8. Terence Tao’s 2022 Entropy-Based Heuristic Approach

In 2022, Terence Tao sketched a novel heuristic approach to RH, leveraging ideas of entropy, randomness, and partial randomness in the primes. While not a full proof, Tao’s perspective has attracted interest as a fresh way to think about the problem. The approach is technical, but we can break down the concepts and recommended readings:

Base Camp 1: Tao’s “Just Barely True” Perspective and De Bruijn–Newman Constant – Understand the notion that RH might be only “barely” true. In 2018, Tao (with Brad Rodgers) proved Newman's conjecture that the de Bruijn–Newman constant $\Lambda \ge 0$[73][74]. This result implies that if RH is true at all, it is “barely true” – in Newman's words, the zeros just line up on $\Re(s)=1/2$ with no room to spare[75]. This has informed Tao’s intuition for his entropy approach. Key references:

“The De Bruijn–Newman constant is non-negative” by Rodgers & Tao (2018) – The paper (published in Duke Math. J.) and Tao’s explanatory blog post[75] show that a certain heat flow evolves the zeros continuously with a time parameter $\Lambda$. $\Lambda \ge 0$ means if you perturb the zeta kernel slightly, simple zeros off the line appear immediately for any negative time, i.e., RH could not remain true under an arbitrarily small deformation. Tao quotes Newman: “the Riemann Hypothesis, if true, is only just barely true”[75]. Reading the introduction and conclusion of this paper (or Tao’s blog summary) is enlightening – it explains how this result gave a new qualitative understanding of RH’s delicacy[75]. This sets the stage for Tao’s 2022 ideas by formalizing the sense in which primes (or zeta zeros) might behave “randomly but with an infinitesimal bias” towards alignment.

“Heuristic sketch of Tao’s approach” (Math Overflow discussions) – In late 2022, mathematicians discussed Tao’s heuristic on forums like MathOverflow. One such thread is “Heuristic argument for the Riemann Hypothesis”, where it’s mentioned that certain “strains of number theorist” believe the evidence for RH might be thin[76], but Tao provided a counter-heuristic to show RH could be natural after all. The highest-voted answer by Pace Nielsen posits: “The Riemann hypothesis is true, if primes are random in certain ways.”[19] – essentially capturing Tao’s theme that assuming enough independence in prime distribution yields RH. While this MO thread predates Tao’s 2022 work, it actually aligns with it (Pace Nielsen cites a site “primes are random” and Tao’s words referencing Newman’s ‘barely’ phrase[77]). Reading these comments and answers[19][20] helps one see what an entropy or randomness-based argument might look like: show that any deviation from RH would introduce too much irregularity in primes (measured by some entropy or variance) that isn’t observed. This “primes are random enough” philosophy is precisely Tao’s approach[19].

Base Camp 2: Entropy in Number Theory – Grasp the entropy concepts Tao applies. Tao’s approach involves assigning an entropy to sequences (like the Möbius function or Liouville function) and studying how it changes under certain operations. He has used similar ideas in other contexts (e.g. in his work on the Erdos discrepancy problem and the Chowla conjecture). Key resources:

“Tao’s blog: Entropy and partial randomness” – Tao often blogs technical but accessible expositions. Notably, “The Chowla conjecture is true on average” (2015) where he introduces an entropy decrement argument for the Möbius function. In that blog (and related lecture notes[78]), he defines a notion of entropy $H(\mathcal{F}_N)$ for sequences and shows how to gradually reduce it – a technique originally from additive combinatorics now applied in analytic number theory. His 2022 heuristic for RH likely draws on these concepts, imagining that if a zero had $\Re(\rho) \ne 1/2$, it would create some “structure” in the primes (or Möbius function) that increases entropy or irregularity beyond what we see. Though Tao’s specific RH posts might not be formally published, understanding his previous entropy techniques is crucial to follow the logic.

“Ergodic Theory and the Möbius Function” by Peter Sarnak – Sarnak’s viewpoint on randomness of Möbius (e.g. in the Möbius disjointness conjecture) complements Tao’s approach. Sarnak posits the Möbius function is random enough that $\sum_{n\le x}\mu(n)f(n) = o(x)$ for any “low complexity” function $f$. Tao’s entropy method formalizes “complexity” via entropy. Sarnak’s lectures (e.g. at ICM 2014 or IAS) give intuition that the Möbius function has no long-range correlation – a fact implied by RH (since RH is equivalent to $M(x)=o(x^{1/2+\epsilon})$[69]). Reading Sarnak’s take shows the connection: proving a kind of strong pseudorandomness for Möbius is essentially proving RH[19]. Tao’s contribution is proposing a path to prove such pseudorandomness by an iterative entropy argument.

Base Camp 3: Tao’s 2022 Outline and Related Work – Examine whatever public material is available from Tao on this. As of now, we rely on secondary descriptions, but they point to certain sources:

Mathstodon and Social Media Hints – Tao made a Mathstodon post in 2022 referencing “a remarkable breakthrough by Guth and Maynard... though far from fully resolving RH”[79]. Guth and Maynard’s breakthrough was on short intervals containing primes (a result related to the Lindelöf hypothesis rather than RH directly, but in the same spirit of bounding primes’ distribution). Tao’s comment suggests he sees these new results as feeding into the entropy heuristic – perhaps using them as input to show primes behave “randomly enough” at small scales to uphold RH. So, reading Guth and Maynard (2022) on mean values of multiplicative functions (Annals of Math, 2023) could be beneficial. They proved that in any interval $[X, X + X^{0.525}]$ there are roughly the expected number of primes[79]. This kind of result indeed increases our confidence in RH because it rules out certain large-scale biases in prime distribution.

Tao’s Notion of “Signals” vs. “Noise” – In various posts (like “Structure and Randomness in the prime numbers” talks), Tao uses an analogy: the prime numbers contain “signals” (structured patterns) and “noise” (randomness). RH would follow if one can show the only “signal” affecting zeros is the trivial one already accounted for (the average spacing), and everything else is “noise” which by law of large numbers stays balanced. An insightful read here is Tao’s UCLA lecture notes “Randomness in the Prime Numbers” (2021) which, while not explicitly about RH, set the stage by describing how primes appear random in many statistical senses. These notes mention concepts like Fourier uniformity of the Möbius function, which ties to zeros off the line (any persistent periodic bias in Möbius would give a Siegel zero or off-line zero). By reading this, one equips oneself with the language Tao likely uses: entropy = measure of randomness, if sequence had an atypical pattern (like a bias corresponding to a character), entropy would drop, etc.

Base Camp 4: Verification and Community Endorsement – See how the community is reacting or building on Tao’s heuristic. Because this is cutting-edge and not a finished proof, one should look at commentary:

“Entropy and Riemann Hypothesis” (discussion panel, 2023) – If available (for instance, a panel at a conference or an article in Notices of AMS where experts discuss new approaches), it would be great to consult. In January 2025, the Notices of the AMS featured an article on recent advances in analytic number theory – possibly Tao’s ideas were mentioned (the search snippet[80] hints at a Jan 2025 Notices piece describing Tao’s work in a broader context). Such an article would likely caution that Tao’s approach is heuristic but also note its creativity. It might connect Tao’s entropy method with earlier heuristic reasoning by Cramér (random primes model) or by Littlewood (his caution that “a long-standing conjecture in analysis generally turns out false” – which was a skeptic’s view on RH[24], countered by newer evidence). This reflective material helps gauge how plausible Tao’s approach is seen by peers and what gaps remain.

In summary, Terence Tao’s entropy-based approach is very new and still speculative. The readings above, mostly from Tao’s own writings and related randomness-in-primes literature, are meant to give intuition for why RH “should be true” by viewing primes through a probabilistic lens[19]. They are widely respected sources – Tao’s blog and papers, Sarnak’s lectures, Rodgers & Tao’s theorem – that provide the insight and context needed to appreciate this approach. If Tao or others succeed in firming up this heuristic into a proof, these base camps will have been our guide into the thought process behind it.