Research proof index

Let $f$ be holomorphic on a punctured neighborhood of $p$. If $(s-p)f(s)\longrightarrow A\qquad(s\to p),$ then $f$ differs from its principal part $A/(s-p)$ by a bounded function on some punctured neighborhood of $p$. In asymptotic notation, $f(s)=\frac{A}{s-p}+O(1).$

Interval form of sin_div_error_pointwise_bound: integrating the pointwise bound over [-1, 1] controls the windowed error integral by (D / π) · vol (Ioc (-1) 1).

Pointwise bound on the sin-div windowed error term: scaling by the normalized sinc kernel and a mean-value estimate on g (x - u) - g x gives a bound D / π that is independent of the height T.

Given $a<b$ and $c<d$, there is a smooth compactly supported function $\Psi:\mathbb R\to\mathbb R$ with $\mathbf 1_{[b,c]}\le\Psi\le\mathbf 1_{(a,d)}$ and support exactly $(a,d)$. In the usual ordered case $a<b\le c<d$, this is a smooth cutoff equal to $1$ on $[b,c]$, strictly supported in $(a,d)$, and taking values between $0$ and $1$.

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.

There exists a smooth nonnegative function $\nu:\mathbb R\to\mathbb R$ supported in $[1/2,2]$ and normalized to have multiplicative mass one: $\int_0^\infty \nu(x)\,\frac{dx}{x}=1.$