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I noticed something interesting when playing around with Mathematica. Consider the sum $$x(N)= \frac{1}{N^2} \sum_{i=1}^{N-1} \frac{1}{1-\cos(2\pi i/N)}$$ this sequence will converge to $1/6$ as $N$...

I am wondering if there is a uniform bound $C$ (independent of $\lambda>10$): $$\sum_{k=-\infty}^{-1}\Big|\int_{2^k}^{2^{1+k}}\frac{\sin(\lambda t^3)}{t}dt\Big|\le C.$$ Remark: (1) An easy upper ...