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3 votes

Estimates for the first coefficient of the Taylor expansion of $\zeta$ around a zero

Since $$c_1=\zeta'(s_0)\qquad\text{and}\qquad c_1'=\zeta'(1-\overline{s_0})=\overline{\zeta'(1-s_0)},$$ the question concerns $|\zeta'(s_0)/\zeta'(1-s_0)|$. The functional equation for $\zeta(s)$ can ...
GH from MO's user avatar
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8 votes

Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $

The question is whether this inequality on the Dirichlet eta function holds: $$\Re\eta(\alpha+i\beta)\leq 2\eta(\alpha).$$ It does not hold. Here is a plot of $\Delta(\alpha,\beta)=2\eta(\alpha)-\Re\...
Carlo Beenakker's user avatar
4 votes

Does transforming a periodic function imply periodicity

A simple counterexample: $$f(x,y)=\begin{cases} \cos(x\sqrt{y})&\text{for}\;1\leq y\leq 2,\\ \text{arbitrary}&\text{for}\; 2<y<3,\\ 0&\text{for}\;y\geq 3, \end{cases}$$ $$\Rightarrow ...
Carlo Beenakker's user avatar

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