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SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  08 April 2019

ALESSANDRO LANGUASCO Open the ORCID record for ALESSANDRO LANGUASCO [Opens in a new window]  and
ALESSANDRO ZACCAGNINI Open the ORCID record for ALESSANDRO ZACCAGNINI [Opens in a new window]
ALESSANDRO LANGUASCO*
Affiliation:
Università di Padova, Dipartimento di Matematica ‘Tullio Levi-Civita’, Via Trieste 63, 35121Padova, Italy
ALESSANDRO ZACCAGNINI
Affiliation:
Università di Parma, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Parco Area delle Scienze 53/a, 43124Parma, Italy e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

Keywords

Waring–Goldbach problemHardy–Littlewood method

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11P55: Applications of the Hardy-Littlewood method 11P05: Waring's problem and variants
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 3 , December 2020 , pp. 351 - 370
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by I. Shparlinski

References

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