A397823 Decimal expansion of Product_{p prime} (1-1/p)^4 * (1 + 4/p + 1/p^2).

0, 4, 9, 3, 2, 1, 6, 7, 3, 5, 7, 9, 4, 0, 0, 0, 9, 1, 7, 6, 1, 9, 7, 5, 9, 1, 0, 0, 8, 6, 9, 7, 9, 9, 8, 9, 1, 5, 3, 1, 9, 2, 9, 2, 1, 7, 0, 0, 6, 0, 3, 6, 8, 5, 3, 3, 6, 4, 9, 3, 3, 9, 6, 8, 1, 8, 6, 8, 1, 4, 9, 0, 0, 6, 9, 9, 2, 8, 3, 4, 0, 7, 4, 6, 3, 1, 1, 2, 8, 4, 8, 4, 3, 0, 1, 9, 6, 8, 1, 0, 1, 8, 6, 5, 4

Offset

0,2

Comments

This constant, c, appears in the conjectured asymptotic formula for the sixth power moment of the Riemann zeta function on the critical line: Integral_{t=0..T} |zeta(1/2 + i*t)|^6 dt ~ (42/9!) * c * T * log(T)^9 (Conrey and Gosh, 1989; Keating and Snaith, 1998; Conrey and Gonek, 2001).

Hardy and Littlewood (1918) proved that Integral_{t=0..T} |zeta(1/2 + i*t)|^2 dt ~ T * log(T), and Ingham (1926) proved that Integral_{t=0..T} |zeta(1/2 + i*t)|^4 dt ~ (1/(2*Pi^2)) * T * log(T)^4.

References

John Brian Conrey and Amit Ghosh, Mean values of the Riemann zeta-function, III, in: Proceedings of the Amalfi Conference on Analytic Number Theory (Maiori, Italy, 1989), Univ. Salerno, Salerno, Italy, 1992, pp. 35-59.

Jonathan Keating and Nina Snaith, Random matrix theory and some zeta-function moments, lecture at Erwin Schrodinger Institute, Vienna, Sept. 1998.

Links

Table of n, a(n) for n=0..104.

Alain Connes, The Riemann Hypothesis: Past, Present and a Letter Through Time, arXiv:2602.04022 [math.NT], 2026. See pp. 17-18.

John Brian Conrey and Steven Mark Gonek, High moments of the Riemann zeta-function, Duke Math. J., Vol. 107, No. 3 (2001), pp. 577-604; alternative link.

G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta mathematica, Vol. 41, No. 1 (1916), pp. 119-196; alternative link.

A. E. Ingham, Mean-value theorems in the theory of the Riemann zeta-function, Proc. London Math. Soc. (2), Vol. 27 (1926), pp. 273-300.

Example

0.049321673579400091761975910086979989153192921700603...

Prog

(PARI) prodeulerrat((1-1/p)^4 * (1 + 4/p + 1/p^2))

Crossrefs

Cf. A397824.

Sequence in context: A021957 A096301 A396272 * A196819 A296448 A217316

Adjacent sequences: A397818 A397819 A397820 * A397824 A397825 A397826

Keyword

nonn,cons,new

Author

Amiram Eldar, Jul 11 2026

Status

approved