[Paper Review] Finite generating functions for the sum of digits sequence

Christophe Vignat, Tanay Wakhare|arXiv (Cornell University)|Aug 22, 2017

semigroups and automata theory10 references3 citations

TL;DR

This paper derives finite generating functions for the sum of digits sequence $ s_2(n) $ in base 2, generalizing classical results by Allouche and Shallit. It introduces a finite sum over $ n = 0 $ to $ b^p - 1 $ involving $ s_b(n) $ and a function $ f(n) $, with an extra real parameter $ z $, yielding closed-form expressions in terms of the Hurwitz zeta function and enabling connections to Dirichlet $ L $-functions and stochastic models.

ABSTRACT

We derive some new finite sums involving the sequence $s_{2}\left(n ight),$ the sum of digits of the expansion of $n$ in base $2.$ These functions allow us to generalize some classical results obtained by Allouche, Shallit and others.

Motivation & Objective

• To generalize classical infinite series results involving the sum of digits function $ s_2(n) $ to finite sums over intervals $ [0, b^p - 1] $.
• To extend these results by introducing an auxiliary real parameter $ z $, enabling a broader class of identities through integral transforms.
• To establish connections between the sum of digits function and special functions such as the Hurwitz zeta function and Dirichlet $ L $-functions.
• To provide stochastic representations of the sum of digits function using independent Bernoulli random variables, enabling moment computations.
• To derive explicit expressions for the moments and cumulants of the normalized sum of digits random variable $ Z_N $, and analyze its limiting distribution.

Proposed method

• Derive a finite sum identity involving $ s_b(n) $ and differences of powers $ (z+n)^{-eta} - (z+n+1)^{-eta} $, expressed via the Hurwitz zeta function.
• Use recursive properties of $ s_b(n) $, particularly $ s_b(bn + r) = s_b(n) + r $, to decompose sums over base-$ b $ digit expansions.
• Introduce a parameter $ z $ to generalize identities, allowing extension to families of functions via integral transforms and Laplace-type representations.
• Construct a stochastic model where $ Z_N $ is represented as a sum of independent random variables $ W_k $, each corresponding to digit contributions in base 2.
• Compute the moment generating function of $ Z_N $ using products of hyperbolic cosine terms derived from characteristic functions of $ W_k $.
• Derive cumulant generating functions and explicit formulas for cumulants $ \kappa_{2n}^{(N)} $, showing convergence to a non-Gaussian limit distribution.

Experimental results

Research questions

• RQ1 Can classical infinite series involving $ s_2(n) $, such as those related to the Riemann and Dirichlet zeta functions, be generalized to finite sums with a parameter $ z $?
• RQ2 Does the inclusion of a real parameter $ z $ in the sum $ \sum_{n=0}^{b^p-1} s_b(n) f(n+z) $ allow for closed-form expressions in terms of special functions like the Hurwitz zeta function?
• RQ3 What is the distributional behavior of the normalized sum of digits $ Z_N $, and can its moments and cumulants be explicitly computed?
• RQ4 Does the limiting distribution of $ Z_N $ as $ N \to \infty $ converge to a known distribution, and if so, how does it compare to the uniform distribution on $[-3,3]$?
• RQ5 Can the connection between the sum of digits function and Dirichlet $ L $-functions be formalized through finite generating functions?

Key findings

• The finite sum $ \sum_{n=1}^{b^p-1} s_b(n) \left( \frac{1}{(z+n)^\alpha} - \frac{1}{(z+n+1)^\alpha} \right) $ admits a closed-form expression in terms of differences of Hurwitz zeta functions, as stated in Theorem 1.
• The limit as $ p \to \infty $ at $ z = 0 $ recovers the classical identity involving the Riemann zeta and Dirichlet eta functions, confirming consistency with Allouche and Shallit's result.
• For $ \alpha = 1 $, the sum reduces to $ \sum_{n=1}^{b^p-1} \frac{s_b(n)}{n(n+1)} $, which generalizes the Putnam competition result $ \sum_{n=1}^\infty \frac{s_2(n)}{n(n+1)} = 2\log 2 $.
• The normalized sum of digits $ Z_N $ has mean $ \mu_N = 2^N - \frac{N}{2} - 1 $ and variance $ \sigma_N^2 = \frac{1}{9}\left(2^{2N} - \frac{3}{4}N - 1\right) $, derived from independent Bernoulli components.
• The cumulants of the standardized variable $ \hat{Z}_N $ are given by $ \kappa_{2n}^{(N)} = \frac{9^n}{(4^N - \frac{3}{4}N - 1)^n} \cdot \frac{B_{2n}}{2n} \cdot \frac{4^n(4^{nN} - N - 1) + N}{4^n - 1} $, providing a full moment characterization.
• The limiting cumulants $ \kappa_{2n}^{(\infty)} = \frac{B_{2n}}{2n} \cdot \frac{6^{2n}}{2^{2n} - 1} $ indicate a non-Gaussian limit distribution, asymptotically resembling the uniform distribution on $[-3,3]$.