\u5df4\u585e\u5c14\u95ee\u9898<\/a> \u5c31\u662f\u8457\u540d\u7684\u5e73\u65b9\u5012\u6570\u4e4b\u548c\u3002\u4e4b\u6240\u4ee5\u79f0\u4e4b\u4e3a\u5df4\u585e\u5c14\u95ee\u9898\uff0c\u56e0\u4e3a\u5df4\u585e\u5c14\u4e0d\u4ec5\u662f\u6b27\u62c9\u7684\u5bb6\u4e61\uff0c\u540c\u65f6\u4e5f\u662f\u4f2f\u52aa\u5229\u5bb6\u65cf\u7684\u5bb6\u4e61\uff0c\u6b27\u62c9\u7684\u535a\u5bfc\u662f\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229\uff08Jacob Bernoulli\uff09\u7684\u5f1f\u5f1f\u7ea6\u7ff0\u00b7\u4f2f\u52aa\u5229\uff08Johann Bernoulli\uff09\u3002\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229\u53d1\u73b0\u8fc7\u51e0\u4e2a\u65e0\u7a77\u7ea7\u6570\u7684\u548c\uff0c\u4f46\u662f\u4ed6\u672a\u80fd\u627e\u51fa\u5e73\u65b9\u5012\u6570\u4e4b\u548c $latex \\sum_{n=1}^\\infty \\frac{1}{n^2}$ \u7684\u89e3\uff0c\u96c5\u5404\u5e03\u00b7\u4f2f\u52aa\u5229\u56e0\u800c\u5199\u9053\uff1a\u300c\u5047\u5982\u6709\u4eba\u80fd\u591f\u6c42\u51fa\u8fd9\u4e2a\u6211\u4eec\u76f4\u5230\u73b0\u5728\u8fd8\u4e3a\u6c42\u51fa\u7684\u548c\u5e76\u80fd\u628a\u5b83\u901a\u77e5\u6211\u4eec\uff0c\u6211\u4eec\u5c06\u4f1a\u5f88\u611f\u8c22\u4ed6\u3002\u300d<\/p>\n \u6b27\u62c9\u7684\u53d1\u73b0\u7531\u89c2\u5bdf\u6b63\u5f26\u51fd\u6570\u7684 \u6cf0\u52d2\u5c55\u5f00 \u5f00\u59cb\u3002<\/p>\n $$\\sin x = x – \\frac{x^3}{3!} + \\frac{x^5}{5!} – \\frac{x^7}{7!} + \\cdots $$<\/p>\n \u4e24\u8fb9\u540c\u65f6\u5904\u4ee5 x\uff0c\u5f97\u5230\uff1a<\/p>\n $$ \\frac{\\sin x}{x} = 1 – \\frac{x^2}{3!} + \\frac{x^4}{5!} – \\frac{x^6}{7!} + \\cdots $$<\/p>\n \u663e\u7136 $latex \\frac{\\sin x}{x}$<\/a> \u7684\u96f6\u70b9\u90fd\u5728 $latex \\pi$ \u7684\u6574\u6570\u500d\u3002<\/p>\n \u63a5\u4e0b\u6765\uff0c\u5c31\u662f\u6b27\u62c9\u7684\u505a\u6cd5\u91cc\u6700\u725b\u903c\u7684\u4e00\u6b65\uff0c\u4ed6\u628a\u6709\u9650\u591a\u9879\u5f0f\u7684\u89c2\u5bdf\u63a8\u5e7f\u5230\u65e0\u7a77\u7ea7\u6570\uff0c\u5e76\u5047\u8bbe\u76f8\u540c\u7684\u6027\u8d28\u5bf9\u4e8e\u65e0\u7a77\u7ea7\u6570\u4e5f\u662f\u6210\u7acb\u7684\uff0c\u4e8e\u662f\u5f97\u5230\u3002<\/p>\n $$\\begin{align} \u6700\u540e\u6211\u4eec\u53ea\u8981\u6bd4\u8f83\u4e00\u4e0b\u4e8c\u6b21\u9879\u7684\u7cfb\u6570\uff0c\u5c31\u53ef\u4ee5\u5f97\u5230\uff1a \u6211\u4eec\u53d1\u73b0\u7b49\u5f0f\u53f3\u8fb9\u51fa\u73b0\u4e86\u5e73\u65b9\u53cd\u6bd4\uff0c\u5316\u7b80\u4e00\u4e0b\u5c31\u53ef\u4ee5\u5f97\u5230\u6211\u4eec\u5f00\u5934\u7684\u516c\u5f0f\uff1a<\/p>\n $$\\sum_{n=1}^\\infty \\frac{1}{n^2} = \\frac{\\pi^2}{6}$$<\/p>\n $$\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}$$<\/p>\n \u9700\u8981\u6307\u51fa\u7684\u662f\uff0c\u6b27\u62c9\u4e2d\u95f4\u7684\u90a3\u4e00\u6b65\u867d\u7136\u662f\u5bf9\u7684\uff08\u53c2\u89c1 \u9b4f\u5c14\u65bd\u7279\u62c9\u65af\u5206\u89e3\u5b9a\u7406\uff08Weierstrass factorization theorem\uff09<\/a>\uff09\uff0c\u4f46\u5728\u5f53\u65f6\u662f\u5e76\u4e0d\u4e25\u5bc6\u7684\uff0c\u800c\u90a3\u6837\u7684\u6027\u8d28\u4e5f\u5e76\u4e0d\u662f\u603b\u662f\u6210\u7acb\u3002\u4f46\u662f\u6b27\u62c9\u786e\u5b9e\u7528\u8fd9\u79cd\u542f\u53d1\u5f0f\u7684\u65b9\u6cd5\u63d0\u524d\u5f97\u5230\u4e86\u6b63\u786e\u7684\u7ed3\u679c\uff0c\u5e76\u4e14\u7528\u7c7b\u4f3c\u7684\u65b9\u6cd5\u8fd8\u53ef\u4ee5\u505a\u5230\u4e00\u7cfb\u5217\u7684\u63a8\u5e7f\uff0c\u5f97\u5230\u66f4\u591a\u5f53\u65f6\u4eba\u4eec\u6240\u4e0d\u77e5\u9053\u7684\u7ed3\u679c\uff0c\u4f8b\u5982\u53ef\u4ee5\u5f97\u5230 $latex \\zeta(4) = \\sum_{n=1}^\\infty \\frac{1}{n^4} = \\frac{\\pi^4}{90}$\u3002<\/p>\n \u800c\u8fd9\u53ea\u8981\u628a\u66f4\u6362\u6700\u540e\u8003\u5bdf\u7684\u7cfb\u6570\u5373\u53ef\uff0c\u66f4\u8fdb\u4e00\u6b65\uff0c\u6211\u4eec\u8fd8\u80fd\u6709\uff1a<\/p>\n $$\\zeta(2n) = \\frac{(2\\pi)^{2n}(-1)^{n+1}B_{2n}}{2\\cdot(2n)!}$$<\/p>\n \u5e76\u4e14\u5f53\u7528\u8fd9\u4e2a\u65b9\u6cd5\u6765\u8003\u5bdf $latex 1-sin(x)$ \u65f6\uff0c\u8fd8\u53ef\u4ee5\u5f97\u5230\u83b1\u5e03\u5c3c\u5179\u7ea7\u6570\uff0c\u800c\u8fd9\u5728\u5f53\u65f6\u662f\u4e00\u4e2a\u5df2\u7ecf\u88ab\u8bc1\u660e\u7684\u7ed3\u679c\u3002<\/p>\n \u6b27\u62c9\u4e94\u5e74\u540e\u56de\u5230\u4e86\u8fd9\u4e2a\u95ee\u9898\uff0c\u5e76\u7ed9\u51fa\u4e86\u4e00\u4e2a\u4e25\u683c\u7684\u8bc1\u660e\u3002<\/p>\n \u6b27\u62c9\u7684\u60f3\u6cd5\u540e\u6765\u88ab\u9ece\u66fc\u5728 1859 \u5e74\u7684\u8bba\u6587\u300a\u8bba\u5c0f\u4e8e\u7ed9\u5b9a\u5927\u6570\u7684\u7d20\u6570\u4e2a\u6570\u300b\uff08On the Number of Primes Less Than a Given Magnitude\uff09\u4e2d\u6240\u91c7\u7528\uff0c\u5e76\u4e14\u8bba\u6587\u4e2d\u5b9a\u4e49\u4e86\u9ece\u66fc\u03b6\u51fd\u6570\uff0c\u8fd9\u7bc7\u8bba\u6587\u4e2d\u4e5f\u7ed9\u51fa\u4e86\u8457\u540d\u7684 \u9ece\u66fc\u731c\u60f3<\/a>\u3002<\/p>\n![\"\"](\"https:\/\/www.shuizilong.com\/house\/wp-content\/uploads\/2020\/01\/Basel.jpg\")
<\/a><\/p>\n\u7531\u7c7b\u6bd4\u4f5c\u51fa\u7684\u53d1\u73b0 \u2014\u2014 \u6b27\u62c9\u6700\u521d\u7684\u63a8\u5bfc<\/h2>\n
![\"\"](\"https:\/\/www.shuizilong.com\/house\/wp-content\/uploads\/2020\/01\/sinxx.gif\")
<\/a><\/p>\n
\n\\frac{\\sin x}{x} &= \\left(1 – \\frac{x}{\\pi}\\right)\\left(1 + \\frac{x}{\\pi}\\right)\\left(1 – \\frac{x}{2\\pi}\\right)\\left(1 + \\frac{x}{2\\pi}\\right)\\left(1 – \\frac{x}{3\\pi}\\right)\\left(1 + \\frac{x}{3\\pi}\\right) \\cdots \\newline \\
\n&= \\left(1 – \\frac{x^2}{\\pi^2}\\right)\\left(1 – \\frac{x^2}{4\\pi^2}\\right)\\left(1 – \\frac{x^2}{9\\pi^2}\\right) \\cdots
\n\\end{align}$$<\/p>\n
\n$$
\n\\frac{1}{3!} = \\frac{1}{6} = -\\left(\\frac{1}{\\pi^2} + \\frac{1}{4\\pi^2} + \\frac{1}{9\\pi^2} + \\cdots \\right) = -\\frac{1}{\\pi^2}\\sum_{n=1}^{\\infty}\\frac{1}{n^2}.
\n$$<\/p>\n\u9ece\u66fc Zeta \u51fd\u6570<\/h2>\n
![\"\"](\"https:\/\/www.shuizilong.com\/house\/wp-content\/uploads\/2020\/01\/\u5c4f\u5e55\u5feb\u7167-2020-01-06-\u4e0b\u534811.51.51.png\")
<\/a><\/p>\n