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A Goldbach-Waring problem for unequal powers of primes. (English) Zbl 1232.11101

The author proves that all even numbers \(n\leq x\) having the representation \[ n=p_1^2+p_2^3+p_3^4+p_4^5\;, \]
where the \(p_i\) are prime numbers, with at most \(\ll x^{1633/1680+\varepsilon}\) exceptions (i.e., the cardinality of the, say, “exceptional set”, is bounded in this way, \(\varepsilon>0\) arbitrarily small, as usual). This is, in fact, an instance of the Goldbach-Waring problem (i.e., representation of natural numbers \(n\) as sums of powers of primes, with congruence conditions for \(n\), here \(n\equiv 0\pmod 2\), of course), with unequal powers (because, without further specification, we assume all powers to be the same). See that the abstract contains a serious misprint (exchanging up & down indices for the fourth prime-power!).
The paper contains an extensive literature of the subject and, as the author explains clearly, his improvement of the previous record for the exponent above (i.e., \(1633/1680\) for the exceptional set) stems from an exponential sums estimate due to A. V. Kumchev [Mich. Math. J. 54, 243–268 (2006; Zbl 1137.11054)] for his new treatment of the minor arcs (of course, he works in the usual environment of the circle method, distinguishing major and minor arcs).

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11L07 Estimates on exponential sums

Citations:

Zbl 1137.11054
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Full Text: DOI

References:

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