Linnik's large sieve and the $L^{1}$ norm of exponential sums
Abstract
The method of proof of Balog and Ruzsa and the large sieve of Linnik are used to investigate the behaviour of the $L^{1}$ norm of a wide class of exponential sums over the square-free integers and the primes. Further, a new proof of the lower bound due to Vaughan for the $L^{1}$ norm of an exponential sum with the von Mangoldt $\Lambda$ function over the primes is furnished. Ramanujan's sum arises naturally in the proof, which also employs Linnik's large sieve.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2019
- DOI:
- 10.48550/arXiv.1908.06946
- arXiv:
- arXiv:1908.06946
- Bibcode:
- 2019arXiv190806946E
- Keywords:
-
- Mathematics - Number Theory;
- 11L03;
- 11L07;
- 11L20;
- 11N36;
- 42A05
- E-Print:
- 15 pages