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?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.