[seqfan] Exponentially S-numbers
Vladimir Shevelev
shevelev at bgu.ac.il
Wed Oct 21 15:27:30 CEST 2015
Dear Seq Fans,
When I learned about A209061 (exponentially squarefree
numbers, or the numbers having all squarefree exponents
in their prime power factorization), I paid attention on a
remarkable formula in Toth's link (Theorem 3, 2007): if RH is true,
then
Sum_{a(n)<=x} 1= hx+O(x^(0.2+eps)), (1)
for every eps>0,
h=Prod{p}(1+Sum_{k>=4} (mu^2(k)-mu^2(k-1))/p^k) (2),
where product is over all primes, mu is the MÃ¶bius function.
Later I found many other papers, beginning with 1972, but
in all of them the authors considered only exponentially squarefree
numbers and, using or not RH, tried improve the remainder term.
Indeed, a deep M. V. Subbarao's 1972-paper,
where he gave some general constructions and for a first time
intoduced the notion " exponentially squarefree
numbers" made this topic classical. Formula (1) has practically
the best known remaider term.
I had a dream: to find a generalization of (1)-(2), maybe with
less good remainder term, but is suitable for the
exponentialy S-numbers for every fixed increasing sequence S
of positive integers. My approach is quite another, than all these
authors. I based on my paper "Compact numbers and factorials",
Acta Arith. 126(2007), no.3 , 195-236, Theorem 1.
I understood that "compact numbers" one can also
name exponentially 2^n-numbers. This was the first clue.
Thank God, I was able to generalize this very
special case. I obtained a common for exponentially
S-numbers (i.e., independent on S) remainder term equals
O(sqrt(x)*log(x)*e^(c*sqrt(log(x))/log(log(x)))), where
c=4*sqrt(2.4/log(2))=7.4430...,
i.e., O(x^((1/2)+eps) with concretizing eps.
Now I am pleased to inform Seq Fans that to-day my paper
with a general case has been appeared in arxiv :
http://arxiv.org/pdf/1510.05914v1.pdf
Best regards,
Vladimir
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