random hyperharmonic series

2012-02-11T16:23:35Z

The Harmonic Series is defined as: $\sum_{n} \frac{1}{n}$ where $n=1,2,3,4....$. This series is known to be divergent.

A generalization of this series can be made by raising each term to $p$: $\sum_{n} \frac{1}{n^p}$ which is also known as the hyperharmonic series and is known to be convergent when $p>1$.

On the other hand, for $p=1$, if the signs of the terms are alternating the sum: $\sum_{n} \frac{(-1)^n}{n}$ is convergent and approaches $\ln{2}$.

A natural extension would be to introduce randomness in the sign of each terms. $\sum_{n} \frac{\epsilon_{n}}{n}$ where $\epsilon_{n}$ is defined by the probability of its outcome: $P(\epsilon_{j} = 1)=P(\epsilon_{j} = -1)=1/2$. This is called the random harmonic series in Schmuland (http://www.stat.ualberta.ca/people/schmu/preprints/rhs.pdf).

My question is: What would happen if we generalize this to the case of the random HYPERharmonic series? What would be the distribution of the result of the summation?

Answer by Noam D. Elkies for random hyperharmonic series

2012-02-11T18:00:42Z

The case $p=2$ is treated briefly in the final section of the cited Schmuland paper, which gives a picture of the distribution. The observations in the paper's first few sections adapt to arbitrary $p>1$: the distribution is not expected to have a simple formula, but the moment-generating function has a product formula (exhibited below), which we can use to compute each power moment ${\mathbb E}(X_p^{2k})$ of this random variable $X_p = \sum_n \epsilon_n/n^p$ as a polynomial in $\zeta(2p), \zeta(4p), \zeta(6p), \ldots, \zeta(2kp)$. [The odd-order moments vanish by symmetry.] This is because the distribution of $X_p$ is the convolution of an infinite series of distributions, the $n$-th of which is supported on $\pm 1 /n^p$ each with probability $1/2$; therefore $$ {\mathbb E}(\exp(tX_p)) = \prod_{n=1}^\infty \left(\frac12 e^{t/n^p} + \frac12 e^{-t/n^p} \right) = \prod_{n=1}^\infty \cosh(t/n^p). $$ We can recover the power moments from the expansion of ${\mathbb E}(\exp(tX_p))$ in a Taylor series about $t=0$, writing $$ \begin{eqnarray} \log {\mathbb E}(\exp(tX_p)) &=& \sum_{n=1}^\infty \log(\cosh(t/n^p)) \cr &=& \sum_{n=1}^\infty \frac12 \left(\frac{t}{n^p_{\phantom1}}\right)^2 - \frac1{12} \left(\frac{t}{n^p_{\phantom1}}\right)^4 + \frac1{45} \left(\frac{t}{n^p_{\phantom1}}\right)^6 - \frac{17}{2520} \left(\frac{t}{n^p_{\phantom1}}\right)^8 + - \cdots \cr &=& \frac{\zeta(2p)}{2} t^2 - \frac{\zeta(4p)}{12} t^4 + \frac{\zeta(6p)}{45} t^6 - \frac{17\zeta(6p)}{2520} t^8 + - \cdots \end{eqnarray} $$ and exponentiating.

While the random variables $X_p$ may be no more than objects of curiosity, similar constructions arise naturally in analytic number theory, such as the value at a fixed $s>1$ of the Dirichlet $L$-series associated to a random real character $\chi$. In this example, the terms $\chi(n)/n^s$ in the Dirichlet series are correlated, but the Euler product $L(s,\chi) = \prod_l (1 - \chi(l)/l^s)^{-1}$ has independent factors, so $\log L(s,\chi)$ is an infinite convolution of the same kind. Likewise if the $\epsilon_n$ were random complex numbers of unit length: the analogue could be $\log L(s,\chi)$ for a random Dirichlet charater $\chi$ that need not be real, or $\log \zeta(\sigma + it)$ for fixed $\sigma>1$ and random real $t$. For these sums of complex-valued random variables, the factors in the moment generating function get more complicated than hyperbolic cosines, but are still tractable.

random hyperharmonic series - MathOverflow

random hyperharmonic series

Answer by Noam D. Elkies for random hyperharmonic series