Open problem, Arithmetic and number theory, posed 1734

Irrationality of ζ(5) and odd zeta values

Open

Is $\zeta(5) = \sum_{n=1}^\infty \frac{1}{n^5}$ irrational? More generally, is $\zeta(2k+1)$ irrational (and transcendental) for every integer $k \ge 2$ ?

Research frontier as of 2026

As of 2026, $\zeta(3)$ remains the only specific odd zeta value proved to be irrational, and it is still unknown whether $\zeta(3)$ is transcendental or whether $\zeta(5)$ is irrational. Refinements of the Ball–Rivoal and Zudilin hypergeometric constructions, together with Fischler–Sprang–Zudilin (2019) elimination techniques, have produced explicit lower bounds for the number of irrational values among $\zeta(3), \zeta(5), \dots, \zeta(2s+1)$ (such as at least $2$ irrational values among $\zeta(3), \zeta(5), \zeta(7), \zeta(9), \zeta(11)$ ), yet no known linear-form construction eliminates $\zeta(7), \zeta(9), \zeta(11)$ while keeping denominators small enough to isolate $\zeta(5)$ alone.

Best known results

Ball and Rivoal (2001): for odd $a \ge 3$ , the $\mathbb{Q}$ -span of $1, \zeta(3), \zeta(5), \dots, \zeta(a)$ has dimension at least $\frac{1+o(1)}{1+\ln 2} \ln a$ .
Zudilin (2001): at least one of the four numbers $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational.
Fischler, Sprang, and Zudilin (2019): among $\zeta(3), \zeta(5), \dots, \zeta(2s+1)$ , at least $2^{(1-o(1)) \frac{\ln s}{\ln \ln s}}$ values are irrational.

Tools and where they stop

Tool	Achieved	Where it stops
Well-poised hypergeometric series and Nesterenko's linear independence criterion (Ball–Rivoal–Zudilin)	Produces small linear forms in $1, \zeta(3), \zeta(5), \dots, \zeta(2s+1)$ with controlled common denominators $\mathrm{lcm}(1, 2, \dots, n)^d$ , proving dimension lower bounds and Zudilin's four-value theorem	The symmetry of the rational functions needed to cancel $\zeta(2), \zeta(4), \dots$ forces the exponent $d$ (and hence the denominator growth $e^{dn}$ ) to exceed the analytic decay when only $1$ and $\zeta(5)$ are retained
Hankel determinants and holomorphic modular / Padé methods (Apéry–Beukers)	Explains Apéry's recurrence for $\zeta(2)$ and $\zeta(3)$ via triple integrals and modular forms of weight $2$ and $4$	For $\zeta(5)$ , natural $5$ -fold Beukers-type integrals have arithmetic denominators $\mathrm{lcm}(1, \dots, n)^5$ that outgrow the geometric convergence rate of the integral

Open questions

Is $\zeta(5)$ irrational?
Are $1, \pi, \zeta(3), \zeta(5), \zeta(7), \dots$ algebraically independent over $\mathbb{Q}$ ?

References

Roger Apéry (1979). Irrationalité de ζ(2) et ζ(3)
Keith Ball, Tanguy Rivoal (2001). Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs · DOI:10.1007/s002220100168
Wadim Zudilin (2001). One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational · DOI:10.1070/RM2001v056n04ABEH000427
Stéphane Fischler, Johannes Sprang, Wadim Zudilin (2019). Many odd zeta values are irrational · DOI:10.1112/S0010437X1900722X · arXiv:1803.08905