A109507 Let x be a positive number, Lambda(d) = Moebius(d)*[log(x/d)]^2, f(m) = Sum_{d|m} Lambda(d), S(x) = Sum_{m <= x} f(m). Sequence gives nearest integer to S(n).

0, 1, 3, 7, 11, 15, 20, 25, 31, 35, 43, 46, 55, 60, 66, 71, 81, 85, 95, 100, 106, 112, 124, 127, 137, 143, 151, 156, 169, 171, 185, 192, 199, 205, 214, 217, 232, 238, 246, 250, 266, 268, 284, 290, 296, 302, 319, 323, 336, 340, 349, 354, 372, 376, 386, 390, 399

Offset

1,3

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.

T. Nagell, "Introduction to Number Theory", Chelsea Pub., New York, 1964, Chap. VIII.

Links

Table of n, a(n) for n=1..57.

J. J. O'Connor and E. F. Robertson, Atle Selberg.

Atle Selberg, An elementary proof of the prime number theorem for arithmetic progressions, Canad. J. Math., 2, (1949), 66-78.

Atle Selberg, An elementary proof of Dirichlet's theorem about primes in an arithmetic progression, Annals Math., 50, (1949). 297-304.

Eric Weisstein's World of Mathematics, Selberg's Formula.

Formula

Selberg proved that S(x) = 2*x*log(x) + o(x*log(x)).

Mathematica

lmbd[d_, x_] := MoebiusMu[d]*Log[x/d]^2; f[n_, x_] := Block[{d = Divisors[n]}, Plus @@ lmbd[d, x]]; s[x_] := Sum[f[n, x], {n, x}]; Table[ Floor[ s[n]], {n, 57}]

Crossrefs

Cf. A109508.

Sequence in context: A310211 A310212 A310213 * A340699 A160802 A170888

Adjacent sequences: A109504 A109505 A109506 * A109508 A109509 A109510

Keyword

nonn

Author

Jonathan Vos Post and Robert G. Wilson v, Jun 30 2005

Status

approved