A086811 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.

0, 3, 6, 16, 45, 126, 224, 1344, 684, 1116, 4752, 23760, 56784, 286944, 164664, 281472, 2449224, 7371648, 27086400, 160392960, 49635936, 68277888, 1049956992, 6077306880, 1252224000, 3240801792, 2083408128, 4066530048, 35225729280, 142745587200, 717382656000, 6279166033920, 2442775449600, 2080906813440, 2251759104000

Offset: 1

Author

Pieter Moree (moree(AT)mpim-bonn.mpg.de), Aug 05 2003

Keywords

• frac
• nonn

Comments

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.

Links

• Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.
• Tom M. Apostol, The resultant of the cyclotomic polynomials F_m(ax) and F_n(bx), Math. Comp. 29 (1975), 1-6.
• Gennady Bachman, On the coefficients of cyclotomic polynomials, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1991, 86 pp.
• Gennady Bachman, On the coefficients of cyclotomic polynomials, Mem. Amer. Math. Soc. 106 (1993), no. 510, 80 pp.
• Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
• A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.
• M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
• H. G. Gadiyar and R. Padma, Linking the circle and the sieve: Ramanujan-Fourier series, arXiv:math/0601574 [math.NT], 2006.
• Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
• Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, arXiv:0803.2483 [math.NT], 2008.
• Yves Gallot, Pieter Moree, and Huib Hommersom, Value distribution of cyclotomic polynomial coefficients, Unif. Distrib. Theory 6 (2011), 177-206.
• Sherry Gong, On a problem regarding coefficients of cyclotomic polynomials, J. Number Theory 129 (2009), 2924-2932.
• Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.
• G. S. Kazandzidis, On the cyclotomic polynomial: Coefficients, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 1-11.
• G. S. Kazandzidis, On the cyclotomic polynomials: Morphology-Estimates, Bull. Soc. Math. Grèce (N.S.) 4A (1963), 50-73.
• D. H. Lehmer, Some properties of the cyclotomic polynomial, J. Math. Anal. Appl. 15 (1966), 105-117.
• H. Möller, Über die i-ten Koeffizienten der Kreisteilungspolynome, Math. Ann. 188 (1970), 26-38.
• Pieter Moree and Huib Hommersom, Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients, arXiv:math/0307352 [math.NT], 2003.
• K. Motose, Ramanujan's sums and cyclotomic polynomials, Math. J. Okayama U. 47, no 1, (2005), Article 5.
• C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
• C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
• K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
• Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
• Wikipedia, Cyclotomic polynomial.
• Wikipedia, Ramanujan's sum.
• Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Crossrefs

Cf. A013595, A013596, A054532, A054533, A054534, A054535.

Programs

Maple

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:

Formula

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).]

Extensions

More terms from Petros Hadjicostas, Aug 01 2019 using the author's Maple program