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%I #25 Jul 28 2018 11:48:48
%S 1,6,92,406,1549,5361,12546,41908,141121,431822,1628048,4657633,
%T 12917328,40604969,109205859,366567325
%N 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).
%C 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).
%H E. Grosswald, <a href="https://doi.org/10.1090/S0002-9947-1967-0202685-7">Oscillation theorems of arithmetical functions</a>, Trans. AMS 126 (1967), 1-28.
%H Greg Hurst, <a href="https://doi.org/10.1090/mcom/3275">Computations of the Mertens function and improved bounds on the Mertens conjecture</a>, Mathematics of Computation, Volume 87, No. 310 (2018), pp. 1013-1028, <a href="https://arxiv.org/abs/1610.08551">arXiv:1610.08551 [math.NT]</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MertensFunction.html">Mertens Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mertens_function">Mertens function</a>
%H <a href="/index/Mo#MOEBIUS">Index entries for sequences related to Moebius transforms</a>
%e For k = [1,..,10], a(1) = 1.
%e For x = [1,..,100], a(2) = 6.
%e For x = [1, ..., 1000], a(3) = 92.
%t 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 *)
%o (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
%Y Cf. A002321, A008683, A028442.
%K nonn,more
%O 1,2
%A _Michel Lagneau_, Dec 31 2009
%E a(10)-a(16) added by _Amiram Eldar_, Jun 19 2018 from the paper by Hurst
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