

A171910


a(n) = number of zeros of the Mertens function M(x) in the interval 0 < x < 10^n (M(x) is the matching summatory function for the Moebius function).


1



1, 6, 92, 406, 1549, 5361, 12546, 41908, 141121, 431822, 1628048, 4657633, 12917328, 40604969, 109205859, 366567325
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OFFSET

1,2


COMMENTS

a(k) is the number of x in interval[1,10^k] such that M(x) = 0, for k >= 1, and is equal to {1,6,92,...}. It is well known that the function M oscillates infinitely around 0 when x tends towards infinity. M(x) = Sum_{n < = x} moebius(n).


LINKS

Table of n, a(n) for n=1..16.
E. Grosswald, Oscillation theorems of arithmetical functions, Trans. AMS 126 (1967), 128.
Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, Mathematics of Computation, Volume 87, No. 310 (2018), pp. 10131028, arXiv:1610.08551 [math.NT].
Eric Weisstein's World of Mathematics, Mertens Function
Eric Weisstein's World of Mathematics, Moebius Function
Wikipedia, Mertens function
Index entries for sequences related to Moebius transforms


EXAMPLE

For k = [1,..,10], a(1) = 1.
For x = [1,..,100], a(2) = 6.
For x = [1, ..., 1000], a(3) = 92.


MATHEMATICA

s={}; sum=0; count=0; Do[ Do[ sum+=MoebiusMu[n]; If[sum==0, count++], {n, 10^k, 10^(k+1)1}]; AppendTo[s, count], {k, 0, 5}]; s (* Amiram Eldar, Jun 19 2018 *)


PROG

(PARI) c = 0; s = 0; for(k = 0, 5, for(n = 10^k, 10^(k+1)1, s+=moebius(n); if(s==0, c++)); print(c)) \\ Amiram Eldar, Jun 19 2018


CROSSREFS

Cf. A002321, A008683, A028442.
Sequence in context: A006151 A005327 A182263 * A278683 A280214 A113266
Adjacent sequences: A171907 A171908 A171909 * A171911 A171912 A171913


KEYWORD

nonn,more


AUTHOR

Michel Lagneau, Dec 31 2009


EXTENSIONS

a(10)a(16) added by Amiram Eldar, Jun 19 2018 from the paper by Hurst


STATUS

approved



