Logarithm of the Kernel Function. 1 Introduction and Preliminary Results

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1 Iteratioal Mathematical Forum, Vol. 3, 208, o. 7, HIKARI Ltd, htts://doi.org/0.2988/imf Logarithm of the Kerel Fuctio Rafael Jakimczuk Divisió Matemática Uiversidad Nacioal de Lujá Bueos Aires, Argetia Coyright c 208 Rafael Jakimczuk. This article is distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial work is roerly cited. Abstract I this ote we study the logarithm of the kerel fuctio ad related fuctios. Mathematics Subject Classificatio: A99, B99 Keywords: Kerel fuctio, logarithm Itroductio ad Prelimiary Results A square-free umber is a umber without square factors, a roduct of differet rimes. The first few terms of the iteger sequece of square-free umbers are, 2, 3, 5, 6, 7, 0,, 3, 4, 5, 7, 9, 2, 22, 23, 26, 29, 30,... Let us cosider the rime factorizatio of a ositive iteger 2 = q s q s 2 2 qt st where q, q 2,... q t are the differet rimes i the rime factorizatio. We have the followig two arithmetical fuctios u = q q 2 q t The arithmetical fuctio u is well-kow i the literature, it is called kerel of, radical of, etc. Note that u is the largest squarefree umber that divides. There exist may aers dedicated to this fuctio. v = u = qs q s 2 2 qt st

2 338 Rafael Jakimczuk We call v the remaider of. Note that v = if ad oly if is a square-free. I this ote we obtai asymtotic formulae for the followig sums. i log ui, i log vi i log ui log, i log vi log As usual, x deotes the iteger art of x, {x} = x x deotes the fractioal art of x, deotes a ositive rime, the simbol mea that the sum ru o all ositive rimes. We eed the followig well-kow lemmas. Lemma. The followig asymtotic formula holds i i = log + γ + o where γ = 0, is called Euler s costat see [2, chater ]. Lemma.2 The followig asymtotic formula holds 2 x log = log x + C + o where C = γ log =, See [2, chater 2] Lemma.3 Prime umber theorem The followig asymtotic formula holds 2 x log = x + ox Lemma.4 The followig asymtotic formula holds i log i = log + o Proof. It is a weak cosequece of the Stirlig s formula! 2π e.

3 Logarithm of the kerel fuctio Mai Results Theorem 2. The followig asymtotic formulae hold log ui = log + A + o where A = i log i = 0, log vi = A + o 2 Proof. Equatio 2 is a immediate cosequece of equatio ad Lemma.4, sice uivi = i. Therefore we shall rove equatio. Note that if j is a arbitrary but fixed ositive iteger ad the rime satisfies the iequality < the j+ j = j. Let ɛ > 0, we choose the fixed ositive iteger s such that s < ɛ 3 ad i the equatio see Lemma. we have We have i log ui = 2 j+ < j s log j = log s + γ + o = o < ɛ s j We have by Lemma.3 ad Lemma. s s j log = j j + o = s 2 s j+ < j j + + o log + 2 s s = j log j j + 4 j + o = log s + γ + o + o 2 5 O the other had, we have by Lemma.2 log = log = log + C + o 3 s 2 s 2 s 2 s log log { } { } 6

4 340 Rafael Jakimczuk where by Lemma.3 ad equatio 3 That is 0 2 s log { } 0 2 s 2 s log { } log = s + o 4 2ɛ 2ɛ 7 We have choose 0 such that if 0 the o 2 < ɛ, o 3 < ɛ ad o 4 < ɛ. Substitutig equatios 5 ad 6 ito equatio 4 we fid that i log ui = log 2 s log { } Therefore see equatio 7 + log + o + o 2 + o 3 i log ui log + + log o + o 2 + o s log { } 5ɛ 0 That is i log ui log + + log = o That is, equatio, sice ɛ ca be arbitrarily small. The theorem is roved. A immediate cosequece of Stirlig s formula see above is the well-kow limit! e Note that! = i = uivi = ui vi i= i= i= i= A immediate cosequece of Theorem 2. is the followig corollary.

5 Logarithm of the kerel fuctio 34 Corollary 2.2 The followig limits hold i= ui e +A i= vi e A Theorem 2.3 The followig asymtotic formulae hold 2 x log v log = A x x log x + o log x 8 2 x log u log = x A x x log x + o log x 9 Proof. Note that x + o = 2 x x = x = log log x 2 x log = log u log + 2 x log v log 2 x loguv log 0 Therefore 9 is a immediate cosequece of 8 ad 0. The roof of 8 is by artial summatio see [3, chater XXII]. We have by Theorem 2. 2 x log v = Ax + ox Cosequetly, if we use the fuctio fx = log x the log v x 2 x log = Ax + ox fx At + ot f t dt 2 = A x x x log x + o x + A log x 2 log 2 t dt + o 2 log 2 t dt = A x x log x + o log x That is, equatio 8. The theorem is roved.

6 342 Rafael Jakimczuk 3 A Alicatio to Sums of Fractioal Parts Dirichlet 849 roved the asymtotic formula k { k } = γ + O De la Vallée Poussi [] roved the asymtotic formula { } = γ log + o log Note that a immediate cosequece of Lemma.2, ad 4 is the asymtotic formula { } log = γ + o Ackowledgemets. The author is very grateful to Uiversidad Nacioal de Lujá. Refereces [] C. de la Vallée Poussi, Sur les valeurs moyees de certaies foctios arithmétiques, Aales de la Société Scietifique de Bruxelles, , [2] S. Fich, Mathematical Costats, Cambridge Uiversity Press, [3] G. H. Hardy ad E. M. Wright, A Itroductio to the Theory of Numbers, Oxford, 960. Received: May 23, 208; Published: Jue 2, 208