Lihat sumber Hipotesis Riemann

[[Berkas:RiemannCriticalLine.svg|jmpl|Bagian nyata (merah) dan bagian imajiner (biru) dari fungsi zeta Riemann sepanjang garis kritis Re(''s'') = 1/2. Nol ''non-trivial'' pertama terdapat di Im(''s'') = ±14.135, ±21.022 dan ±25.011.]]
{{Masalah Millenium}}

Dalam [[matematika]], '''hipotesis Riemann''' merupakan [[Konjektur|dugaan]] bahwa [[fungsi zeta Riemann]] memiliki akar-akar hanya pada bilangan genap negatif dan pada [[bilangan kompleks]] dengan bagian nyata {{sfrac|2}}. Banyak yang mengganggap hipotesis ini merupakan [[Daftar masalah yang belum terpecahkan dalam matematika|pertanyaan belum terjawab]] paling penting dalam [[matematika murni]].{{sfnp|Bombieri|2000}} Hipotesis ini memiliki peran penting dalam [[teori bilangan]] karena mengimplikasi hasil-hasil mengenai distribusi [[bilangan prima]]. Hipotesis ini diusulkan {{harvs|txt|first=Bernhard|last=Riemann|year=1859|author-link=Bernhard Riemann}}, dalam tesisnya mengenai distribusi [[bilangan prima]].

Hipotesis Rieman dan beberapa perumumannya, seperti [[konjektur Goldbach]] dan [[Bilangan prima kembar|konjektur prima kembar]], membentuk [[masalah Hilbert kedelapan]] dalam daftar dua puluh tiga masalah belum terjawab [[David Hilbert]]. Hipotesis ini juga termasuk dalam [[Masalah Milenium|daftar masalah Milenium Prize]], yang menawarkan satu juta dollar AS untuk siapapun yang dapat menyelesaikan masalah tersebut. 
[[Berkas:ParametricZeta.svg|jmpl|Fungsi zeta Riemann pada garis kritis Re(''s'') = 1/2 (nilai real pada sumbu horizontal dan nilai imajiner pada sumbu vertikal): Re(&#x3B6;(1/2 + ''it''), Im(&#x3B6;(1/2 + ''it'') dengan nilai ''t'' berkisar antara &#x2212;30 dan 30.]]
Persamaan zeta Riemann ζ(''s'') adalah sebuah [[Fungsi (matematika)|fungsi]] dengan [[Argumen dari sebuah fungsi|argumen]] berupa sembarang bilangan kompleks selain 1, dan nilai fungsi tersebut juga berupa bilangan kompleks. Fungsi ini memiliki akar-akar pada bilangan genap negatif; yakni <math>\zeta(s) = 0</math> ketika <math>s</math> bernilai −2, −4, −6,&nbsp;.... Akar-akar ini disebut ''akar-akar sederhana'' (trivial). Fungsi zeta juga memiliki akar pada nilai-nilai <math>s</math> yang lain, yang disebut dengan akar-akar tak-sederhana (nontrivial). Hipotesis Riemann memperhatikan lokasi dari akar-akar tak-sederhana ini, dan menyatakan bahwa:{{bquote|Bagian real dari setiap akar tak-sederhana dari fungsi zeta Riemann adalah &nbsp;{{sfrac|2}}.}}Akibatnya, jika hipotesis ini benar, semua akar tak-sederhana akan terletak pada garis kritis <math>\frac{1}{2} + it</math>, dengan <math>t</math> merupakan [[bilangan real]] dan <math>i</math>adalah [[unit imajiner]].

== Fungsi zeta Riemann ==
{{Main|Fungsi zeta Riemann}}
Fungsi zeta Riemann terdefinisi pada bilangan kompleks <math>s</math> dengan bagian real lebih besar dari 1, lewat [[deret takhingga]] yang [[Konvergensi absolut|konvergen absolut]]<math display="block">\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots.</math>[[Leonhard Euler]] telah mempelajari deret ini pada tahun 1730-an untuk nilai real <math>s</math>, bersamaan dengan solusi mengenai [[masalah Basel]]. Ia juga membuktikan deret itu sama dengan [[Perkalian Euler|darab (perkalian) Euler]]

: <math>\zeta(s) = \prod_{p \text{ prima}} \frac{1}{1-p^{-s}}= \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}} \cdot \frac{1}{1-11^{-s}} \cdots</math>

dengan [[darab takhingga]] dilakukan atas semua bilangan prima <math>p</math>.<ref>Leonhard Euler.  [http://eulerarchive.maa.org/docs/originals/E072.pdf Variae observationes circa series infinitas.] Commentarii academiae scientiarum Petropolitanae 9, 1744, pp. 160–188, Theorems 7 and 8. In Theorem 7 Euler proves the formula in the special case <math>s=1</math>, and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that   <math>\zeta(1)=\log\infty</math>, and makes use of this latter result in his Theorem 19, in order to show that the sum of the inverses of the prime numbers is <math>\log\log\infty</math>.</ref>

Hipotesis Riemann membahas akar-akar diluar [[daerah konvergensi]] dari deret itu dan darab Euler. Untuk dapat memahami maksud dari hipotesis, diperlukan [[Kontinuasi analitik|kontinuasi (perluasan) analitik]] dari fungsi untuk mendapatkan bentuk yang valid untuk semua bilangan kompleks <math>s</math>. Karena fungsi zeta termasuk [[Fungsi meromofik|meromofik]], semua pilihan cara untuk melakukan kontinuasi analitik ini akan menghasilkan bentuk yang sama, sebagai akibat dari [[teorema identitas]]. Langkah pertama dalam proses kontinuasi ini adalah pengamatan bahwa fungsi zeta dan [[fungsi eta Dirichlet]] memenuhi hubungan

: <math>\left(1-\frac{2}{2^s}\right)\zeta(s) = \eta(s) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \cdots,</math>

pada daerah konvergensi mereka masing-masing. Akan tetapi, fungsi deret eta pada ruas kanan tidak hanya konvergen untuk bilangan kompleks dengan bagian real lebih besar dari 1, tapi juga untuk sembarang <math>s</math> dengan bagian real positif. Akibatnya, fungsi zeta dapat didefinisikan ulang sebagai <math>\eta(s)/(1-2/2^s)</math>, memperluas domain dari <math>\text{Re}(s)>1</math> menjadi <math>\text{Re}(s)>0</math>, kecuali untuk titik-titik yang menyebabkan <math>1-2/2^s</math> bernilai nol. Titik-titik tersebut memiliki bentuk <math>s = 1 + 2\pi in/\log 2</math> dengan <math>n</math> dapat berupa sembarang [[bilangan bulat]] bukan-nol. Fungsi zeta dapat diperluas lebih lanjut untuk titik-titik tersebut dengan menggunakan limit, menghasilkan nilai yang hingga untuk sembarang nilai <math>s</math> dengan bagian real positif; kecuali untuk [[kutub sederhana]] <math>s=1</math>.

Pada daerah berbentuk pita <math>0<\text{Re}(s)<1</math>, perluasan dari fungsi zeta ini akan memenuhi persamaan fungsional

: <math>\zeta(s) = 2^s\pi^{s-1}\ \sin\left(\frac{\pi s}{2}\right)\ \Gamma(1-s)\ \zeta(1-s).</math>

Fungsi <math>\zeta(s)</math> juga dapat didefinisikan untuk bilangan kompleks <math>s</math> yang tersisa (yakni <math>\text{Re}(s)\leq0</math> dan <math>s\neq0</math>) dengan menggunakan fungsi ini di luar pita, lalu membuat <math>\zeta(s)</math> bernilai sama dengan ruas kanan kapanpun <math>s</math> memenuhi <math>\text{Re}(s)\leq0</math> (dan <math>s\neq0</math>).

Jika <math>s</math> merupakan bilangan genap negatif, maka <math>\zeta(s)=0</math> karena faktor <math display="inline">\sin(\frac{\pi s}{2})</math> bernilai nol; ini adalah ''akar-akar sederhana'' dari fungsi zeta. Argumen ini tidak berlaku ketika <math>s</math> berupa bilangan genap positif karena akar dari fungsi [[Sinus dan kosinus|sinus]] tercoret dengan kutub-kutub dari [[fungsi gamma]]. Nilai <math display="inline">\zeta(s)=-\frac{1}{2}</math> tidak terdefinisi lewat [[persamaan fungsional]], namun lewat nilai limit <math>\zeta(s)</math> ketika <math>s</math> menuju nol. Persamaan fungsional juga menyimpulkan bahwa fungsi zeta tidak memiliki akar pada titik-titik selain akar-akar sederhana; mengartikan semua akar-akar tak-sederhana terletak pada pita kritis <math>0<\text{Re}(s)<1</math>.

== Catatan ==
{{Reflist|30em}}

== Referensi ==
Terdapat beberapa buku nonteknis mengenai hipotesis Riemann, seperti {{harvtxt|Derbyshire|2003}}, {{harvtxt|Rockmore|2005}}, {{harvs|txt|last=Sabbagh|year=2003a|year2=2003b}}, {{harvtxt|du Sautoy|2003}}, dan {{harvtxt|Watkins|2015}}. Buku-buku seperti {{harvtxt|Edwards|1974}}, {{harvtxt|Patterson|1988}}, {{harvtxt|Borwein|Choi|Rooney|Weirathmueller|2008}}, {{harvtxt|Mazur|Stein|2015}} dan {{harvtxt|Broughan|2017}} memberikan pengenalan secara matematis, sedangkan {{harvtxt|Titchmarsh|1986}}, {{harvtxt|Ivić|1985}} dan {{harvtxt|Karatsuba|Voronin|1992}} berupa [[monografi]] yang lebih lanjut.
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{{refend}}

=== Eksposisi populer ===

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