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Riemann conjectured that all the zeros of the Riemann $\Xi$-function arereal, which is now known as the Riemann Hypothesis (RH). In this article weintroduce the study of the zeros of the truncated sums $\Xi_N(z)$ in Riemann'suniformly convergent infinite series expansion of $\Xi (z)$ involvingincomplete gamma functions. We conjecture that when the zeros of $\Xi_N (z)$ inthe first quadrant of the complex plane are listed by increasing real part,their imaginary parts are monotone nondecreasing. We show how this conjectureimplies the RH, and discuss some computational evidence for this and otherrelated conjectures.

Some conjectures on the zeros of approximates to the Riemann $\Xi$-function and incomplete gamma functions