Theorem 2. Let $s_0 = \sigma_0 + i t_0$ be a fixed complex number

Expert intro: This is a theorem statement in complex analysis / analytic number theory style. The best support here is to organize the hypotheses, the conclusion, and the role of each constant clearly.

Detailed walkthrough

What the theorem says

The statement gives a set of hypotheses on a function $g$ defined on a closed disk

$K=\{s\in\mathbf{C}:|s-s_0|\le r\},$

where $s_0=\sigma_0+it_0$ and $\sigma_0\in\left(\frac12,1\right)$.

Under these assumptions, and for suitable choices of the parameters $\delta_0$, $\varepsilon$, and a sufficiently large $T$, the theorem guarantees the existence of a shift $\tau\in[T-t_0,\,T+H-t_0]$ such that the translated function $K(s+i\tau)$ approximates $g(s)$ uniformly on the circle $|s-s_0|=r$:

$\max_{|s-s_0|=r}|K(s+i\tau)-g(s)|<\varepsilon.$

It also imposes a further smallness condition involving

$\max_{|s-s_0|=r}|K(s+i\tau)|\frac{\delta^N}{1-\delta}<\frac{\varepsilon}{3}.$

How to read the assumptions

The theorem requires:

• $g$ to be continuous and nonvanishing on $K$
• $g$ to be analytic in the interior of $K$
• a numerical condition on $N$ ensuring that

$\left(\max_{|s-s_0|=r}|g(s)|\right)\frac{\delta_0^N}{1-\delta_0}<\frac{\varepsilon}{3}$

• a lower bound on $T$ involving the constants $C_2$ and $B$

Meaning of the conclusion

The conclusion is an approximation statement: after shifting by $i\tau$, the function $K$ can be made arbitrarily close to the target function $g$ on the boundary of the disk. The theorem is therefore a controlled existence result, with explicit constants and explicit dependence on the parameters.

Structure of the bound

The bound on $T$ has the form

$T\ge \max\left\{\exp_2\left(C_2(N,\sigma_0)\left(B(N,g,\sigma_0,\delta_0,r,\varepsilon)\right)^{\frac{1}{\eta_0+\sigma_0+\frac{1}{\omega}}}\right),r\right\}.$

This says that the theorem becomes effective once $T$ is large enough relative to the complexity of the target function and the chosen approximation tolerance.

Final takeaway

The theorem provides a quantitative approximation result for a complex-analytic function after vertical translation, with explicit control through $\varepsilon$, $N$, and the constants $C_2$ and $B$.

💡 Pitfall guide

A common mistake is to lose track of where the maximum is taken. The inequalities are on the boundary circle $|s-s_0|=r$, not on the whole disk. Another pitfall is treating $C_2$, $B$, and $N$ as interchangeable; each has a distinct role in the bound.

🔄 Real-world variant

If the target tolerance $\varepsilon$ is made smaller, the condition on $N$ becomes stricter and the required lower bound on $T$ typically grows. If $g$ were simpler or smaller on the boundary, the quantity $B(N,g,\sigma_0,\delta_0,r,\varepsilon)$ could also become smaller, improving the bound.

🔍 Related terms

complex analysis, uniform approximation, analytic function