Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {R}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the number of representations of the even integer n as the sum of two squares, a cube and three biquadrates of primes and we write <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {E}(N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">E</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the number of positive integers n satisfying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n \le N,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mi>N</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n \not \equiv \ 2 \pmod 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≢</mo> <mspace width="4pt" /> <mn>2</mn> <mspace width="4.44443pt" /> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that the expected asymptotic formula for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {R}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> fails to hold. In this paper, we prove <Equation ID="Equ19"> <EquationSource Format="TEX">$\mathcal {E}(N)\ll N^{\frac{5}{12}+\varepsilon }$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">E</mi> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> <mo>≪</mo> <msup> <mi>N</mi> <mrow> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> <mo>+</mo> <mi>ε</mi> </mrow> </msup> </mrow> </math></EquationSource> </Equation>for any <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon > 0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The exponent <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\frac{5}{12}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>5</mn> <mn>12</mn> </mfrac> </math></EquationSource> </InlineEquation> constitutes a refinement of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\frac{61}{144}$</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mn>61</mn> <mn>144</mn> </mfrac> </math></EquationSource> </InlineEquation> due to Liu.

Waring-Goldbach problem: On the sum of unlike powers of primes

期刊论文
发表日期：

2026年6月26日

查看全文

Shuangrui Tian

摘要

Let $\mathcal {R}(n)$ $R (n)$ denote the number of representations of the even integer n as the sum of two squares, a cube and three biquadrates of primes and we write $\mathcal {E}(N)$ $E (N)$ for the number of positive integers n satisfying $n \le N,$ $n \leq N,$ $n \not \equiv \ 2 \pmod 3$ $n ≢ 2 (mod 3)$ such that the expected asymptotic formula for $\mathcal {R}(n)$ $R (n)$ fails to hold. In this paper, we prove $\mathcal {E}(N)\ll N^{\frac{5}{12}+\varepsilon }$ $E (N) ≪ N^{\frac{5}{12} + ε}$ for any $\varepsilon > 0.$ $ε > 0 .$ The exponent $\frac{5}{12}$ $\frac{5}{12}$ constitutes a refinement of $\frac{61}{144}$ $\frac{61}{144}$ due to Liu.