|
%I #47 Aug 01 2019 09:07:04
%S 0,3,6,16,45,126,224,1344,684,1116,4752,23760,56784,286944,164664,
%T 281472,2449224,7371648,27086400,160392960,49635936,68277888,
%U 1049956992,6077306880,1252224000,3240801792,2083408128,4066530048,35225729280,142745587200,717382656000,6279166033920,2442775449600,2080906813440,2251759104000
%N Average (scaled by a certain explicit factor) over all integers k of a_k(n), the n-th coefficient of the k-th cyclotomic polynomial.
%C When n is odd the n-th term is an integer. If n is even then twice the n-th term is an integer. Conjecturally (Y. Gallot) the n-th term is always an integer. For n <= 128 this has been verified numerically by Yves Gallot. It is also an unproved conjecture due to H. Möller (1970) that no term of this sequence is negative.
%H Tom M. Apostol, <a href="https://projecteuclid.org/download/pdf_1/euclid.pjm/1102968273">Arithmetical properties of generalized Ramanujan sums</a>, Pacific J. Math. 41 (1972), 281-293.
%H Tom M. Apostol, <a href="https://www.jstor.org/stable/2005456">The resultant of the cyclotomic polynomials F_m(ax) and F_n(bx)</a>, Math. Comp. 29 (1975), 1-6.
%H Gennady Bachman, <a href="https://search.proquest.com/docview/303936346">On the coefficients of cyclotomic polynomials</a>, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1991, 86 pp.
%H Gennady Bachman, <a href="http://dx.doi.org/10.1090/memo/0510">On the coefficients of cyclotomic polynomials</a>, Mem. Amer. Math. Soc. 106 (1993), no. 510, 80 pp.
%H Eckford Cohen, <a href="https://dx.doi.org/10.1073/pnas.41.11.939">A class of arithmetic functions</a>, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
%H A. Elashvili, M. Jibladze, and D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10 (1999), 173-188.
%H M. L. Fredman, <a href="https://doi.org/10.1016/0097-3165(75)90008-4">A symmetry relationship for a class of partitions</a>, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
%H H. G. Gadiyar and R. Padma, <a href="https://arxiv.org/abs/math/0601574">Linking the circle and the sieve: Ramanujan-Fourier series</a>, arXiv:math/0601574 [math.NT], 2006.
%H Emiliano Gagliardo, <a href="http://www.bdim.eu/item?id=BUMI_1953_3_8_3_269_0">Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità</a>, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
%H Yves Gallot, Pieter Moree, and Huib Hommersom, <a href="https://arxiv.org/abs/0803.2483">Value distribution of cyclotomic polynomial coefficients</a>, arXiv:0803.2483 [math.NT], 2008.
%H Yves Gallot, Pieter Moree, and Huib Hommersom, <a href="https://math.boku.ac.at/udt/vol06/no2/13GaMoHo11-2.pdf">Value distribution of cyclotomic polynomial coefficients</a>, Unif. Distrib. Theory 6 (2011), 177-206.
%H Sherry Gong, <a href="https://doi.org/10.1016/j.jnt.2009.04.008">On a problem regarding coefficients of cyclotomic polynomials</a>, J. Number Theory 129 (2009), 2924-2932.
%H Otto Hölder, <a href="http://matwbn.icm.edu.pl/ksiazki/pmf/pmf43/pmf4312.pdf">Zur Theorie der Kreisteilungsgleichung K_m(x)=0</a>, Prace mat.-fiz. 43 (1936), 13-23.
%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&i=50">On the cyclotomic polynomial: Coefficients</a>, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 1-11.
%H G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&i=52">On the cyclotomic polynomials: Morphology-Estimates</a>, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 50-73.
%H D. H. Lehmer, <a href="https://doi.org/10.1016/0022-247X(66)90144-2">Some properties of the cyclotomic polynomial</a>, J. Math. Anal. Appl. 15 (1966), 105-117.
%H H. Möller, <a href="https://doi.org/10.1007/BF01435412">Über die i-ten Koeffizienten der Kreisteilungspolynome</a>, Math. Ann. 188 (1970), 26-38.
%H Pieter Moree and Huib Hommersom, <a href="https://arxiv.org/abs/math/0307352">Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients</a>, arXiv:math/0307352 [math.NT], 2003.
%H K. Motose, <a href="http://escholarship.lib.okayama-u.ac.jp/mjou/vol47/iss1/5">Ramanujan's sums and cyclotomic polynomials</a>, Math. J. Okayama U. 47, no 1, (2005), Article 5.
%H C. A. Nicol, <a href="https://dx.doi.org/10.1073/pnas.39.9.963">On restricted partitions and a generalization of the Euler phi number and the Moebius function</a>, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
%H C. A. Nicol and H. S. Vandiver, <a href="https://dx.doi.org/10.1073/pnas.40.9.825 ">A von Sterneck arithmetical function and restricted partitions with respect to a modulus</a>, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
%H K. G. Ramanathan, <a href="https://www.ias.ac.in/article/fulltext/seca/020/01/0062-0069">Some applications of Ramanujan's trigonometrical sum C_m(n)</a>, Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram21.pdf">On certain trigonometric sums and their applications in the theory of numbers</a>, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cyclotomic_polynomial">Cyclotomic polynomial</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_sum">Ramanujan's sum</a>.
%H Aurel Wintner, <a href="https://www.jstor.org/stable/2371672">On a statistics of the Ramanujan sums</a>, Amer. J. Math., 64(1) (1942), 106-114.
%F Let M_k = k * Product_{prime p<=k} p. Let q be any prime > k. Then the k-th term (for k >= 2) is M_k * Sum_{d|M_k} ( a_d(k) + a_{d*q}(k) )/(2*d). The average of the k-th coefficient of the n-th cyclotomic polynomial is given by the k-th coefficient of this sequence divided by Zeta(2) * k * Product_{p<=k} (p+1). (Zeta(2) = Pi^2/6.) [See Section 8.3 in Moree and Hommerson (2003).]
%p with(numtheory):for k from 1 to 50 do; v := 1: w := 1:j := 1:z := 1:while ithprime(j)<=k do; v := v*ithprime(j); w := w*(1+1/ithprime(j)); z := z*(ithprime(j)+1); j := j+1; end do: v := v*k:z := z*k:q := ithprime(j):te := 0:for i from 1 to nops(divisors(v)) do; d := divisors(v)[i]; kl(x) := 1; for j from 1 to k do; if modp(d,j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(d/j),x,k+1); end if; end do: te := te+coeff(kl(x),x,k)/d; kl(x) := 1; for j from 1 to k do; if modp(q*d,j)=0 then kl(x) := taylor(kl(x)*(1-x^j)^mobius(q*d/j),x,k+1); end if; end do: te := te+coeff(kl(x),x,k)/d; end do: zr := te/(2*w):print(k,zr*z):end do:
%Y Cf. A013595, A013596, A054532, A054533, A054534, A054535.
%K frac,nonn
%O 1,2
%A Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003
%E More terms from _Petros Hadjicostas_, Aug 01 2019 using the author's Maple program
|
