浣浼版板锛1+2+3= -1/12 锛

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杩涓╀跺锛板锛Numberphile锛涓甯浜涓涓棰锛澹扮Оㄩㄦ存扮-1/12

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璇烘璁猴Zenosparadox 锛璇达浠界姝ｈ揪1锛浣瑙搴锛浠浠ユ版杩1杩灏辨板瀹朵滑璁ㄨ绌风骇版舵ㄥ扮瀹涔锛瀹璺浠磋涓璁や负绛浜姒蹇垫烘寸

浣苟绾ф伴芥舵锛浠О涓舵绾ф颁负ｇ骇帮涓浜绾ф帮姣濡1-1+1-1...锛褰浠坊椤规讹跺n椤圭浼ㄤ奸村ㄣ涓浜锛姣濡1+2+3+4...藉寰绌峰ぇ剧讹褰ㄦ姒蹇靛涔绾ф扮舵ф讹绾ф1+2+3+4...舵涓哄濡璇磋涓骇扮涓L锛ｄ浠ヨ交惧扮哄板灏椤逛绾ф扮灏变瓒杩L锛浠璇存涓舵浜L

板-1/12 浠ュ绾ф1+2+3拌绯昏捣ワ浣惧浜-1/12姝ｆ存扮瑙ｅ宠涓涓涓规瑕ㄥ板涓В寤舵锛analyticcontinuation锛炽

璁炬浠涓涓涔ㄥ骞抽㈢芥f(z)锛璇ュ芥扮瀹涔涓U浣浠ユ惧颁涓规ユ涓涓涔ㄦ村箍瀹涔涓芥F(z)锛婊¤冻瀵逛换浜Uz锛f(z)=F(z)浠ユ版芥F(z)ㄦ澶f(z)瀹涔版逛f(z)涓达骞朵f(z)瀹涔涔澶逛涔瀹涔芥F(z)灏卞f(z)瑙ｆ寤舵锛娉ㄦ澶娌℃涔涓锛涓轰涓芥版瑙ｆ寤舵锛

瑙ｆ寤舵ㄧ锛涓哄芥板浠ョ充z绌风骇帮澶у拌绉绌风骇板褰z浜兼舵舵锛璇娉宠澶芥颁ㄦ浜瑰瀹涔锛锛濡浠浠ヨ璇ュ芥板ㄦ村箍版规瀹涔浼姣杈濂斤涓涓芥扮瑙ｆ寤舵灏卞浠ヨ芥板ㄦ绌风骇扮舵瑰涔瀹涔褰浠ㄤ涓芥扮瑙ｆ寤舵兼ヤ唬垮涔芥扮绌风骇版讹浠氨浠ヨ1+2+3=-1/12 浜

瑕ㄤ涓芥版圭瑙ｆ寤舵间涓ㄥ朵版瑰涔绌风骇颁存句涓涓

ㄩ瑕璁ㄨ嘉跺芥 锛the Riemann zeta function锛锛瀹浠ヤ扮甯寰澶ц绯昏绉般褰s瀹ㄥぇ浜1锛榛嘉跺芥拔(s) 琚涔涓猴浠甯哥ㄥ姣z浠ｈ〃澶芥颁腑浣涓轰娌跨ㄩ煎味芥扮瀹涔[瑙榛1859骞寸璁烘]锛杩浠sヨ〃绀鸿杩涓绌风骇板s=-1朵舵锛浣浠ョ帮褰浠护s=-1讹榛嘉跺芥板氨绛浜骇1+2+3浠ラ嘉跺芥版涓芥板ㄦ翠釜澶骞抽やs=1澶锛涓瑙ｆ寤舵褰s=-1, 味(s)=-1/12杩ㄎ(-1)涓ㄥ骞抽㈢朵版瑰涔芥扮绌风骇颁存句绛锛浠氨寰轰1+2+3=-1/12缁璁恒

瑙ｆ寤舵骞朵板-1/12绾ф1+2+3绯昏捣ョ寰瑕充瑙ｅ朵㈣灏藉涓娑澶濂芥规锛瑙锛

http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/

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濡板浠ユ村娓板拌В绯绘板间绌风骇扮朵寰锛灏变浼浠浠璁╀汉浠寰板瀹舵绘ㄥ锋㈡蹇点ㄩｆ瑙棰熬锛剁浜哄疯开路浼Brady Haran锛╃瀛灏悸峰╀锛Tony Padilla锛锛濡姘歌涓板ㄨ绠ㄤ绱姝ｆ存帮涓=讹浼涓浼寰-1/12锛甯╀瀣绗璇达寰涓板锛锛 浣瀹涓绛妗搴璇ユ杩锛璁や负浠浠浠ュ瑙浼婢娓浠浠ㄤ涓绉涓规ョ绌风骇拌绠板硷杩峰浠ュぇ澶у瑙棰璇э浣浠澶变轰

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锛浣锛 Evelyn-Lamb锛逛澶у澹缈昏锛讹瀹℃★筹