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A New Lower Bound for the L 1 Mean of the Exponential Sum with the Möbius Function
Author(s) -
Balog Antal,
Ruzsa Imre Z.
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609398005414
Subject(s) - mathematics , exponential function , function (biology) , combinatorics , upper and lower bounds , möbius function , pure mathematics , mathematical analysis , evolutionary biology , biology
The L 1 means of various exponential sums with arithmetically interesting coefficients have been investigated in many recent papers. For example, Balog and Perelli proved in [ 1 ] thatexp (c log x log log x) ≪ ∫ 0 1| ∑ n ⩽ xμ ( n ) e ( n α )| d α ≪ x 1 / 2 ,for a suitable positive number c . The method of proving the lower bound in [ 1 ] is rather flexible and can work well with many multiplicative functions in place of μ( n ), the Möbius function, whose Dirichlet series have a suitable expression by the Riemann ζ‐function. In this short note we improve on the above lower bound. 1991 Mathematics Subject Classification 11L03, 42A70.