Let \(q_1, \dots, q_k\) be natural numbers and set \(G = {\mathbb Z}/q_1{\mathbb Z} \times \cdots \times {\mathbb Z}/q_k{\mathbb Z}\). For a complex-valued function \(f\) on \(G\), the author considers the multiple Dirichlet series \[ F(s_1, \dots, s_k) = \sum_{n_1, \dots, n_k = 1}^\infty {f(n_1, \dots, n_k) \over n_1^{s_1} \cdots n_k^{s_k}}. \] The series converges absolutely if each \(\operatorname{Re}(s_i) >1\), \(1\leq i \leq k\). By using the theory of the Hurwitz zeta function \[ \zeta(s,x) = \sum_{n=0}^\infty {1 \over (n+x)^s}, \] the author shows that the series can be re-written as \[ {1 \over q_1^{s_1} \cdots q_k^{s_k} }\sum_{(a_1, \dots, a_k)\in G} f(a_1, \dots, a_k) \zeta(s_1, a_1/q_1) \cdots \zeta(s_k, a_k/q_k) \] thereby obtaining a meromorphic continuation of \(F(s_1, \dots, s_k)\). To proceed further, the author imposes the following convergence conditions: \[ \sum_{n_i=1}^{q_i} f(a_1, \dots, n_i, \dots, a_k) = 0 \] for all \(a_j \in {\mathbb Z}/q_j{\mathbb Z}\). Under these conditions, he shows that one can evaluate \(F(1, \dots, 1)\) as \[ {(-1)^k \over q_1 \cdots q_k} \sum_{(a_1, \dots, a_k) \in G} f(a_1, \dots, a_k)\psi(a_1/q_1) \cdots \psi(a_k/q_k), \] where \(\psi(x)\) is the logarithmic derivative of the gamma function \(\Gamma(x)\) (often called the digamma function).
This generalizes a formula derived by the reviewer and N. Saradha [“Transcendental values of the digamma function”, J. Number Theory 125, No. 2, 298–318 (2007; Zbl 1222.11097)]. At the same time, using the theory of the Fourier transform, the author shows that this sum is essentially equal to \[ \sum_{(a_1, \dots, a_k) \in G, a_i\neq 0} \widehat{f}(a_1, \dots, a_k) \log (1 - \zeta_{q_1}^{a_1}) \cdots \log (1 - \zeta_{q_k}^{a_k}). \] If \(k=1\), then this is a linear form in logarithms and one can use Baker’s theory to deduce the transcendental nature of this sum, as was done in the Murty-Saradha paper cited above. But for \(k >1\), there is no corresponding theory that one can invoke. Therefore, the author appeals to Schanuel’s conjecture which states that if \(z_1, \dots, z_n\) are complex numbers linearly independent over \({\mathbb Q}\), then the transcendence degree of the field \[ {\mathbb Q}(z_1, \dots, z_n, e^{z_1}, \dots, e^{z_n}) \] is at least \(n\). This conjecture implies that if \(\alpha_1, \dots, \alpha_n\) are algebraic numbers such that \(\log \alpha_1, \dots \log \alpha_n\) are linearly independent over \({\mathbb Q}\), then these numbers are algebraically independent over \(\mathbb Q\). Therefore, assuming Schanuel’s conjecture for \(k > 1\), the author shows that \(F(1, \dots, 1)\) is either zero or transcendental. At the end of the paper, he also discusses the more general sum of the form \[ \sum_{n_1, \dots, n_k=1}^\infty {f(n_1, \dots, n_k)A_1(n_1) \cdots A_k(n_k) \over B_1(n_1) \cdots B_k(n_k)} \] where the \(A_i(x)\) and \(B_i(x)\) are polynomials with algebraic coefficients and discusses when this is transcendental (again, modulo Schanuel’s conjecture).