Improving the lower bound for the number of irrational odd zeta values

For $N(s)=\#\{\text{odd } i\in[3,s]:\zeta(i)\notin\mathbb{Q}\}$ , the best established lower bound is Lai's $N(s)\ge 1.284579\sqrt{s/\log s}$ for all sufficiently large $s$ (arXiv preprint arXiv:2501.05321, 2025), obtained by combining Zudilin's $\Phi_n$ factor method with the Fischler–Sprang–Zudilin elimination technique. The open problem is to strictly improve this asymptotic lower bound: prove $N(s)\ge C\sqrt{s/\log s}$ for an explicit $C>1.284579$ , or prove a bound of strictly larger order of growth. Conjecturally every odd zeta value is irrational, so the expected truth is $N(s)\sim s/2$ , but even $s^{1/2+\varepsilon}$ is out of reach of current methods.

Background

The Riemann zeta function is $\zeta(s)=\sum_{m=1}^{\infty}m^{-s}$ for $\operatorname{Re}(s)>1$ , and its values at odd integers $\zeta(3),\zeta(5),\zeta(7),\ldots$ (odd zeta values) are famously mysterious arithmetic constants: $\zeta(2k)$ is a rational multiple of $\pi^{2k}$ and hence transcendental, but for no single odd $s>3$ is the arithmetic nature of $\zeta(s)$ known. Apéry (1979) proved $\zeta(3)$ irrational. Rivoal (2000), with Ball–Rivoal (2001), proved that infinitely many odd zeta values are irrational: the dimension of the $\mathbb{Q}$ -linear span (the number of rationally independent combinations) of $1,\zeta(3),\zeta(5),\ldots,\zeta(s)$ is at least $(1-o(1))\log s/(1+\log 2)$ . Zudilin (2001) proved that at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational; in 2018 he extended this to: at least one of $\zeta(5),\ldots,\zeta(25)$ . These results rest on the linear-forms method: one constructs explicit rational linear combinations $L=\sum a_i\zeta(i)+a_0$ (linear forms) that are simultaneously very small in absolute value and have controlled denominators, then applies a linear-independence criterion (small value plus small common denominator forces many coefficients-weighted irrationality). Zudilin's key device is the $\Phi_n$ factor method: multiplying the forms by arithmetic factors built from Euler's totient function $\varphi$ (collections of denominators $b$ with small $\varphi(b)$ ) to reduce the size of common denominators. Fischler–Sprang–Zudilin (2019) introduced an elimination technique that removes unwanted terms from the linear combinations, converting span information into a count of irrational values; Lai–Yu (2020) used it to prove $N(s)\ge 1.192507\sqrt{s/\log s}$ , the first bound of order $\sqrt{s/\log s}$ for the count. Fischler (2021) gave a different construction based on Siegel's lemma (the classical principle that an underdetermined system of homogeneous linear equations with integer coefficients admits a nonzero integer solution of small height), producing non-explicit linear forms by linear algebra and proving that the dimension of the span of $1,\zeta(3),\ldots,\zeta(s)$ is at least $0.21\sqrt{s/\log s}$ . Lai (2025, arXiv:2501.05321) combined the elimination technique with the $\Phi_n$ factor method and computer optimization of parameters ( $M$ , shifts $\delta_j$ ) to reach $N(s)\ge 1.284579\sqrt{s/\log s}$ for all sufficiently large $s$ , the current record. The same paper recalls that the irrationality of each individual $\zeta(5),\zeta(7),\ldots$ remains open. Conjecturally all odd zeta values are irrational, so $N(s)$ should grow like $s/2$ ; the gap between $\sqrt{s/\log s}$ and linear growth is wide open, and no lower bound of order $s^{1/2+\varepsilon}$ is known. The obstacle in the current machinery is a structural trade-off: the number of available linear forms scales like $B^2$ under the range constraint $s\ge 10(2r+1)MB^2$ , capping the count near $\sqrt{s}$ , while the denominator arithmetic inserts the $1/\sqrt{\log s}$ loss.

Problem Statement

Let $\zeta(s)=\sum_{m\ge 1}m^{-s}$ and let $N(s)=\#\{\text{odd } i\in[3,s]:\zeta(i)\notin\mathbb{Q}\}$ count the irrational odd zeta values up to $s$ . Lai (2025) proved that $N(s)\ge 1.284579\sqrt{s/\log s}$ for all sufficiently large integers $s$ . Improve this asymptotic lower bound rigorously: prove that $N(s)\ge C\sqrt{s/\log s}$ for all sufficiently large $s$ for some explicit constant $C>1.284579$ , or prove a lower bound for $N(s)$ of strictly larger order of growth (for example $N(s)\ge s^{\alpha}$ for some $\alpha>1/2$ , or $N(s)\ge c\,s$ for a constant $c>0$ ). The ultimate conjectural truth, $N(s)\sim s/2$ (all odd zeta values irrational), remains the guiding target.

References

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T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 4, 267–270. doi:10.1016/S0764-4442(00)01624-4, https://doi.org/10.1016/S0764-4442(00)01624-4
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