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Let $\pi(x)$ be the prime-counting function, I'm curious about if a suitable variant of the second Hardy–Littlewood conjecture (this corresponding Wikipedia) $$\pi(x^a+y^b)^\alpha\leq \pi(x^c)^\beta+\pi(y^d)^\gamma\tag{1}$$ can be proved, where the constants $0<a,b,c,d\leq 1$ and the constants $0< \alpha,\beta,\gamma\leq 1$ are very close to our upper limit $1$, for all real numbers $x<y$ with $L<x$ for a suitable choice of a constant $L$.

Question. Is it possible to prove any statement of the type $(1)$ under the cited requirements, for constants $0<a,b,c,d\leq 1$ and constants $0< \alpha,\beta,\gamma\leq 1$ all these (all together/ simultaneously) very close to $1$, for all real numbers $x<y$ for a suitable $L<x$? Many thanks.

I don't know if this type of proposals $(1)$ are in the literature, or are essentially the same original second Hardy–Littlewood conjecture, when we require that those constants are very close to $1$.

If there is relevant literature answer my question as a reference request and I try to search and read those statements from the literature.

References:

[1] G. H. Hardy and J. E. Littlewood, Some problems of ‘Partitio numerorum’ III: On the expression of a number as a sum of primes, Acta Math. (44): 1–70 (1923).

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  • $\begingroup$ I'm waiting to know if this question is potentitally interesting, feel free to criticize it, it is very useful to me with the purpose to avoid ask bad questions. Many thanks for help and the patience of all users. $\endgroup$
    – user142929
    Commented Dec 29, 2019 at 20:01
  • $\begingroup$ I've undeleted this post, that I've deleted two years ago. Can you provide feedback a about if the question is potentially interesting? Many thanks. $\endgroup$
    – user142929
    Commented May 6, 2022 at 19:03

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