In [this question.][1], a very inefficient, yet rigorous analytic approach for finding the next prime was established. I wondered whether a similar approach could exist to find the next non-trivial zero ($\rho_k$) of the the Riemann $\zeta$-function.

The series:

$$\sigma_r = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right) \quad r \in \mathbb{N}$$

has a couple of related closed forms, see f.i. [Lehmer's groundwork on this][2], or [using the coefficients of the Taylor expansion of the Riemann $\xi$-function][3] or applying the [Stieltjes constants][4].

Now define the function:

$$f(r,N,x)= \sigma_r -\left( \sum_{k=1}^{N} \left( \frac{1}{\rho_k^r} + \frac{1}{(1-\rho_k)^r}\right)+ \left( \frac{1}{x^r} + \frac{1}{(1-x)^r}\right)\right)$$

where $r, N \in \mathbb{N}$ and $x$ is the unknown next non-trivial zero ($\rho_{N+1}$).

In Maple: 

    for N from 0 to 6 do N, fsolve(f(33, N, x), x = 0 + 12*I .. 1 + 42*I, complex) end do;

yields the encouraging list for $\rho_{N+1}$:

      0, 0.50000000000000000000 + 14.134724467544674288 I
      1, 0.50000000000000000000 + 21.020287719482273773 I
      2, 0.50000000000000000000 + 25.009821593878642551 I
      3, 0.50000000000000000000 + 30.365139393045860165 I
      4, 0.50000000000000000000 + 32.923423713014869349 I
      5, 0.50000000000000000000 + 37.517148219902925704 I
      6, 0.50000000000000000000 + 40.750950425647022339 I

Accuracy improves for higher $r$, let's test $N=2$ for increasing $r$: 

    for r from 5 by 5 to 35 do r, fsolve(h(r, 2, x), x = 0 + 12*I .. 1 + 42*I, complex) end do;

       5, 0.50000000000000000000 + 22.755757318210846210 I
      10, 0.50000000000000000000 + 24.477994717471646165 I
      15, 0.50000000000000000000 + 24.922438734940328122 I
      20, 0.50000000000000000000 + 24.979676684194649394 I
      25, 0.50000000000000000000 + 25.004076475496886667 I
      30, 0.50000000000000000000 + 25.008087373823152633 I
      35, 0.50000000000000000000 + 25.010200234483713493 I

Computations quickly require higher precision since $\sigma_r$ becomes very small. 

Now, contrary to the primes that are integers and have a minimal distance of $2$ between the odd primes, we are now dealing with probably irrational numbers and no known minimal distance between them (nothing stops [Lehmer-pairs][5] from becoming infinitely small).

**Q:** Could the above approach of analytically 'recovering' the next zero be made rigorous up to a fixed $n$-digits accuracy?  


  [1]: https://mathoverflow.net/questions/365323/could-computing-the-next-prime-in-a-finite-euler-product-be-made-rigorous
  [2]: https://pdfs.semanticscholar.org/d4ae/e1f704e73711e38ed0645beacb0c90218551.pdf?_ga=2.196818921.451340533.1601978797-1416226021.1596734321
  [3]: https://www.sciencedirect.com/science/article/pii/S0022314X14002455
  [4]: https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html
  [5]: https://en.wikipedia.org/wiki/Lehmer_pair