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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 (list; graph; refs; listen; history; text; internal format)
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), 1-28.

Greg Hurst, Computations of the Mertens function and improved bounds on the Mertens conjecture, Mathematics of Computation, Volume 87, No. 310 (2018), pp. 1013-1028, 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

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Last modified November 26 01:22 EST 2020. Contains 338631 sequences. (Running on oeis4.)