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

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: A005327 A182263 A360826 * 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