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From some older context I am re-considering the following variant of the geometric series $$ s_b(p)=\sum_{k=1}^\infty b^{k^p} $$ for the convergent cases $0 \lt b \lt 1$ and $ 0 \lt p$ first. I'm looking at it via formal power series expansions (because I hope, that this will later allow to extend the consideration to parameters b and/or p, where the series is no more convergent). Let $\beta = \log(b)$ denote the natural logarithm of b, then the k'th term of the series can be expressed by an exponential series $$ \begin{eqnarray} b^{k^p} &=& \exp( \beta k^p ) &=& 1+ (\beta k^p) + {(\beta k^p)^2\over 2!} + \ldots \\ &&&=&1+ \beta k^p + {\beta^2 \over 2!}k^{2p} + \ldots \end{eqnarray} $$and the complete series as a double series where I change the order of summation and arrive at an expression in terms of a constant and zeta's at negative arguments just similar to the Ramanujan-summation (where I replace the bernoulli-numbers by zeta at negative arguments, which can then also be fractional): $$ \begin{eqnarray} s_b(p) &=&C_{b,p} \\ &&&+& 1&+ \beta 1^p &+ {\beta^2 \over 2!}1^{2p} &+ {\beta^3 \over 3!}1^{3p} &+\ldots \\ &&&+& 1&+ \beta 2^p &+ {\beta^2 \over 2!}2^{2p} &+ {\beta^3 \over 3!}2^{3p} &+\ldots \\ &&&+& 1&+ \beta 3^p &+ {\beta^2 \over 2!}3^{2p} &+ {\beta^3 \over 3!}3^{3p} &+\ldots \\ &&&+& \ldots \\ \hline &=&C_{b,p}&+&\zeta(0)&+ \beta \zeta(-p) &+ {\beta^2 \over 2!}\zeta(-2p) &+ {\beta^3 \over 3!} \zeta(-3p) &+\ldots \\ \end{eqnarray} $$ As long as the parameters b,p establish a convergent series this all seems to be fine with some example-computations if the constant $C_{b,p}$ equals the integral $$ C_{b,p} = \int_0^\infty b^{x^p} dx $$ However, I would like to express the integral in some form which extends the pattern of the double series, and I think I've seen something like that the exponential series was extended to the left - which is usually irrelevant because in the denominators appear then the singularities of the factorials at negative arguments. On the other hand, if p equals the reciprocal of a natural number, say p=1/2 then the expression $$ \beta^{-2} {\zeta(2p) \over (-2)! }$$ could make sense and evaluate to a finite value which becomes then a part of the $C_{b,p}$ - constant.
update: In fact for some checked bases b and exponents $p=\frac1n$ the sum, when computed via the serial summation, and that sum, when computed via the double series differ by that simple form of the "constant" if we set $$\lim_{\delta \to 0} {\zeta(1+\delta)\over \Gamma(\delta)} = 1 \qquad \text{ and } (-2)!={(-1)! \over -1} , (-3)!={(-1)! \over (-1)(-2)} , \quad \ldots $$
We get then $$C_{b,p}=\lim_{\delta \to 0} {\zeta(n \cdot p + \delta) \over \Gamma(\delta)} \cdot n! \cdot (- \beta)^{-n} $$ and thus $$C_{b,p}={n! \over (- \beta)^n} $$ [end update]

So I have two questions:

  • Q1: I've heuristically seen, that the upper limit of the integral must be $\infty$ - is that true? (I've seen other examples of Ramanujan-summation, where it is -for instance- 1 and I do not understand the why's and when's...)
  • Q2: Is a meaningful representation of the constant $C_{b,p}$ in terms of the zeta at positive arguments possible? and if, which?
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