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Investigating the Subadditivity of the Prime Counting Function π(z) and Its Implications for the Second Hardy–Littlewood Conjecture

This paper investigates the subadditive properties of the prime counting function π(z) and its relationship with the Second Hardy–Littlewood Conjecture, which suggests that the prime counting function satisfies the inequality π(x + y) ≤ π(x) + π(y). We analyze this conjecture through an exploration of specific properties of prime k-tuples and their ...
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THE POSSIBILITY OF TSCHEBYCHEFF QUADRATURE ON INFINITE INTERVALS. [PDF]

Proc Natl Acad Sci U S A, 1961
Wilf HS.
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Violations of transitivity: Implications for a theory of contextual choice. [PDF]

J Exp Anal Behav, 1993
Grace RC.
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Proof of the Hardy-Littlewood K-tuple Conjecture in the Distribution of Numbers Coprime with the Primorial

In the symmetries in the numbers that are coprime with the primorial we find proof of the existence of infinitely many twin primes and prime k-tuples.
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On the conjecture of Hardy & Littlewood concerning the number of primes of the form 𝑛²+𝑎 [PDF]

Mathematics of Computation, 1960
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On the Hardy-Littlewood prime tuples conjecture and higher convolutions of Ramanujan sums

Functiones et Approximatio Commentarii Mathematici, 2023
In the paper under review, the authors generalize the method of \textit{H. G. Gadiyar} and \textit{R. Padma} [Physica A. 269, 503--510 (1999)], which is based on a simple orthogonality principle for Ramanujan sums originally discovered by \textit{R. D. Carmichael} [Proc. Lond. Math. Soc.
Chaubey, Sneha, Goel, Shivani, Murty, M. Ram   +2 more
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The Hardy-Littlewood conjecture. An algebraic approach

Journal of Mathematical Sciences, 1996
See the review in Zbl 0805.11073.
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