 First integral of the Chebyshev function

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 $\psi (x/n)$.