Is it true that sum_(p<= x)(log p)^2 ~ x log x ?

Brooke Richard

Brooke Richard

Answered question

2022-11-07

Is it true that
p x ( log p ) 2 x log x       ? ( 1 )

Answer & Explanation

Patrick Arnold

Patrick Arnold

Beginner2022-11-08Added 21 answers

For some η ( n , n + 1 )
n x π ( n ) ( log ( n + 1 ) 2 log ( n ) 2 ) = n x π ( n ) 2 log ( η ) η = n x ( n log ( n ) + O ( n log ( n ) 2 ) ) ( 2 log ( n ) n + O ( log ( n ) n 2 ) ) = 2 x + O ( x log ( x ) )
Therefore, summing by parts,
p x log ( p ) 2 = n x log ( n ) 2 ( π ( n ) π ( n 1 ) ) = log ( x ) 2 π ( x ) n x π ( n ) ( log ( n + 1 ) 2 log ( n ) 2 ) = log ( x ) 2 ( x log ( x ) + x log ( x ) 2 + O ( x log ( x ) 3 ) ) 2 x + O ( x log ( x ) ) = x log ( x ) x + O ( x log ( x ) )

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