Evaluation of Integral for Recovering Second Chebyshev Function $\psi[y]$

The context of this post is the relationship between evaluation of the partial integral for $\psi(x)$ defined in (1) below and partial evaluation of von Mandgoldt’s explicit formula for $\psi(x)$ defined in (2) below.

(1) $\quad\psi(x)=\frac{1}{2\ \pi\ i}\int_{a-b_m\ i}^{a+b_m\ i}\left(−\frac{\zeta′[s]}{\zeta[s]}\right)\frac{y^s}{s}\ ds$

(2) $\quad\psi(x)=x-\sum_\rho\frac{x^\rho}{\rho}-\log(2\ \pi)-\frac{1}{2}\log(1-x^{-2})$

I’ve been wondering if it’s true that for some value of $b_m$ where $Im[ZetaZero[m]]<b_m<Im[ZetaZero[m+1]]$, evaluation of integral (1) above is equivalent to evaluating von Mangoldt’s formula (2) above over the first $m$ conjugate pairs of zeta zeros, and if so whether theres a way to determine the intermediate value of $b_m$ other than using successive approximation.

The plots below illustrate evaluation of the partial integral (1) above approximates partial evaluation of von Mandoldt’s explicit formula (2) above through the 10th and 11th zeta zeros, and the difference between these two approximates the contribution of the 11th zeta zero in von Mandoldt’s explicit formula (2) above. I’m not sure how the two small error terms in von Mandoldt’s formula evolve over the range of integral (1) above, but I suspect they evolve fairly quickly, and if this is the case it seems evaluation of integral (1) above through $b_m$ will approximate partial evaluation of von Mandgoldt’s explicit formula (2) above through zeta zero $m$ with increasing accuracy as $m$ increases.

The orange curves in the following two plots illustrate partial evaluation of von Mangoldt’s explicit formula (2) above through the 10th and 11th zeta zeros respectively. Note there is a slight increase in sharpness of the second plot below since it includes evaluation of an additional zeta zero. The blue curves in the two plots below illustrate evaluation of integral (1) above using integration bounds corresponding to an estimate of $b_{10}$ in the first plot below and an estimate of $b_{11}$ in the second plot below, but the evaluations of (1) and (2) are so similar that at this scale the evaluations of (1) in blue are mostly hidden by the evaluations of (2) in orange, and therefore the green curves were added to the following two plots to illustrate the difference between the evaluations of (1) and (2).

Partial evaluation of (1) (blue) and (2) (orange) through $b_{10}$ and 10th zeta zero respectively and difference between the two (green)
Partial evaluation of (1) (blue) and (2) (blue) through $b_{11}$ and 11th zeta zero respectively and difference between the two (green)

The orange curve in the following plot illustrates evaluation of the contribution of the 11th zeta zero in von Mangoldt’s formula (2) above. The blue curve in the following plot illustrates evaluation of formula (3) below using estimates of $b_{11}$ and $b_{10}$ in the first and second integrals of formula (3) respectively, but the evaluations of (2) and (3) are so similar that at this scale the evaluation of (3) in blue is mostly hidden by the evaluation of (2) in orange, and therefore the green curve was added to the following plot to illustrate the difference between the evaluations of (3) and (2).

(3) $\quad \Re\left(\frac{1}{2\ \pi\ i}\left(\int_{a-b_{11}\ i}^{a+b_{11}\ i}\left(−\frac{\zeta′[s]}{\zeta[s]}\right)\frac{x^s}{s}\ ds-\int_{a-b_{10}\ i}^{a+b_{10}\ i}\left(−\frac{\zeta′[s]}{\zeta[s]}\right)\frac{x^s}{s}\ ds\right)\right)$

Evaluation of (3) (blue), contribution of 11th zeta zero in (2) (orange), and difference between the two (green)

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