over Nontrivial zeros of the Riemann zeta function is absolutely convergent. This integral differs from the integrated Chebyshev sum, which denotes a sum of the values $\psi (x/n)$.

For $x>1$, moving the Mellin contour past the pole at one, the Nontrivial zeros of the Riemann zeta function, and the remaining poles givesZeros 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 $x^2$ and dominated convergence give ${\Psi }_1(x)\sim x^2/2$. Monotonicity of $\psi$ then recovers $\psi (x)\sim x$ by upper and lower difference quotients.