General Stirling-Ramanujan Constants are exponential periods and applications
Abstract
For n≥ 0, Stirling-Ramanujan constants Sn are the Ramanujan summation of the divergent series ∑k≥ 1 knlog k. These constants are exponential periods over the exponential base field 𝔼=ℚ(t,e−t). We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials P and Q, with ℜ Q(s)>0 for ℜ s >0, the constant S(P,Q) is the Ramanujan summation of the series ∑k≥ 1 P(k)log Q(k). They are exponential periods over 𝕂P((ω)) (t,e−t, (e−ωt)) where 𝕂P is the field of definition of P and (ω) are the zeros of Q. These constants are related to the derivatives ζ'H(−n,w) of Hurwitz zeta function at negative integers, which we prove are exponential periods over 𝔼 ( w,e−wt ). They can be expressed using Bendersky Gamma functions Γ̂n and the values log Γ̂n(w) for ℜ w >0 are exponential periods over 𝔼 ( w,e−wt ). The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over 𝔼. The values ζ(2n+1)/π2n are exponential periods over 𝔼. For a periodic function χ:ℤ→ ℂ, we extend Ramanujan summation to the twisted series ∑k≥ 1 χ(k) P(k)log Q(k) and define General Twisted Stirling-Ramanujan constants Sχ(P,Q). We derive integral formulas proving that they are exponential periods over 𝔼(χ)=ℚ(χ)(t,e−t). For n≥ 0, Lχ'(−n) and Lχ(n+1)/πn are given in terms of these constants and are exponential periods over 𝔼(χ).
Related
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- Supercharacter theory and applications to Ramanujan sums over a finite Frobenius ring
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BibTeX
@misc{hajli2026general,
title = {{General Stirling-Ramanujan Constants are exponential periods and applications}},
author = {Mounir Hajli and Ricardo Perez-Marco},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22113},
eprint = {2608.22113},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22113},
}
RIS
TY - UNPB
TI - General Stirling-Ramanujan Constants are exponential periods and applications
AU - Mounir Hajli
AU - Ricardo Perez-Marco
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2608.22113
UR - https://arxiv.org/abs/2608.22113
AB - For n≥ 0, Stirling-Ramanujan constants S_n are the Ramanujan summation of the divergent series ∑_k≥ 1 k^nlog k. These constants are exponential periods over the exponential base field 𝔼=ℚ(t,e^−t). We generalize this result to a broad class of General Stirling-Ramanujan constants. Given polynomials P and Q, with ℜ Q(s)>0 for ℜ s >0, the constant S(P,Q) is the Ramanujan summation of the series ∑_k≥ 1 P(k)log Q(k). They are exponential periods over 𝕂_P((ω)) (t,e^−t, (e^−ωt)) where 𝕂_P is the field of definition of P and (ω) are the zeros of Q. These constants are related to the derivatives ζ'_H(−n,w) of Hurwitz zeta function at negative integers, which we prove are exponential periods over 𝔼 ( w,e^−wt ). They can be expressed using Bendersky Gamma functions Γ̂_n and the values log Γ̂_n(w) for ℜ w >0 are exponential periods over 𝔼 ( w,e^−wt ). The logarithms of the determinants of the Laplacian on spheres and lens spaces are exponential periods over 𝔼. The values ζ(2n+1)/π^2n are exponential periods over 𝔼. For a periodic function χ:ℤ→ ℂ, we extend Ramanujan summation to the twisted series ∑_k≥ 1 χ(k) P(k)log Q(k) and define General Twisted Stirling-Ramanujan constants S_χ(P,Q). We derive integral formulas proving that they are exponential periods over 𝔼(χ)=ℚ(χ)(t,e^−t). For n≥ 0, L_χ'(−n) and L_χ(n+1)/π^n are given in terms of these constants and are exponential periods over 𝔼(χ).
ER -