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On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values

Oleg Petrushov (2015)

Acta Arithmetica

We consider the behavior of the power series 0 ( z ) = n = 1 μ 2 ( n ) z n as z tends to e ( β ) = e 2 π i β along a radius of the unit circle. If β is irrational with irrationality exponent 2 then 0 ( e ( β ) r ) = O ( ( 1 - r ) - 1 / 2 - ε ) . Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that 0 ( e ( β ) r ) = Ω ( ( 1 - r ) - 1 + δ ) .

On the linear independence of p -adic L -functions modulo p

Bruno Anglès, Gabriele Ranieri (2010)

Annales de l’institut Fourier

Let p 3 be a prime. Let n such that n 1 , let χ 1 , ... , χ n be characters of conductor d not divided by p and let ω be the Teichmüller character. For all i between 1 and n , for all j between 0 and ( p - 3 ) / 2 , set θ i , j = χ i ω 2 j + 1 if χ i is odd ; χ i ω 2 j if χ i is even . Let K = p ( χ 1 , ... , χ n ) and let π be a prime of the valuation ring 𝒪 K of K . For all i , j let f ( T , θ i , j ) be the Iwasawa series associated to θ i , j and f ( T , θ i , j ) ¯ its reduction modulo ( π ) . Finally let 𝔽 p ¯ be an algebraic closure of 𝔽 p . Our main result is that if the characters χ i are all distinct modulo ( π ) , then 1 and the series f ( T , θ i , j ) ¯ are linearly independent over a certain...

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