On some lower bounds of some symmetry integrals
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چکیده
X iv :1 00 3. 45 53 v2 [ m at h. N T ] 6 J un 2 01 0 ON SOME LOWER BOUNDS OF SOME SYMMETRY INTEGRALS by G.Coppola Abstract. We study the “symmetry integral”, say If , of some arithmetic functions f : N → R; we obtain from lower bounds of If (for a large class of arithmetic functions f ) lower bounds for the “Selberg integral” of f , say Jf (both these integrals give informations about f in almost all the short intervals [x−h, x+ h], when N ≤ x ≤ 2N ). In particular, when f = dk, the divisor function (having Dirichlet series ζ , with ζ the Riemann zeta function), where k ≥ 3 is integer, we give lower bounds for the Selberg integrals, say Jk = Jdk , of the dk. We apply elementary methods (Cauchy inequality to get Large Sieve type bounds) in order to give If lower bounds.
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