Dirichlet kernel harmonic lower bound (${\Lambda }_N\ge \frac{4}{{\pi }^2}{\sum }_{j=1}^{N}1/j$)

For $D_N(t)=\sin ((N+1/2)t)/\sin (t/2)$, define ${\Lambda }_N=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }|D_N|$. Evenness and $\sin (t/2)\le t/2$ on $(0,\pi ]$ give ${\Lambda }_N\ge (2/\pi ){\int }_0^{(N+1/2)\pi }|\sin v|/vdv$. The first $N$ complete sine periods of length $\pi$ have absolute sine integral two, and on the $j$th such interval $1/v\ge 1/(j\pi )$. This proves the displayed unbounded lower bound. The real functional $f\mapsto S_N(f,0)$ on continuous functions has norm exactly ${\Lambda }_N$: the upper bound is its kernel integral, and the continuous functions $D_N/\sqrt{D_N^2+{\delta }^2}$ approach that bound as $\delta \downarrow 0$ by dominated convergence.