The generalized Euler–Briggs constants γ(Ω,a,q)\gamma(\Omega,a,q) unify Briggs's constants for arithmetic progressions and Diamond–Ford's generalized Euler constants. Gun, Murty and Saha proved unconditional linear-independence and infinite-dimensionality results for the spaces these constants span, but their algebraic-independence theorems (Theorems 17 and 18) for the infinite families of normalized residues γ∗(Ω,a,q)−γ−∑χ≠χ0αχ,Ω,q∗L(1,χ)\gamma^*(\Omega,a,q)-\gamma-\sum_{\chi\neq\chi_0}\alpha^*_{\chi,\Omega,q}L(1,\chi) are conditional on the Weak Schanuel conjecture. By the closed formula for γ(Ω,a,q)\gamma(\Omega,a,q), each such residue is an explicit Q‾\overline{\mathbb Q}-linear combination of logarithms of primes, so removing the hypothesis subsumes the famous open problem of algebraic independence of logarithms of multiplicatively independent rational numbers (e.g. log⁡2\log 2 and log⁡3\log 3). The problem is to give an unconditional proof of these algebraic independence statements.

Background

Euler's constant γ=lim⁡x→∞(∑n≤x1/n−log⁡x)\gamma=\lim_{x\to\infty}(\sum_{n\le x}1/n-\log x) is expected to be transcendental, yet it is not even known to be irrational. A fruitful strategy in transcendence theory is to study γ\gamma not in isolation but within infinite families of related constants. The Euler–Briggs constants γ(a,q)=lim⁡x→∞(∑n≤x,  n≡a  (mod  q)1n−1qlog⁡x),(a,q)=1,\gamma(a,q)=\lim_{x\to\infty}\Big(\sum_{n\le x,\; n\equiv a\; (\mathrm{mod}\; q)}\frac{1}{n}-\frac{1}{q}\log x\Big),\qquad (a,q)=1, introduced in the 1960s and studied systematically by Lehmer (1975), reduce to γ\gamma when q=1q=1. Diamond and Ford (2008), motivated by the Riemann hypothesis, introduced for each finite set of primes Ω\Omega the generalized Euler constant γ(Ω)=lim⁡x→∞(∑n≤x,(n,PΩ)=11/n−δΩlog⁡x)\gamma(\Omega)=\lim_{x\to\infty}(\sum_{n\le x,(n,P_\Omega)=1}1/n-\delta_\Omega\log x), where PΩ=∏p∈ΩpP_\Omega=\prod_{p\in\Omega}p and δΩ=∏p∈Ω(1−1/p)\delta_\Omega=\prod_{p\in\Omega}(1-1/p); they proved γ(Ω)=δΩ(γ+∑p∈Ωlog⁡p/(p−1))\gamma(\Omega)=\delta_\Omega(\gamma+\sum_{p\in\Omega}\log p/(p-1)). Gun, Murty and Saha (2016) unified the two constructions: for a finite set of primes Ω\Omega and coprime natural numbers a,qa,q with (q,PΩ)=1(q,P_\Omega)=1, the generalized Euler–Briggs constant is γ(Ω,a,q)=lim⁡x→∞(∑n≤x, n≡a (mod q), (n,PΩ)=11/n−δΩlog⁡x/q)\gamma(\Omega,a,q)=\lim_{x\to\infty}(\sum_{n\le x,\, n\equiv a\,(\mathrm{mod}\,q),\,(n,P_\Omega)=1}1/n-\delta_\Omega\log x/q).

A closed formula (due to Gun–Saha–Sinha, 2014) splits γ(Ω,a,q)\gamma(\Omega,a,q) into an 'LL-part' 1φ(q)∑χ≠χ0χˉ(a)L(1,χ)∏p∈Ω(1−χ(p)/p)\frac{1}{\varphi(q)}\sum_{\chi\neq\chi_0}\bar{\chi}(a)L(1,\chi)\prod_{p\in\Omega}(1-\chi(p)/p) and a 'logarithmic part' δΩq(γ+∑p∣qlog⁡p/(p−1)+∑p∈Ωlog⁡p/(p−1))\frac{\delta_\Omega}{q}(\gamma+\sum_{p\mid q}\log p/(p-1)+\sum_{p\in\Omega}\log p/(p-1)). Here χ\chi ranges over the non-principal Dirichlet characters modulo qq (multiplicative characters of (Z/qZ)×(\mathbb Z/q\mathbb Z)^\times, extended by 00 to all integers), L(s,χ)L(s,\chi) is the associated Dirichlet LL-function (the Dirichlet series ∑n≥1χ(n)/ns\sum_{n\ge1}\chi(n)/n^s), and φ\varphi is Euler's totient function. Throughout, Q‾\overline{\mathbb Q} denotes the field of all algebraic numbers; a set of complex numbers is algebraically independent over Q‾\overline{\mathbb Q} if no nonzero polynomial in several variables with coefficients in Q‾\overline{\mathbb Q} vanishes at its elements, and linear independence over Q‾\overline{\mathbb Q} is defined analogously with linear polynomials.

Prior unconditional results concern 'at most one algebraic element' statements: Murty and Saradha (2010) proved that among γ\gamma and the γ(a,q)\gamma(a,q) with 1≤a≤q1\le a\le q, (a,q)=1(a,q)=1, at most one number is algebraic; Murty and Zaytseva (2013) proved the analogue for the γ(Ω)\gamma(\Omega); Gun–Saha–Sinha (2014) proved the corresponding statements for the generalized Euler–Briggs constants using cyclotomic units (algebraic units in cyclotomic fields, that is, fields generated by roots of unity) and Baker's theorem on linear forms in logarithms, which states that a nonzero linear form with algebraic coefficients in logarithms of algebraic numbers is transcendental, and which therefore controls only linear relations. Gun, Murty and Saha (2016) then established unconditional linear-independence results: the Q\mathbb Q- and number-field-KK-spans of the constants have dimensions growing at least linearly with a cutoff (hence are infinite-dimensional), and the Q‾\overline{\mathbb Q}-span of {γ(Ω,a,q)∣Ω∈C(q)}\{\gamma(\Omega,a,q)\mid \Omega\in C(q)\} is infinite-dimensional, where C(q)C(q) denotes the collection of finite sets of primes coprime to qq. All of these tools stop at linear independence: Baker's theorem handles only linear forms in logarithms, and no unconditional method is known that produces algebraic independence of logarithms of multiplicatively independent algebraic numbers.

For algebraic independence, Gun–Murty–Saha assume the Weak Schanuel conjecture: if nonzero algebraic numbers α1,…,αn\alpha_1,\dots,\alpha_n have Q\mathbb Q-linearly independent logarithms, then log⁡α1,…,log⁡αn\log\alpha_1,\dots,\log\alpha_n are algebraically independent (a consequence of Schanuel's conjecture, the central unproved conjecture of transcendence theory). Under it they prove (Theorems 17 and 18) that two infinite families of normalized residues are algebraically independent. Writing γ∗(Ω,a,q)=qγ(Ω,a,q)/δΩ\gamma^*(\Omega,a,q)=q\gamma(\Omega,a,q)/\delta_\Omega and αχ,Ω,q∗=χˉ(a)∏p∈Ω(1−χ(p)/p)(1−1/p)−1∏p∣q(1−1/p)−1\alpha^*_{\chi,\Omega,q}=\bar{\chi}(a)\prod_{p\in\Omega}(1-\chi(p)/p)(1-1/p)^{-1}\prod_{p\mid q}(1-1/p)^{-1}, these residues are γ∗(Ωn,a,q)−γ−∑χ≠χ0αχ,Ωn,q∗L(1,χ)\gamma^*(\Omega_n,a,q)-\gamma-\sum_{\chi\neq\chi_0}\alpha^*_{\chi,\Omega_n,q}L(1,\chi) for Ωn\Omega_n ranging over an infinite 'irreducible' sequence TT of finite prime sets (irreducible means no finite subunion ∪i=1kΩni\cup_{i=1}^k\Omega_{n_i} equals ∪j∈JΩnj\cup_{j\in J}\Omega_{n_j} for a proper subcollection JJ; any sequence of distinct singleton sets qualifies), and analogously for fixed Ω\Omega with an infinite irreducible sequence of moduli qq. By the closed formula, each such residue equals ∑p∈Ωnlog⁡p/(p−1)+∑p∣qlog⁡p/(p−1)\sum_{p\in\Omega_n}\log p/(p-1)+\sum_{p\mid q}\log p/(p-1) — a Q‾\overline{\mathbb Q}-linear combination of logarithms of primes. The unconditional status of these algebraic independence statements is the open problem; the motivating question of the arithmetic nature of γ\gamma itself (not even irrationality is known) remains the famous standalone problem behind the whole circle of ideas.

Problem Statement

Prove, without any unproved hypothesis such as the Weak Schanuel conjecture or Schanuel's conjecture, that the elements of S1={γ∗(Ωn,a,q)−γ−∑χ≠χ0αχ,Ωn,q∗ L(1,χ)  ∣  Ωn∈T}S_1=\Big\{\gamma^*(\Omega_n,a,q)-\gamma-\sum_{\chi\neq\chi_0}\alpha^*_{\chi,\Omega_n,q}\,L(1,\chi)\;\Big|\;\Omega_n\in T\Big\} are algebraically independent over Q‾\overline{\mathbb Q}, where a,qa,q are coprime natural numbers, T={Ωn}T=\{\Omega_n\} is an infinite irreducible sequence of non-empty finite sets of primes coprime to qq, γ∗(Ω,a,q)=qγ(Ω,a,q)/δΩ\gamma^*(\Omega,a,q)=q\gamma(\Omega,a,q)/\delta_\Omega, and αχ,Ω,q∗=χˉ(a)∏p∈Ω(1−χ(p)/p)(1−1/p)−1∏p∣q(1−1/p)−1\alpha^*_{\chi,\Omega,q}=\bar{\chi}(a)\prod_{p\in\Omega}(1-\chi(p)/p)(1-1/p)^{-1}\prod_{p\mid q}(1-1/p)^{-1} — i.e. remove the hypothesis from Theorem 17 of Gun–Murty–Saha (arXiv:1604.02896); and likewise for the family S2S_2 of Theorem 18 (fixed finite prime set Ω\Omega, infinite irreducible sequence of moduli qq coprime to PΩP_\Omega, where irreducibility for moduli means that for every finite subcollection the set of prime divisors of the subcollection's product is not the union of the prime-divisor sets of a proper subcollection, and fixed aa coprime to every modulus). Since by the closed formula each element of S1S_1 is the explicit Q‾\overline{\mathbb Q}-linear combination ∑p∈Ωnlog⁡p/(p−1)+∑p∣qlog⁡p/(p−1)\sum_{p\in\Omega_n}\log p/(p-1)+\sum_{p\mid q}\log p/(p-1) of logarithms of primes, the stated goal already subsumes the two-element case: in the special case q=1q=1 (where the character sum is empty) with singleton sets Ωn={pn}\Omega_n=\{p_n\} for distinct primes pnp_n, the residues are the nonzero rational multiples log⁡pn/(pn−1)\log p_n/(p_n-1) of single prime logarithms, so this subsumed instance is exactly the algebraic independence over Q‾\overline{\mathbb Q} of logarithms of two multiplicatively independent rational numbers, such as log⁡2\log 2 and log⁡3\log 3; settling that two-element instance alone would not constitute a solution of the full statement, which concerns the entire families S1S_1 and S2S_2 for all admissible parameters. A resolution in the opposite direction must prove that nontrivial algebraic relations with algebraic coefficients among such residues exist, exhibiting the relations explicitly.

References

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