Alkan: A Generalization of the Hardy-Littlewood Conjecture

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The retained folder-name PDF is the INTEGERS 22 (2022) #A53 article, 21 pages numbered 1–21; a Markdown reading copy sits beside it. The journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02), and the journal's Zenodo deposit of the article records a Creative Commons license (https://doi.org/10.5281/zenodo.10987450, read 2026-10-02): the Creative Commons Attribution 4.0 license.

E. Alkan, "A Generalization of the Hardy-Littlewood Conjecture," Integers 22 (2022), #A53, 21 pp.; Zenodo deposit https://doi.org/10.5281/zenodo.10987450

Overview

Alkan studies Hardy–Littlewood subadditivity not only for the ordinary prime-counting function but for the counting function $\pi_P(x)$ of a prescribed set of primes $P$. Writing $\langle P\rangle$ for the multiplicative semigroup generated by $P$, the paper also treats convexity phenomena and second-order approximations to weighted prime counts. The original assertion $\pi(x+y)\leq\pi(x)+\pi(y)$ is Conjecture B, equation (1.1) (p. 2); the prime $k$-tuples conjecture is separately stated as Conjecture A (p. 2). The introduction distinguishes the paper’s results from cited background, including Dusart’s validity range $2\leq x\leq y\leq (7/5)x\log x\log\log x$ (p. 3).

Generalized subadditivity. Theorem 1 (p. 4) assumes

$$\pi_P(x)=c\pi(x)+O\!\left(\frac{x}{(\log x)^3}\right),\qquad 0<c\leq1,$$

as equation (1.2). It proves that there are effective constants $K,x_0>0$, depending on $P$, such that

$$\pi_P(x+y)<\pi_P(x)+\pi_P(y)$$

whenever $x,y\geq x_0$ and $Kx/\log x\leq y\leq x$; this is equation (1.3). In §2 (pp. 9–13), (1.2) and the classical prime-number-theorem error term (2.1) yield the two-term expansion (2.3). For $Kx/\log x\leq y\leq x/2$, the substitution $\alpha=1+y/x$ gives the uniform expansion (2.10); the negative term is made larger than the error by (2.11)–(2.13). Equations (2.16)–(2.23) then show $\pi_P(y)>(\alpha-1)\pi_P(x)$. The remaining range $x/2\leq y\leq x$ is handled by the uniform negative bound (2.26) and inequalities (2.27)–(2.28).

Corollary 1 (pp. 4–5) turns Theorem 1 into an almost-all result. For $S_X$ defined by (1.4), at most $O_P(X^2/\log X)$ pairs $(m,n)\in S_X$ violate subadditivity. The proof in §3 bounds the possible exceptions by the two sums in (3.1), estimated in (3.2)–(3.3) (p. 13). The corollary applies, in particular, to primes in a fixed reduced residue class, using the cited prime number theorem for progressions in (3.4); to the complement of prime starting points of a fixed admissible $k$-tuple with $k\geq3$, where the needed upper bound is supplied by the cited Selberg sieve; and to sets satisfying the semigroup asymptotic (1.5), by the cited theorem of Nyman recorded as (3.5) (pp. 13–14).

Convexity and conditional large deviations. Theorem 2 (p. 5) proves unconditionally that, for every infinite $P$ and fixed $0<\lambda<1$, the step function $\pi_P$ is neither $\lambda$-convex nor $\lambda$-concave on any terminal interval. The proof uses the jump relation (4.1) at a prime of $P$ (p. 14). Its quantitative second assertion is conditional on Conjecture A: for every $\epsilon>0$, infinitely many $x,y\to\infty$ satisfy the lower bound (1.6) for

$$\pi((1-\lambda)x+\lambda y)-(1-\lambda)\pi(x)-\lambda\pi(y).$$

The proof (§4, pp. 14–15) introduces the maximal interval counts $\rho$ and $\rho^*$ in (4.2)–(4.3). It invokes, rather than proves, Hensley–Richards’ conditional equality $\rho(N)=\rho^*(N)$ and their unconditional estimate (4.4). Under Conjecture A these give the subadditivity violations (4.5). Rescaling by $x'=x/(1-\lambda)$ and $y'=N/\lambda$, together with the asymptotic calculations (4.6)–(4.9), produces (1.6). The paper explicitly leaves unusually large negative deviations, corresponding to strong failure of $\lambda$-concavity, unresolved (p. 6).

Tauberian uniqueness of Legendre-type approximations. Theorem 3(i) (pp. 6–7) says that if nonnegative prime weights $a_p$ obey the Tauberian condition (1.8), and their summatory function has the precision (1.9), then necessarily its constants are $c=c'$ and $A=1$. Part (ii) gives the analogous result for $\pi_P$: condition (1.10) together with approximation (1.11) forces $c=c'$ and $A=1$. In §5 (pp. 15–19), partial summation gives (5.7) and (5.8); comparison with the Mertens-type hypothesis through (5.9)–(5.13) proves part (ii). For part (i), positivity and the dyadic identity (5.14) yield the growth estimate (5.17); equations (5.18)–(5.21) then force the same two constants. This theorem is a uniqueness statement conditional on the stated approximation, not an existence theorem for such an approximation.

Relation to E855

This source bears on Problem 855.

For E855, take $P$ to be the set of all primes. Then $\pi_P=\pi$, and hypothesis (1.2) holds trivially with $c=1$. If

$$M=\max(x,y),\qquad m=\min(x,y),$$

Theorem 1 therefore proves E855 unconditionally in the region

$$x,y\geq x_0,\qquad m\geq \frac{KM}{\log M}.$$

Thus it covers pairs whose smaller coordinate is not too small relative to the larger one. Its proof also identifies the limitation of this method: near $m\asymp M/\log M$, the negative main term in (2.10) is only of order $M/(\log M)^3$, the same scale as the error allowed by (1.2), so choosing $K$ is what makes the sign detectable. The theorem gives no information for $m<KM/\log M$, a region containing pairs with both coordinates arbitrarily large.

Corollary 1 proves that among $2\leq m\leq M\leq X$, only $O(X^2/\log X)$ pairs can fail the inequality. This is useful as a density reduction, but it cannot establish E855: the latter quantifies over every sufficiently large pair, while an exceptional set of density zero may still contain infinitely many arbitrarily large pairs. The stronger Dusart range quoted in §1 (p. 3) is cited background, not a theorem proved by Alkan.

The sharp obstruction appears in equation (4.5). Assuming the prime $k$-tuples conjecture, for every fixed $0<\epsilon<\log 2$ there are arbitrarily large $u,v$ such that

$$\pi(u+v)\geq\pi(u)+\pi(v)+(\log 2-\epsilon)\frac{v}{(\log v)^2}.$$

Consequently Conjecture A would imply infinitely many counterexamples to E855, with both variables tending to infinity. The logical dependence is essential: (4.5) uses the Hensley–Richards identity $\rho(v)=\rho^*(v)$ only under Conjecture A. The paper proves neither Conjecture A nor an unconditional instance of (4.5), so it does not resolve E855. The weighted convexity estimate (1.6) is obtained from the same conditional interval excess after rescaling; the unconditional nonconvexity part of Theorem 2 merely reflects jumps of a counting function and does not imply failure of subadditivity.

Theorem 3 has no direct implication for E855. It can constrain a proposed second-order asymptotic used in a future subadditivity argument—forcing the denominator shift to be $A=1$ under (1.8) or (1.10)—but it supplies neither the short-interval control needed to construct a violation nor uniform error estimates sufficient to cover the missing range $m<KM/\log M$.