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We derive the asymptotic behavior of weighted quadratic variations offractional Brownian motion $B$ with Hurst index $H=1/4$. This completes theonly missing case in a very recent work by I. Nourdin, D. Nualart and C. A.Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzyand J. Swanson on the asymptotic behavior of the Riemann sums with alternatingsigns associated to $B$.

Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$