For a finite set of primes Ω\Omega and coprime integers a,qa,q with q>1q>1 and (q,∏p∈Ωp)=1(q,\prod_{p\in\Omega}p)=1, the generalized Euler–Briggs constant γ(Ω,a,q)\gamma(\Omega,a,q) is the limit of the harmonic sum over n≡a(modq)n\equiv a\pmod{q} with no prime factor in Ω\Omega, suitably renormalized by a logarithm; it simultaneously generalizes the Euler–Lehmer (Euler–Briggs) constants γ(a,q)=γ(∅,a,q)\gamma(a,q)=\gamma(\varnothing,a,q) and the Diamond–Ford generalized Euler constants γ(Ω)\gamma(\Omega). Gun, Saha and Sinha (J. Number Theory 145, 2014) proved that the family {γ(Ω,a,q)}\{\gamma(\Omega,a,q)\} with Ω\Omega varying over all admissible finite prime sets is infinite yet contains at most one algebraic number, and dually that for fixed Ω\Omega and an infinite pairwise-coprime sequence of moduli qiq_i the family {γ(Ω,a,qi)}\{\gamma(\Omega,a,q_i)\} contains at most one algebraic number. The open problem is whether this single possible algebraic exception ever occurs: does there exist any admissible parameter for which γ(Ω,a,q)\gamma(\Omega,a,q) is algebraic and, if so, which parameter and what algebraic value; or, conversely, is every member of each family transcendental? Because the admissible sets include Ω=∅\Omega=\varnothing, a negative answer for even one family would prove the Euler–Lehmer constant γ(a,q)\gamma(a,q) transcendental — no such constant is currently known to be irrational — while an affirmative answer with nonempty Ω\Omega would force Euler's constant γ\gamma itself to be transcendental.

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 the most famous number whose arithmetic nature is unknown: it is not even known to be irrational. A standard strategy in transcendence theory is to study a constant as a member of a family rather than in isolation. Briggs (1961) and Lehmer (1975) introduced the Euler–Briggs (or Euler–Lehmer) constants of arithmetic progressions, defined for coprime integers a,qa,q with q>1q>1 by γ(a,q)=lim⁡x→∞(∑n≤x, n≡a(modq)1/n−1qlog⁡x)\gamma(a,q)=\lim_{x\to\infty}\big(\sum_{n\le x,\,n\equiv a\pmod q}1/n-\tfrac{1}{q}\log x\big), so that each residue class carries its own Euler-type constant. Diamond and Ford (2008), motivated by the Riemann hypothesis, introduced the generalized Euler constants γ(Ω)=lim⁡x→∞(∑n≤x, (n,∏p∈Ωp)=11/n−δΩlog⁡x)\gamma(\Omega)=\lim_{x\to\infty}\big(\sum_{n\le x,\,(n,\prod_{p\in\Omega}p)=1}1/n-\delta_\Omega\log x\big), where Ω\Omega ranges over finite sets of primes, 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) (with P∅=δ∅=1P_\varnothing=\delta_\varnothing=1); these are the constants obtained by sifting the harmonic sum by the primes in Ω\Omega. Gun, Saha and Sinha (2014) unified the two constructions: for a finite set of primes Ω\Omega with (q,PΩ)=1(q,P_\Omega)=1 they defined the generalized Euler–Briggs constants γ(Ω,a,q)=lim⁡x→∞(∑n≤xn≡a(modq)(n,PΩ)=11n−δΩqlog⁡x).\gamma(\Omega,a,q)=\lim_{x\to\infty}\Big(\sum_{\substack{n\le x\\ n\equiv a\pmod q\\ (n,P_\Omega)=1}}\frac{1}{n}-\frac{\delta_\Omega}{q}\log x\Big). Taking Ω=∅\Omega=\varnothing recovers γ(a,q)\gamma(a,q), and taking q=1q=1 recovers γ(Ω)\gamma(\Omega). The arithmetic of these constants is studied through a closed formula expressing γ(Ω,a,q)\gamma(\Omega,a,q) as a linear combination of values L(1,χ)L(1,\chi) of Dirichlet LL-functions (the Dirichlet series L(s,χ)=∑n≥1χ(n)n−sL(s,\chi)=\sum_{n\ge1}\chi(n)n^{-s} attached to Dirichlet characters χ\chi, i.e. multiplicative periodic functions on the integers), of logarithms of primes, and of γ\gamma itself. The transcendence engine is Baker's theorem — a nonvanishing linear form in logarithms of algebraic numbers with algebraic coefficients is transcendental — combined with the multiplicative independence of certain units (Ramachandra units) in cyclotomic fields, i.e. the fields Q(ζq)\mathbb{Q}(\zeta_q) obtained by adjoining a primitive qq-th root of unity ζq\zeta_q, and the theorem of Baker–Birch–Wirsing on nonvanishing of ∑f(n)/n\sum f(n)/n for algebraic-valued periodic functions ff. With these tools Murty and Saradha (2010) proved that among the φ(q)+1\varphi(q)+1 numbers γ\gamma and γ(a,q)\gamma(a,q) (1≤a≤q1\le a\le q, (a,q)=1(a,q)=1, φ\varphi being Euler's totient) at most one is algebraic, and Murty and Zaytseva (2013) proved that among all γ(Ω)\gamma(\Omega) at most one is algebraic. Gun, Saha and Sinha (2014) proved the corresponding and unifying statements: (Theorem 2) for fixed coprime a,qa,q with q>1q>1, the family T(a,q)={γ(Ω,a,q):Ω finite, (q,PΩ)=1}T(a,q)=\{\gamma(\Omega,a,q):\Omega\text{ finite},\,(q,P_\Omega)=1\} is infinite and contains at most one algebraic element; and (Theorem 3) for a fixed finite set of primes Ω\Omega and an infinite sequence S={q1,q2,… }S=\{q_1,q_2,\dots\} of pairwise coprime integers qi>1q_i>1 with (qi,PΩ)=1(q_i,P_\Omega)=1, and any aa coprime to every qiq_i, the family {γ(Ω,a,qi):qi∈S}\{\gamma(\Omega,a,q_i):q_i\in S\} contains at most one algebraic element. They also proved that for nonempty Ω\Omega the difference γ(Ω,a,q)−δΩqγ\gamma(\Omega,a,q)-\tfrac{\delta_\Omega}{q}\gamma is transcendental. These dichotomy results, however, identify no specific member as transcendental: the methods structurally cannot decide which member, if any, is algebraic.

Problem Statement

For a finite set of primes Ω\Omega write 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), with P∅=δ∅=1P_\varnothing=\delta_\varnothing=1, and for coprime integers a,qa,q with q>1q>1 and (q,PΩ)=1(q,P_\Omega)=1 define γ(Ω,a,q)=lim⁡x→∞(∑n≤xn≡a(modq)(n,PΩ)=11n−δΩqlog⁡x).\gamma(\Omega,a,q)=\lim_{x\to\infty}\Big(\sum_{\substack{n\le x\\ n\equiv a\pmod q\\ (n,P_\Omega)=1}}\frac{1}{n}-\frac{\delta_\Omega}{q}\log x\Big). Gun, Saha and Sinha (2014) proved: (Theorem 2) for fixed coprime a,qa,q with q>1q>1, the family T(a,q)={γ(Ω,a,q) : Ω a finite set of primes with (q,PΩ)=1}T(a,q)=\{\gamma(\Omega,a,q)\,:\,\Omega\text{ a finite set of primes with }(q,P_\Omega)=1\} is infinite and contains at most one algebraic number; and (Theorem 3) for a fixed finite set of primes Ω\Omega and an infinite sequence of pairwise coprime integers qi>1q_i>1 with (qi,PΩ)=1(q_i,P_\Omega)=1, and any aa coprime to all qiq_i, the family {γ(Ω,a,qi)}\{\gamma(\Omega,a,q_i)\} contains at most one algebraic number. The open problem left by this work is to determine whether the single possible algebraic exception actually occurs in any of these families. Concretely: (i) does there exist any parameter triple (Ω,a,q)(\Omega,a,q) as above — including Ω=∅\Omega=\varnothing, i.e. the classical Euler–Lehmer constants γ(a,q)\gamma(a,q) — for which γ(Ω,a,q)\gamma(\Omega,a,q) is algebraic, in which case one would of course want to identify the exceptional Ω\Omega (respectively the exceptional modulus qiq_i) and the algebraic value and its degree; or (ii) prove unconditionally that no exception occurs, i.e. that no parameter triple (Ω,a,q)(\Omega,a,q) as above has γ(Ω,a,q)\gamma(\Omega,a,q) algebraic — equivalently, that every member of every family T(a,q)T(a,q) and of every family {γ(Ω,a,qi)}\{\gamma(\Omega,a,q_i)\} is transcendental. Note that a negative answer for a single family T(a,q)T(a,q) — partial progress toward (ii) — would in particular prove the Euler–Lehmer constant γ(a,q)\gamma(a,q) transcendental, while a positive answer with nonempty Ω\Omega would prove Euler's constant γ\gamma transcendental; both are far beyond current methods, and the source paper's linear-forms techniques establish only the 'at most one' dichotomy.

References

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