川田, 浩一

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Title素数の3乗の和で表せない自然数の密度について (解析的整数論とその周辺)Author(s)川田, 浩一Citation数理解析研究所講究録 (2009), 1665: 175-184Issue Date2009-10URL http://hdl.handle.net/2433/141041RightType Departmental Bulletin PaperTextversionpublisherKyoto University3 .Koichi KAWADA ( )Faculty of Education, Iwate University( )1. .1937 , 3I. M. Vinogradov , ,. ,, Riemann, unconditional , , 1938Hua [2] . 3, [2] , 93 .Waring-Goldbach( , 1 Goldbach ), ,. ,, ,, ,., 9 3$Hua$ , . $n$ 93 , 9, 1 2. , $n$ 9 3 , $n-2^{3}$8 3 . ,$n=p_{1}^{3}+p_{2}^{3}+\cdots+p_{9}^{3}$$p_{j}$ , “”9 , $n$ ,, 9 3 $n$ , $n$1665 2009 175-184 175., $s$ 3 $n$ ,$q$ ,$n\equiv x_{1}^{3}+x_{2}^{3}+\cdots+x_{s}^{3}$ $(mod q)$$(x_{1}x_{2}\ldots x_{s},q)=1$ $x_{1},$ $x_{2},$ $\ldots,$ $x_{s}$ $n$. $n$ $\mathcal{N}_{s}$ .(Vaughan [9], Lemma 2.14 ) ,4 $s\}_{c}^{}$ $\mathcal{N}_{s}$;$\mathcal{N}_{4}=\{n\in \mathbb{N}:n\equiv 0(mod 2), n\not\equiv\pm 1, \pm 3(mod 9), n\not\equiv\pm 1(mod 7)\}$ ,$\mathcal{N}_{5}=\{n\in N:n\equiv 1(mod 2), n\not\equiv 0, \pm 2(mod 9), n\not\equiv O(mod 7)\}$ ,$\mathcal{N}_{6}=\{n\in \mathbb{N}:n\equiv 0(mod 2), n\not\equiv\pm 1(mod 9)\}$,$\mathcal{N}_{7}=\{n\in \mathbb{N}:n\equiv 1(mod 2), n\not\equiv 0(mod 9)\}$ ,$\mathcal{N}_{s}=\{n\in N:n\equiv s(mod 2)\}$ $(\forall s\geq 8)$ ., 4 $s$ , $n\in \mathcal{N}_{s}$ $s$ 3, . 3 3$x$ $O(X(l\circ gX)^{-3})$, $s\}_{c}^{}$ 4 . Hua, $s\geq 9$ , ., $Hua$ [2] 70 ,$s=8$ $*$ . ,$n\in \mathcal{N}_{s^{\rfloor}}$ , $n\in \mathcal{N}_{s}$ $s$ 3, $5\leq s\leq 8$ $Hua[2]$. , $x$ $n\in \mathcal{N}_{s}$ , $s$3 $n$ $E_{s}(X)$ . $X$ $\mathcal{N}_{s}$$x$ , $E_{S}(X)=\circ(X)(Xarrow\infty)$ $\dagger$ ,$n\in \mathcal{N}_{s}$ $s$ 3 , ,$*$ , , 9 3, ( ). Hua [2], $s\geq 10$ , , , $s$ 3.\dagger , $s\geq 4$ $E_{s}(X)\ll 1$ . Hua [2]$s\geq 9$ , $s$ .176Hua [2] , $5\leq s\leq 8$ , $A$ ,$E_{s}(X)\ll X(\log X)^{-A}$ (1). , $E_{4}(X)=o(X)$ ,circle method , ,$E_{8}(X)\ll 1$ , \ddagger .Hua [2] , (1) $A$\S Schwarz [8] 1961 , Sb Ren [7]2000 ,$E_{5}(X)\ll X^{152/153+\epsilon}$’. Ren [7] $6\leq s\leq 8$ $E_{S}(X)$, circle method $,$ minor arcBessel ,, , $5\leq s\leq 8$:$E_{s}(X)\ll X^{1-(s-4)/153+\epsilon}$ .$x$ , $s=8$ 0.97386 .$s\geq 9$ $E_{S}(X)\ll 1$ , $E_{8}(X)$, , Bessel., 2002 Wooley [10]. Bessel , minor arc 2, , , , 1, . ,, . $E_{s}(X)$, Wooley [10] :$E_{5}(X)\ll X^{35/36+\epsilon}$ , $E_{6}(X)\ll X^{17/18+\epsilon}$ ,(2)$E_{7}(X)\ll X^{23/36+\epsilon}$ , $E_{8}(X)\ll X^{11/36+\epsilon}$ .$\iota_{E_{4}(X)=o(X)(X}arrow\infty)$ $E_{8}(X)\ll 1$ , ., . $E_{4}(X)$, , Kawada [3], Br\"udern-Kawada[1] .\S , , (1) Vinogradov $A$ ., $\epsilon$ . , $\epsilon$.177$23/36=0.63\dot{9},$ $11/36=0.30\dot{5}$ , $s=7$ $s=8$., circle method minor arcWooley [10], , [10]3 ,Wooley $E_{S}(X)$ . ,,wooley $E_{S}(X)$ ,2003([4] ). , [4] , Kumchev,, $E_{S}(X)$ Kumchev.Kumchev [6] .,. , ,$E_{5}(X)\ll X^{79/84}$ , $E_{6}(X)\ll X^{31/35}$ ,$(3)$$E_{7}(X)\ll X^{17/28}$ , $E_{8}(X)\ll X^{23/84}$Kumchev[5] . Wooley , $E_{5}(X)$ ,$E_{7}(X),$ $E_{8}(X)$ 0.03 , $E_{6}(X)$ 0.06 ,. 2003 [4] ,Kumchev , $s=5,7,8$ 0.0065 $,$ $s=6$0.02 , , ., Kumchev ,, .,(alrnost prime) ,( ),. Kumchev[5] , $E_{5}(X)$ , 79/84 79/84-1 $0^{}$, 79/84-1$0^{}$., 79/84-1 $0^{}$, $\searrow$178, , $E_{5}(X)$, 79/84–1.8 $\cdot 10^{-3}$ .$E_{5}(X)\ll X^{107/114}$ , $E_{6}(X)\ll X^{50/57}$ ,$E_{7}(X)\ll X^{23/38}$ , $E_{8}(X)\ll X^{31/114}$ ., Kumchev , $s=$$5,7,8$ 0.0018 , $s=6$ 0.0085 ,. , Kumchev [5], $E_{6}(X)$ ,Wooley [10]. , Kumchev[5], ,.2. circle method ., $5\leq s\leq 8$$s$ , $s=5$ , $E_{5}(X)$.$X$ , $P= \frac{1}{2}X^{1/3}$ , $P<p\leq 2P$$p$$\mathcal{P}$ . , $\mathcal{A}$ ,$f( \alpha;\mathcal{A})=\sum_{x\in A}e(x^{3}\alpha)$$($ , $e(\alpha)=\exp(2\pi i\alpha))$. , $s=5$ , Wooley [10] ,Bessel , , circlemethod :$\mathcal{A}\subset(P, 2P]$ , $1\leq q\leq P^{3/2},$ $(a, q)=1,$ $|q\alpha-a|\leq P^{-3/2}$$q,$ $a$ ,$f(\alpha;\mathcal{A})\ll P^{1-\rho+\epsilon}+P(\log P)^{c}\kappa(q)^{1/2}q^{\epsilon}(1+P^{3}|\alpha-a/q|)^{-1/2}$ (4)( $c$ , $\kappa(q)$ ),$n=x^{3}+p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$ $(x\in \mathcal{A},$ $p_{j}\in \mathcal{P})$$n \in(\frac{1}{2}X,X]\cap \mathcal{N}_{5}$ , $O(X^{1-\frac{2}{3}\rho+\epsilon})$ .“ ” ,, $f(\alpha;\mathcal{A})$ major arc179, , ., (4) , 1 $P$ $\rho$. $\kappa(q)$ , $\kappa(q)$ $q$, $q$, $0$ $u$ ,$\kappa(p^{3u+1})=3p^{-u-1/2}$ , $\kappa(p^{3u+2})=\kappa(p^{3u+3})=p^{-u-1}$. $\kappa(q)\ll q^{-1/3}$ ,$(-1/3)$ , $\kappa(q)$ $O(q^{-1/2})$.$E_{5}(X)$ , , $\mathcal{A}=\mathcal{P}$. Wooley [10] , $\mathcal{A}=\mathcal{P}$ , (4) $\rho=1/24$. ,$E_{5}(X)-E_{5}(X/2)\ll X^{1-\frac{2}{3}\rho+\epsilon}\ll X^{35/36+\epsilon}$ ,$E_{5}(X)\ll X^{35/36+\epsilon}$ Wooley(2) ., , $\mathcal{A}=\mathcal{P}$ , (4) $\rho=1/16$, Kumchev [6] $\rho=1/14$ . $1- \frac{2}{3}\cdot\frac{1}{14}=20/21$, $E_{5}(X)\ll X^{20/21+\epsilon}$, Kumchev (3) ., , $\mathcal{A}=\mathcal{P}$ (4).(4) $\mathcal{A}=P$ , Vinogradov, , ,,, .. ,$f( \alpha;\mathcal{P})=\sum_{p\in P}e(p^{3}\alpha)=\sum_{P<x\leq 2P}e(x^{3}\alpha)-\sum_{P<x\leq 2P}e(x^{3}\alpha)$(5), ,1 Weyl , ,. ,180, $\sum_{m}\sum_{n}e((mn)^{3}\alpha)$ , $m$ $n$.Kumchev , (4) $\mathcal{A}=\mathcal{P},$ $p=1/14$, , (5), – $\mathcal{P}’$ –, . , $x\in \mathcal{P}’$ $x$, (5) –$\rho$. , $\mathcal{P}’$ (5) ,$f( \alpha;\mathcal{P}\cup \mathcal{P}’)=\sum_{P<x\leq 2P}e(x^{3}\alpha)-\sum_{P<x\leq 2P}e(x^{3}\alpha)x\not\in P\cup \mathcal{P}$’(6), , $\mathcal{P}’$ ,(5) . , $\mathcal{A}=\mathcal{P}$(4) $p=1/14$ ,$\mathcal{P}’$ , $\mathcal{A}=\mathcal{P}\cup \mathcal{P}’$ , (4) $\rho$1/14 , . ,$\rho$ , (4) $\mathcal{A}=\mathcal{P}\cup \mathcal{P}’$ $\mathcal{P}’$( , $\rho$)., (4) $\mathcal{A}=\mathcal{P}\cup \mathcal{P}’$, $O(P^{1-\frac{2}{3}\rho+\epsilon})$ , $n \in(\frac{1}{2}X, X]\cap \mathcal{N}_{5}$$n=x^{3}+p_{1}^{3}+p_{2}^{3}+p_{3}^{3}+p_{4}^{3}$ $(x\in \mathcal{P}\cup \mathcal{P}’,$ $p_{j}\in \mathcal{P})$, $R_{1}(n)$ ,$R_{1}(n)>C_{1}P^{2}(\log P)^{-5}$, $C_{1}$ .$x$ , $E_{5}(X)$. , $R_{1}(n)$ , $x\in \mathcal{P}’$$R_{2}(n)$ , $R_{1}(n)$ ,$C_{2}$ , $O(P^{1-\frac{2}{3}\rho+\epsilon})$ $n \in(\frac{1}{2}X, X]\cap \mathcal{N}_{5}$ ,$R_{2}(n)<C_{2}P^{2}(\log P)^{-5}$. $C_{1}>C_{2}$ , $n$$R_{1}(n)>R_{2}(n)$ , $n$ 5 3181. $C_{1},$ $C_{2}$ , $\mathcal{P}’$ , $\mathcal{P}’$ $\rho$, $C_{1},$ $C_{2}$ $p$ , $C_{1}>C_{2}$ $p$. $\rho$ , $E_{5}(X)\ll X^{1-\frac{2}{3}\rho+\epsilon}$, . Kumchev $E_{5}(X)$ , $\rho$ 5/56, ,5/56 7/76 .3. ., $\rho$ Kumchev , ,.Kumchev [6] , , , $(P^{2\rho}, P^{1-8\rho}]$$\mathcal{A}$ , (4). , $Type$ II, , Wooley [10]. , Kumchev , (1, $P^{1-10\rho}]$$x\in(P, 2P]$ $\mathcal{A}$ , (4), , M\"obius $\mu(d)$.$z$ (z) $= \prod_{p<z}p$ ,$\sum_{P<x<2P}$ $e(x^{3} \alpha)=\sum_{P<x\leq 2P}e(x^{3}\alpha)$$\sum$ $\mu(d)$$d|(x,\Pi(P^{1-10\rho}))$$(x,\Pi(P^{1-\overline{1}0\rho}))=1$$= \sum_{d|\Pi(P^{1-10\rho})}\mu(d)$$\sum$ $e((dy)^{3}\alpha)$ . (7)$P/d<y\leq 2P/d$$d|\Pi(P^{1-10\rho})$ , $d$ $P^{1-10\rho}$ ,$d>P^{2\rho}$ , $d$ $(P^{2\rho}, P^{1-8\rho}]$ ,Type II , $d$ , (4). $d\leq P^{2\rho}$ , $d$ ,(7) $y$ Weyl, ( Type I). , (4) $,$ $\mathcal{A}=\{P<x\leq 2P, (x, \Pi(P^{1-10\rho}))=1\}$. Kumchev [5]1 , $A$ $(P, 2P]$, ,.182, 1 .Kumchev $I_{1}=(1, P^{1-10\rho}]$, $I_{2}=(P^{\rho}, P^{\frac{1}{2}-4\rho}]$ ., (Pl-10p) ,$\Pi_{\rho}=\prod_{p\in I_{1}\cup I_{2}}p$(4) . . (7),$(x, \Pi_{\rho})=1\sum_{P<x\leq 2P}e(x^{3}\alpha)=\sum_{d|\Pi_{\rho}}\mu(d)\sum_{P/d<y\leq 2P/d}e((dy)^{3}\alpha)$, $d\leq P^{2\rho}$ , Type I .$d>P^{2\rho}$ , $d|\Pi_{\rho}$ ,(i) $d$ 12 , $d$ $I_{1}$ ,$d$ $P^{1-10\rho}$ , ,$d$ $(P^{2\rho}, P^{1-8\rho}]$ , $OK$ .(ii) $d$ $I_{2}$ 2 , $d=p_{1}p_{2}$d’ $(p_{1}, p_{2}\in I_{2})$, $P^{2\rho}<p_{1}p_{2}\leq P^{1-8\rho}$ , $d$ $(P^{2\rho}, P^{1-8\rho}]$, $OK$ .(iii) $d$ $I_{2}$ 1 ,$d=pd’$ ($p\in I_{2},$ $d’$ $I_{1}$ ) , $d’$$(P^{2\rho}/p, P^{1-8\rho}/p]$ $d”$$\Vert$ ,$d$ $(P^{2\rho}, P^{1-8\rho}]$ $pd”$ , $OK$ ., $d>P^{2\rho}$ Type $II$, $\mathcal{A}=\{P<x\leq 2P;(x, \Pi_{\rho})=1\}$ (4). , $\Pi(P^{1-10\rho})$ $\Pi_{\rho}$, $I_{2}$ , 1., Kumchev. , , Kumchev Buch-stab ,$||$ , $d>P^{2\rho}>P^{1}z^{-4\rho}\geq p$ . $\rho>1/12$ , ., .183weighted sieve ,. , ,, . ,weighted sieve ,( – , ). ,, weighted sieve ,., , Kumchev [5]5/56 $\rho$ , , 7/76,, .References.[1] J. Br\"udern and K. Kawada, (On the Waring-Goldbach problem forcubes“, to appear in Glasgow Math. J.[2] L. K. Hua, “On the representation of numbers as the sum of thepowers of primes,” Math. Z. (1938), 335-346.[3] K. Kawada, “Note on the sum of cubes of primes and an almostprime,” Arch. Math. (Basel) 69 (1997), 13-19.[4] , 3 ,1384 (2004 7 ), 215-220.[5] A. V. Kumchev, “On the Waring-Goldbach problem: Exceptionalsets for sums of cubes and higher powers,“ Canad. J. Math. 57(2005), 298-327.[6] A. V. Kumchev, “On Weyl sums over primes and almost primes,”Michigan Math. J. 54 (2006), 243-268.[7] X.-M. Ren, “The Waring-Goldbach problem for cubes“, Acta Arith.94 (2000), 287-301.[8] Schwarz, $($ Zur Darstellung von Zahlen durch Summen von Primzahl-potenzen,” J. Reine Angew. Math. 206 (1961), 78-112.[9] R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., CambridgeUniv. Press, 1997.[10] T. D. Wooley, “Slim exceptional sets for sums of cubes,“ Canad. J.Math. 54 (2002), 418-428.184

ソスウ ノ 3ジョウ ノ ワ デ アラワセナイ シゼンスウ ノ ミツド ニツイテ カイセキテキ セイスウロン ト ソノ シュウヘン

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ソスウノ 3ジョウノワデアラワセナイシゼンスウノミツドニツイテカイセキテキセイスウロントソノシュウヘン

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ソスウ ノ 3ジョウ ノ ワ デ アラワセナイ シゼンスウ ノ ミツド ニツイテ カイセキテキ セイスウロン ト ソノ シュウヘン

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Kyoto University Research Information Repository

ソスウノ 3ジョウノワデアラワセナイシゼンスウノミツドニツイテカイセキテキセイスウロントソノシュウヘン