On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts
Abstract
We prove that for any fixed real numbers y1, y2 not equal to 0, and constant C > 0, there exists a threshold T* = T*(y1, y2, C) > 0 such that for all T >= T*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y1)) != 0, and zeta(1/2 + i (gamma + y2)) != 0. This extends earlier work by Banks (for a single shift y) to two distinct shifts y1, y2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula.
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