We consider Dirichlet series zg,a(s)=ån=1¥ g(na) e-ln s{zeta_{g,alpha}(s)=sum_{n=1}^infty g(nalpha) e^{-lambda_n s}} for fixed irrational α and periodic functions g. We demonstrate that for Diophantine α and smooth g, the line Re(s) = 0 is a natural boundary in the Taylor series case λ n = n, so that the unit circle is the maximal domain of holomorphy for the almost periodic Taylor series ån=1¥ g(na) zn{sum_{n=1}^{infty} g(nalpha) z^n}. We prove that a Dirichlet series zg,a(s) = ån=1¥ g(n a)s{zeta_{g,alpha}(s) = sum_{n=1}^{infty} g(n alpha)^s} has an abscissa of convergence σ 0 = 0 if g is odd and real analytic and α is Diophantine. We show that if g is odd and has bounded variation and α is of bounded Diophantine type r, the abscissa of convergence σ 0 satisfies σ 0 ≤ 1 − 1/r. Using a polylogarithm expansion, we prove that if g is odd and real analytic and α is Diophantine, then the Dirichlet series ζ g,α (s) has an analytic continuation to the entire complex plane.
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机译：我们考虑Dirichlet级数z g，a （s）=å n = 1 ¥ g（na）e -l <sub > n s {zeta_ {g，alpha}（s）= sum_ {n = 1} ^ infty g（nalpha）e ^ {-lambda_n s}}对于固定无理α和周期函数g 。我们证明了对于Diophantineα和光滑g，线Re（s）= 0是泰勒级数情况λ n = n的自然边界，因此单位圆是全纯的最大域几乎周期性的泰勒级数å n = 1 ¥ g（na）z n {sum_ {n = 1} ^ {infty} g （nalpha）z ^ n}。我们证明Dirichlet级数z g，a （s）=å n = 1 ¥ g（na）/ n s {zeta_ {g，alpha}（s）= sum_ {n = 1} ^ {infty} g（n alpha）/ n ^ s}的横坐标为σ 0 =如果g为奇数且为实数解析，且α为Diophantine，则为0。我们证明，如果g为奇数且具有有界变化，并且α为有界丢丢番丁类型r，则收敛的横坐标σ 0 满足σ 0 ≤1-1 / r。使用多对数展开，我们证明如果g为奇数且是实解析，而α为Diophantine，则Dirichlet级数ζ g，α（s）在整个复平面上具有解析连续性。
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