Axler: Some Results on a Conjecture of Hardy and Littlewood

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proposition_2_4: Axler's computer-assisted proposition that pi(m+n) <= pi(m)+pi(n) for all integers m, n >= 2 with m+n <= 39,708,229,123, which is the prime of index 1.7 x 10^9.

proposition_5_1: Axler's criterion turning lower and upper expansions of pi(x) in powers of 1/log x into the inequality pi(x+y) <= pi(x)+pi(y) for real max{5393, cx/(log x)^k} <= y <= x and x beyond explicit thresholds.

theorem_1_1: Axler's theorem that pi(m+n) <= pi(m)+pi(n) for all integers m, n >= 2 with m/1950 <= n <= m, proved from explicit prime-counting bounds, a computation for small m+n and earlier results for n >= m/109.

theorem_1_2: Axler's explicit form of Udrescu's theorem: for real 0 < epsilon <= 1, the inequality pi(m+n) <= pi(m)+pi(n) holds for integers n with epsilon m <= n <= m whenever m >= exp(sqrt(0.3426/log(1+epsilon))).

theorem_1_3: Axler's theorem that pi(m+n) <= pi(m)+pi(n) for all integers m >= n >= 2 with n >= c_0 m/(log m)^2, where c_0 = 0.70881678090424862707121, the widest unconditional range in the paper.

theorem_1_4: Axler's theorem that pi(m+n) <= pi(m)+pi(n) for all integers m >= n >= 2 with m+n <= 10^20 and n >= 2 sqrt(m)(1 - 2c_1/(log m + c_1)), where c_1 = 2(1 - log 2).

theorem_1_5: Axler's conditional theorem that, if the Riemann hypothesis is true, pi(m+n) <= pi(m)+pi(n) for all integers m >= n >= 2 with n >= c_2 sqrt(m) log m log(m log^8 m), where c_2 = 1/(4 pi).

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The copy read for this card is arXiv:1909.12625v2 (30 September 2019), 9 pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1909.12625), every other right reserved.

Christian Axler, "Some Results on a Conjecture of Hardy and Littlewood," arXiv:1909.12625 (2019).

Overview

The paper studies the “second Hardy–Littlewood conjecture” (HLC)

$$\pi(m+n)\le \pi(m)+\pi(n)\qquad(m,n\in\mathbb N\setminus\{1\}),$$

labelled (1.2), and proves it in several explicit regions rather than in full. The introduction distinguishes this global conjecture from Lionnet’s diagonal inequality (1.1). The principal unconditional results are: HLC holds when, after ordering $m\ge n$, one has $n\ge m/1950$ (Theorem 1.1); for fixed $0<\varepsilon\le1$, it holds throughout $\varepsilon m\le n\le m$ once

$$m\ge \exp\!\sqrt{0.3426/\log(1+\varepsilon)}$$

(Theorem 1.2); and it holds in the substantially more unbalanced range

$$n\ge c_0m/\log^2m,\qquad c_0=0.70881678090424862707121$$

(Theorem 1.3). These are the paper’s main uniform explicit advances over the previously cited ranges (1.4) and (1.5).

Section 2 develops the computational component. Segal’s criterion, reproduced as Lemma 2.1, says that the full HLC is equivalent to

$$p_k\ge p_{k-q}+p_{q+1}-1$$

for $k\ge3$ and $1\le q\le(k-1)/2$, equation (2.1). Lemma 2.2 identifies the least counterexample sum with the least prime $p_k$ violating (2.1), while Lemma 2.3 records Panaitopol’s reduction to $k\ge9680$ and $34\le q\le(k-1)/27$. A computer calculation then gives Proposition 2.4: HLC holds whenever

$$m+n\le39\,708\,229\,123=p_{1.7\times10^9}.$$

This is a finite verified statement, not an asymptotic theorem; the acknowledgement credits a C++ program for the verification.

The proof of Theorem 1.1 in Section 3 combines Proposition 2.4 with explicit upper and lower rational approximations to $\pi(t)$. With $f_c(t)=t/(\log t-1-c/\log t)$, equation (3.1), Proposition 3.1 supplies a parameterized criterion for real $x\ge y\ge3$ in ratio bands $x/r\le y\le x/s$, subject to the explicit threshold (3.3). Its proof reduces nonnegativity of $\pi(x)+\pi(y)-\pi(x+y)$ to inequalities (3.4)–(3.6). Theorem 1.1 is obtained by taking $b=1.15$, covering $m/1950\le n\le m/109$ with a table of overlapping ratio bands above approximately $3.83\times10^{10}$, using Proposition 2.4 below that point, and invoking the cited earlier bounds (1.4)–(1.5) for the remaining balanced range.

Section 4 proves Theorem 1.2 directly from explicit estimates for $\pi$. Inequalities (4.1) and (4.3) place the $m$- and $n$-terms under a common denominator that also bounds $\pi(m+n)$. The result is uniform only after $\varepsilon$ has been fixed; its threshold deteriorates as $\varepsilon\to0$.

Section 5 gives a more reusable asymptotic device. Equations (5.1) and (5.2) are cited Panaitopol-type lower and upper expansions for $\pi(x)$. Proposition 5.1 shows, for suitable expansion data and $c>\varepsilon>0$, that $\pi(x+y)\le\pi(x)+\pi(y)$ for real $x,y\ge2$ with

$$\max\{5393,cx/\log^k x\}\le y\le x$$

and $x$ at least four explicit thresholds. The proof uses $\log(1+t)\ge t-t^2/2$, followed by the denominator comparisons (5.3)–(5.5); the lower bound $\pi(t)\ge t/(\log t-1)$ is cited from Dusart [3, p. 55]. Theorem 1.3 results from $k=2$, $a_1=1$, $a_2=2.85$, and explicit $\varepsilon,c$. The proof’s final reference to “Proposition 3.1” is evidently a printed cross-reference error: the substitutions described there are into Proposition 5.1.

For bounded total size, Theorem 1.4 proves HLC for $m\ge n\ge2$, $m+n\le10^{20}$, and

$$n\ge2\sqrt m\left(1-\frac{2c_1}{\log m+c_1}\right), \qquad c_1=2(1-\log2).$$

Section 6 derives this from Dusart’s bounds $\pi(x)\le\operatorname{li}(x)$ and $\operatorname{li}(x)-2\sqrt{x}/\log x\le\pi(x)$, recorded as Proposition 6.1 and equations (6.1)–(6.2). The mean-value theorem leads to the central error comparison (6.4), which is discharged in four size ranges for $n$.

Theorem 1.5 is conditional on the Riemann hypothesis. It gives HLC when

$$n\ge \frac{1}{4\pi}\sqrt m\,\log m\,\log(m\log^8m).$$

The input is Dusart’s RH error estimate in Proposition 7.1. Section 7 compares its two endpoint errors against the contribution from $\pi(n)$; inequality (7.1) is the main reduction, and (7.2)–(7.4) handle three ranges of $n$. This remains a restricted-range implication even under RH.

Finally, Section 8 records a conditional obstruction rather than a theorem proved unconditionally in this paper. Under the Prime $k$-tuples Conjecture, the cited Schinzel–Sierpiński identity

$$\rho^*(m)=\limsup_{n\to\infty}(\pi(m+n)-\pi(n))$$

is equation (8.1), attributed to [18, pp. 204–205]. The cited Hensley–Richards estimate [9, p. 380] gives

$$\rho^*(m)-\pi(m)\ge(\log2-\varepsilon)m/\log^2m$$

for sufficiently large $m$, hence (8.2). Consequently, assuming the Prime $k$-tuples Conjecture, every sufficiently large $m$ participates in infinitely many violations of HLC. This appendix establishes only a conditional incompatibility. The paper's reference [9] is the 1973 Proceedings paper, pp. 123–127, so the locator p. 380 falls outside it; p. 380 is a page of Hensley and Richards's Primes in intervals, Acta Arith. 25 (1973/74), 375–391 (its card).

Result pages

• Theorem 1.1 (p. 2; proof pp. 3–4): the inequality for integers $m,n\ge2$ with $m/1950\le n\le m$.
• Theorem 1.2 (p. 2; proof pp. 4–5): for fixed $0<\varepsilon\le1$, the inequality on $\varepsilon m\le n\le m$ for $m\ge e^{\sqrt{0.3426/\log(1+\varepsilon)}}$.
• Theorem 1.3 (p. 2; proof pp. 5–6): the inequality for integers $m\ge n\ge2$ with $n\ge c_0m/\log^2m$.
• Theorem 1.4 (p. 2; proof pp. 6–7): the inequality for $m\ge n\ge2$, $m+n\le10^{20}$ and $n\ge2\sqrt m(1-2c_1/(\log m+c_1))$.
• Theorem 1.5 (p. 2; proof pp. 7–8): under the Riemann hypothesis, the inequality for $m\ge n\ge2$ with $n\ge\sqrt m\log m\log(m\log^8m)/(4\pi)$.
• Proposition 2.4 (p. 2): the computation for $m+n\le39\,708\,229\,123$.
• Proposition 5.1 (p. 5): the criterion behind Theorem 1.3.

Read status: claims checked. The statements of Theorems 1.1 to 1.5 and Propositions 2.4 and 5.1 were read clause by clause on the pages of the copy named above; the proofs were read but not checked, the computation of Proposition 2.4 was not repeated, and nothing here is independently reviewed.

Relation to E855

This source bears on Problem 855.

Write

$$X=\max\{x,y\},\qquad Y=\min\{x,y\}.$$

Then Axler’s $(m,n)$ is E855’s $(X,Y)$, and the target inequality is unchanged by symmetry. The paper studies the stronger assertion for every integer pair $X,Y\ge2$, whereas E855 asks only for a threshold $T$ such that it holds whenever both $X,Y\ge T$.

The most useful unconditional reduction is Theorem 1.3: any integer counterexample must satisfy

$$Y< c_0X/\log^2X, \qquad c_0=0.70881678090424862707121.$$

Theorem 1.1 additionally excludes the balanced cone $Y\ge X/1950$, and Proposition 2.4 excludes every pair with $X+Y\le39\,708\,229\,123$. Thus the unresolved part of E855 is the highly unbalanced, unbounded region in which the smaller argument is large in absolute terms but can be arbitrarily small relative to $X/\log^2X$. Proposition 5.1 could enter an E855 argument as a general mechanism for converting sharper explicit upper and lower bounds for $\pi$ into a wider admissible region; Proposition 3.1 similarly permits computer-assisted patching of fixed ratio bands. Lemmas 2.1–2.3 provide a prime-index formulation suitable for finite verification, although their stated equivalence concerns the full HLC rather than E855’s eventual version.

Theorem 1.2 does not yield E855: choosing $\varepsilon=Y/X$ makes its lower bound for $X$ depend on the pair, and this bound is not uniform as $Y/X\to0$. Likewise, Theorem 1.4 has the finite restriction $X+Y\le10^{20}$, while Theorem 1.5, even under RH, covers only

$$Y\ge \frac{1}{4\pi}\sqrt X\,\log X\,\log(X\log^8X).$$

None supplies a single $T$ valid for all $X,Y\ge T$.

Section 8 is directly relevant in the opposite direction. Equations (8.1)–(8.2) imply, conditional on the Prime $k$-tuples Conjecture, that for every sufficiently large fixed $Y$ there are infinitely many $X$ with

$$\pi(X+Y)>\pi(X)+\pi(Y),$$

and the excess for infinitely many such $X$ is $\rho^*(Y)-\pi(Y)$, at least $(\log2-o(1))Y/\log^2Y$ by the Hensley–Richards bound. Taking both variables beyond any proposed threshold would then refute E855. This is conditional on an unproved conjecture and therefore is not a counterexample or resolution of E855; unconditionally, the paper chiefly localizes any possible counterexamples and supplies explicit tools for excluding broad regions.

Bears on. #855: the paper proves the problem's inequality on explicit regions of integer pairs $X\ge Y\ge2$, namely $Y\ge X/1950$ (Theorem 1.1), $Y\ge c_0X/\log^2X$ (Theorem 1.3), fixed-ratio cones beyond explicit thresholds (Theorem 1.2) and the bounded ranges of Proposition 2.4 and Theorem 1.4, with a wider region under the Riemann hypothesis (Theorem 1.5); its appendix recalls the conditional incompatibility with the prime $k$-tuples conjecture. It does not decide the problem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.