Research proof index

For the smoothed Chebyshev integrand $F(s)=-\frac{\zeta'(s)}{\zeta(s)}\,\mathcal M(\widetilde{1_\varepsilon})(s)X^s,$ the central part of the normalized vertical integral on $\operatorname{Re}s=\sigma_1$ can be shifted to $\operatorname{Re}s=\sigma_2<\sigma_1$. With the project's contour-piece notation, the complete decomposition is $I_{37}=I_3-I_4+I_5+I_6+I_7,$ where $I_3$ and $I_7$ are the unchanged tails, $I_5$ is the new central vertical side, and $I_4,I_6$ are the horizontal connectors.

For a continuously differentiable smoothing function supported in $[1/2,2]$ and normalized by $\int_0^\infty \nu(x)\,dx/x=1$, the Mellin factor of the smoothed cutoff at $s=1$ is $1+O(\varepsilon)$. Precisely, there is a constant $C$ such that, for every sufficiently small $\varepsilon>0$ and every $X\ge0$, $\left|\mathcal M(\widetilde{1_\varepsilon})(1)X-X\right|\le C\varepsilon X.$