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The method of proof of Balog and Ruzsa and the large sieve of Linnik are usedto investigate the behaviour of the $L^{1}$ norm of a wide class of exponentialsums over the square-free integers and the primes. Further, a new proof of thelower bound due to Vaughan for the $L^{1}$ norm of an exponential sum with thevon Mangoldt $\Lambda$ function over the primes is furnished. Ramanujan's sumarises naturally in the proof, which also employs Linnik's large sieve.

Linnik's large sieve and the $L^{1}$ norm of exponential sums