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Journal of the London Mathematical Society 1998 57(2):275-288; doi:10.1112/S0024610798006000
© 1998 by London Mathematical Society
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© The London Mathematical Society

On the L1 Mean of the Exponential Sum Formed with the Möbius Function

A. Balog and A. Perelli

Mathematical Institute of the Hungarian Academy of Sciences Reàltanoda u. 13-15, H-1364 Budapest, Hungary. E-mail: balog{at}math-inst.hu
Dipartimento di Matematica Via Dodecaneso 35, 16146 Genova, Italy. E-mail: perelli{at}dima.unige.it

Received 10 April 1995.

In this paper we study the L1 mean


Formula
(1)

of the exponential sum M({alpha})={sum}n≤Xµ(n)e(n{alpha}), where µ(n) is the Möbius function and e(x)=e2{pi}ix. From the Cauchy–Schwarz inequality and Parseval's identity, we have


Formula
, (2)

and it is an interesting problem to investigate whether (2) reflects the true order of magnitude of (1).


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