FIYE DA KASHI 83.9% NA SIFILAN ZETA SUN BAMBANTA

Riemann zeta function tana da sifilai “marasa sauƙi” (non-trivial) marasa iyaka, wurare a cikin complex plane inda take zama sifili. Tambayoyi biyu game da su sun rage a buɗe. Shin dukansu suna kan layin muhimmi (critical line), inda ɓangarensu na ainihi (real part) ya yi daidai da rabi? Wannan shi ne Riemann hypothesis. Kuma shin dukansu sauƙaƙa ne — shin kowane sifili sifili ne guda ɗaya maimakon biyu ko fiye da aka tara a wuri ɗaya? Wannan shi ne simple zeros conjecture (hasashen sifilai sauƙaƙa). A cewar takardar, ba wanda ya san ko ɗaya daga cikin waɗannan maganganun yana nufin ɗayan.

Har zuwa tsayin 3 × 10¹², kwamfutoci sun tabbatar da cewa sifilan suna kan layin kuma sauƙaƙa ne. Bayan haka, masana lissafi suna tabbatar da kaso: aƙalla wani kaso na dukan sifilan yana da halin.

Tseren da ake auna shi da lambobin bayan ɗigo

Kristian Muri Knausgård, a Kristiansand, Norway, ya bayyana tarihin kwanan nan dalla-dalla. Kason sifilai masu bambanta da aka tabbatar ya tafi daga kashi 63.9% zuwa 70%, sannan ya yi tsalle zuwa kashi 83.625% da wata hujja da takardar ta danganta ga wani preprint da Claude (Anthropic) ya sa hannu, wanda masana lissafi Alpöge da Furman suka tantance, kuma Lamzouri ya sake tabbatarwa. Ƙananan ci gaba sun biyo baya, har da takardar marubucin ta baya a kashi 83.6993%.

Gabatarwar takardar ta lissafa ƙimomi sama da goma sha biyu da aka sanar ga sifilai masu sauƙi kuma a kan layin muhimmi, daga 0.67300 zuwa 0.673492, da yawa an saka su a ma’ajiyar code ta jama’a tsakanin Agusta da Oktoba 2026 kuma ba a yi musu bita ba. Marubucin ya bayyana cewa ba a tantance waɗannan ƙimomin don takardar ba kuma babu ɗaya daga cikin sakamakonta da ya dogara a kansu.

Sababbin iyakoki

Ba tare da zaton Riemann hypothesis ko wata magana da ba a tabbatar ba, takardar ta tabbatar da cewa:

fiye da kashi 83.900% na sifilan sun bambanta (ainihin iyakar ita ce 1645064/1960733);
saboda haka, fiye da kashi 67.80% sauƙaƙa ne;
fiye da kashi 67.353% sauƙaƙa ne kuma a kan layin muhimmi;
aƙalla kashi 88.93% sauƙaƙa ne ko a kan layin, don haka sifilan da ke duka a wajen layin kuma maimaitattu ne ba su wuce kashi 11.07% ba.

Maimaitattun sifilai suna cin kuzari

Ra’ayin ya dace a cikin hoto ɗaya. Sake daidaita ma’aunin sifilan ta yadda tazarar da ke tsakaninsu ta zama raka’a ɗaya a matsakaici, kamar duwatsun carbi a kan zare. Wani theorem na Montgomery kan yadda nau’i-nau’in sifilai ke rarrabuwa yana kayyade, a hankali har iyaka (asymptotically), jimillar “kuzari” da aka tara a kan dukan nau’i-nau’i. Sifili da aka maimaita sau d yana ba da d² ga wannan kuzarin. Sifilai maƙwabta suma suna amfani da wani ɓangare nasa, ta hanyar haɗuwarsu.

Don haka idan mutum zai iya tabbatar da cewa maƙwabta dole suna amfani da kuzari mai yawa, kaɗan ne ke saura ga maimaitattun sifilai. Hujjar tana yin hakan a matakai uku:

Sifilan da ke riƙe da mafi ƙarancin nisa tsakaninsu ana sarrafa su daidai, saboda wani rashin daidaito (inequality) da aka sani da large sieve.
Wani rashin daidaito a kan sifilai bakwai masu bin juna, wanda aka tabbatar da taimakon kwamfuta, yana ba da lada ga sifilai ninki biyu fiye da sauƙaƙa. Sharuɗɗan gyaransa suna soke juna idan aka tara su a tsawon layin.
Wani aikin gwaji da aka tsara a hankali, “taga” (window), yana sa wannan ladan ya zama babba gwargwadon iko.

Takardar ta kwatanta matsalar da nemo yanayin mafi ƙarancin kuzari na wani iskar gas mai girma ɗaya da ke da nau’ikan ƙwayoyi biyu, sifilai sauƙaƙa da ninki biyu.

Abin da kwamfuta ta tantance, da wanda ba ta yi ba

Rashin daidaiton maki bakwai ya buƙaci binciken akwatuna 60,467,309 na yiwuwa, ba tare da gazawa ba. Sakamakon layin muhimmi ya buƙaci fiye da miliyan 535. Dukan manyan theorems an tsara su a cikin Lean 4, wani software na tantance hujja, a cikin kusan layuka 13,300 da 10,000 a kan wani ɗakin karatu da ke akwai na kusan layuka 102,000, da maganganu game da sifilan zeta kamar yadda aka ayyana a ɗakin karatun lissafi na Lean.

Marubucin ya bayyana giɓi ɗaya a fili. Kowane theorem ya dogara ne kan ƙarin axiom ɗaya da ke rubuta cewa wani shirin bincike ya dawo da “gaskiya” (true). Kernel ɗin Lean ba ya sake gudanar da wannan lissafin; yana amincewa da compiler da runtime na Lean a kansa. Bincike uku masu zaman kansu suna goyon bayan gudanarwar, ciki har da ƙidaya iri ɗaya daga shirye-shiryen C++ da Python daban-daban.

Hanya da ke kusa da ƙarshenta

Takardar kuma ta tabbatar da inda dole ne hanyarta ta tsaya. Da babbar tagarta, ba za ta taɓa kaiwa kashi 84% ba: silin shi ne 0.83998. Ga kowace taga mai tsari mai ma’ana, silin shi ne 0.84093 — ƙasa da 0.002 sama da sabon sakamakon. Ci gaba fiye da haka zai buƙaci wani ra’ayi daban.

Rikicin moriya. Marubucin ya bayyana wannan aiki a matsayin “gwaji a binciken lissafi da AI ke taimakawa”. Marubucin ya bayyana cewa an haɓaka hujjoji, lissafi, tsarawa a Lean, zane-zane da yawancin rubutun tare da babban taimakon OpenAI Codex da Anthropic Claude, ƙarƙashin jagorancin marubucin; Claude ne ya haɓaka tsarawar, nazarin iyakokin hanyar, taga da takardar shaida (certificate) ta theorem na layin muhimmi, zane-zane da yawancin rubutun, kuma tsarin biyu sun yi bitar daftari “a matsayin masu bita (referees)”. An danganta hujjar farawa ga wani preprint da Claude ya sa hannu. Claude ne kuma ya rubuta wannan labarin.