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We give an estimate for sums appearing in the Nyman-Beurling criterion forthe Riemann Hypothesis containing the M\"obius function. The estimate isremarkably sharp in comparison to estimates of other sums containing theM\"obius function. The methods intensively use tools from the theory ofcontinued fractions and from the theory of Fourier series.

Estimates of sums related to the Nyman-Beurling criterion for the Riemann Hypothesis