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Search: id:A057794
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| A057794 |
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(Integer nearest R(10^n)) - pi(10^n), where pi(x) is the number of primes <= x, R(x) = Sum_{ k>=1 } (mu(k)/k * li(x^(1/k))) and li(x) is the Cauchy principal value of the integral from 0 to x of dt/log(t). |
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+0
4
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| 1, 1, 0, -2, -5, 29, 88, 97, -79, -1828, -2318, -1476, -5773, -19200, 73218, 327052, -598255, -3501366, 23884333, -4891825, -86432204, -127132665
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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This is Riemann's remarkable approximation for the number of primes <= x.
Equivalently, R(x) is given by the Gram series, 1 + sum of log(x)^k/(k*k!*zeta(k+1)) for k = 1 to infinity. This series converges more quickly.
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REFERENCES
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John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 146.
M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 90.
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LINKS
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Tomas Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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MATHEMATICA
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R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Abs[Round[R[10^n]-PrimePi[10^n]]]
gram[x_] := 1+Sum[N[Log[x]^k/(k*k!*Zeta[k+1]), 100], {k, 1, 1000}]; a[n_] := Abs[Round[gram[10^n]-PrimePi[10^n]]]
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PROGRAM
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(PARI) A057794=vector(#A006880, i, round(1+suminf(k=1, log(10^i)^k/(k*k!*zeta(k+1)))-A006880[i])) \\ - M. F. Hasler (MHasler(AT)univ-ag.fr), Feb 26 2008
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CROSSREFS
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Cf. A006880, A057752.
Sequence in context: A061351 A126107 A083472 this_sequence A073715 A104083 A132282
Adjacent sequences: A057791 A057792 A057793 this_sequence A057795 A057796 A057797
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KEYWORD
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sign
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AUTHOR
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Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2000
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EXTENSIONS
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First term corrected by David Baugh (dbaugh(AT)owlnet.rice.edu), Nov 15 2002
Signs added by M. F. Hasler, Feb 26 2008
The value of a(23) is not known at present, I believe. - N. J. A. Sloane (njas(AT)research.att.com), Mar 17 2008
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