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$begingroup$ @Alphonse See this answer by Raymond Manzoni: Manzoni's step function /zeta zero counting function, is the same as the last formula on page 128 in the book 'Riemann' zeta function, by H.M. The zeta zero counting function has been know for at least since the year 1912, maybe even longer back to Riemann, I don't know the history behind it.
What I did was to wrap the counting function with the sign function and compute the numerical integral for between 0 and 16, a simple rectangle. Solving the symbolical integral would be more valuable. $endgroup$–May 25 '17 at 6:43.
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$begingroup$ The length of the rectangle is equal to the first zeta zero = 417. And the height of the rectangle is equal to 1. The rectangle starts at 0 and ends at the first zeta zero. The rectangle thereby falls within the interval 0 to 16, but the upper integration limit 16 is arbitrary, (although if one wants to set it closer to the first zeta zero, then one needs to use the mathematica command MinRecursion-12). So one can see with the bare eyes that the integral must be the first zeta zero. Be aware that I used the Quiet command in Mathematica which silences the error messages.
$endgroup$–May 25 '17 at 6:56.