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Jacob's ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations
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abstract
In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth order expression $|\zeta(\frac{1}{2}+i\vp_1(t))|^4|\zf|^2$. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivic. Dedicated to the 75th aniversary of Anatolii Alekseevich Karatsuba.
Forward citations
Cited by 3 Pith papers
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Jacob's ladders, $\zeta$-transformation of the Fourier orthogonal system (2014) and new infinite sets of $\zeta$-equivalents of the Fermat-Wiles theorem
A zeta-weighted integral limit is shown to equal (x^n+y^n)/z^n exactly, making 'limit not equal 1' a verbatim restatement of Fermat's Last Theorem.
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Jacob's ladders and new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theory based on sums of elementary $\zeta$-pulses
The paper restates Moser's ζ-functional as sums over zero-to-zero intervals and derives 'ζ-equivalents' of Fermat's Last Theorem that are tautological consequences of the claimed asymptotic.
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Jacob's ladders, E. C. Titchmarsh's hypothesis (1934) and new $\zeta$-equivalents of the Fermat-Wiles theorem or connections between Fermat's rationals and the Gram's sequence
The paper derives zeta-function expressions that equal x for every x>0, then plugs in Fermat rationals, so its "equivalents" of Fermat's Last Theorem are identities rather than new mathematics.
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