The convergence of an alternating series of Erdos, assuming the Hardy–Littlewood prime tuples conjecture

What's new 2023-08-15

I have just uploaded to the arXiv my paper “The convergence of an alternating series of Erdös, assuming the Hardy–Littlewood prime tuples conjecture“. This paper concerns an old problem of Erdös concerning whether the alternating series ${\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}}$ converges, where ${p_n}$ denotes the ${n^{th}}$ prime. The main result of this paper is that the answer to this question is affirmative assuming a sufficiently strong version of the Hardy–Littlewood prime tuples conjecture.

The alternating series test does not apply here because the ratios ${\frac{n}{p_n}}$ are not monotonically decreasing. The deviations of monotonicity arise from fluctuations in the prime gaps ${p_{n+1}-p_n}$ , so the enemy arises from biases in the prime gaps for odd and even ${n}$ . By changing variables from ${n}$ to ${p_n}$ (or more precisely, to integers in the range between ${p_n}$ and ${p_{n+1}}$ ), this is basically equivalent to biases in the parity ${(-1)^{\pi(n)}}$ of the prime counting function. Indeed, it is an unpublished observation of Said that the convergence of ${\sum_{n=1}^\infty \frac{(-1)^n n}{p_n}}$ is equivalent to the convergence of ${\sum_{n=10}^\infty \frac{(-1)^{\pi(n)}}{n \log n}}$ . So this question is really about trying to get a sufficiently strong amount of equidistribution for the parity of ${\pi(n)}$ .

The prime tuples conjecture does not directly say much about the value of ${\pi(n)}$ ; however, it can be used to control differences ${\pi(n+\lambda \log x) - \pi(n)}$ for ${n \sim x}$ and ${\lambda>0}$ not too large. Indeed, it is a famous calculation of Gallagher that for fixed ${\lambda}$ , and ${n}$ chosen randomly from ${1}$ to ${x}$ , the quantity ${\pi(n+\lambda \log x) - \pi(n)}$ is distributed according to the Poisson distribution of mean ${\lambda}$ if the prime tuples conjecture hold. In particular, the parity ${(-1)^{\pi(n+\lambda \log x)-\pi(n)}}$ of this quantity should have mean asymptotic to ${e^{-2\lambda}}$ . An application of the van der Corput ${A}$ -process then gives some decay on the mean of ${(-1)^{\pi(n)}}$ as well. Unfortunately, this decay is a bit too weak for this problem; even if one uses the most quantitative version of Gallagher’s calculation, worked out in a recent paper of Kuperberg, the best bound on the mean ${|\frac{1}{x} \sum_{n \leq x} (-1)^{\pi(n)}|}$ is something like ${1/(\log\log x)^{-1/4+o(1)}}$ , which is not quite strong enough to overcome the doubly logarithmic divergence of ${\sum_{n=1}^\infty \frac{1}{n \log n}}$ .

To get around this obstacle, we take advantage of the random sifted model ${{\mathcal S}_z}$ of the primes that was introduced in a paper of Banks, Ford, and myself. To model the primes in an interval such as ${[n, n+\lambda \log x]}$ with ${n}$ drawn randomly from say ${[x,2x]}$ , we remove one random residue class ${a_p \hbox{ mod } p}$ from this interval for all primes ${p}$ up to Pólya’s “magic cutoff” ${z \approx x^{1/e^\gamma}}$ . The prime tuples conjecture can then be intepreted as the assertion that the random set ${{\mathcal S}_z}$ produced by this sieving process is statistically a good model for the primes in ${[n, n+\lambda \log x]}$ . After some standard manipulations (using a version of the Bonferroni inequalities, as well as some upper bounds of Kuperberg), the problem then boils down to getting sufficiently strong estimates for the expected parity ${{\bf E} (-1)^{|{\mathcal S}_z|}}$ of the random sifted set ${{\mathcal S}_z}$ .

For this problem, the main advantage of working with the random sifted model, rather than with the primes or the singular series arising from the prime tuples conjecture, is that the sifted model can be studied iteratively from the partially sifted sets ${{\mathcal S}_w}$ arising from sifting primes ${p}$ up to some intermediate threshold ${w<z}$ , and that the expected parity of the ${{\mathcal S}_w}$ experiences some decay in ${w}$ . Indeed, once ${w}$ exceeds the length ${\lambda \log x}$ of the interval ${[n,n+\lambda \log x]}$ , sifting ${{\mathcal S}_w}$ by an additional prime ${p}$ will cause ${{\mathcal S}_w}$ to lose one element with probability ${|{\mathcal S}_w|/p}$ , and remain unchanged with probability ${1 - |{\mathcal S}_w|/p}$ . If ${|{\mathcal S}_w|}$ concentrates around some value ${\overline{S}_w}$ , this suggests that the expected parity ${{\bf E} (-1)^{|{\mathcal S}_w|}}$ will decay by a factor of about ${|1 - 2 \overline{S}_w/p|}$ as one increases ${w}$ to ${p}$ , and iterating this should give good bounds on the final expected parity ${{\bf E} (-1)^{|{\mathcal S}_z|}}$ . It turns out that existing second moment calculations of Montgomery and Soundararajan suffice to obtain enough concentration to make this strategy work.