Hipotesis Riemann

Dalam matematika, hipotesis Riemann merupakan dugaan bahwa fungsi zeta Riemann memiliki akar-akar hanya pada bilangan genap negatif dan pada bilangan kompleks dengan bagian nyata 12. Banyak yang mengganggap hipotesis ini merupakan pertanyaan belum terjawab paling penting dalam matematika murni.^[1] Hipotesis ini memiliki peran penting dalam teori bilangan karena mengimplikasi hasil-hasil mengenai distribusi bilangan prima. Hipotesis ini diusulkan Bernhard Riemann (1859), dalam tesisnya mengenai distribusi bilangan prima.

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.

Hipotesis Rieman dan beberapa perumumannya, seperti konjektur Goldbach dan konjektur prima kembar, membentuk masalah Hilbert kedelapan dalam daftar dua puluh tiga masalah belum terjawab David Hilbert. Hipotesis ini juga termasuk dalam daftar masalah Milenium Prize, yang menawarkan satu juta dollar AS untuk siapapun yang dapat menyelesaikan masalah tersebut.

Fungsi zeta Riemann pada garis kritis Re(s) = 1/2 (nilai real pada sumbu horizontal dan nilai imajiner pada sumbu vertikal): Re(ζ(1/2 + it), Im(ζ(1/2 + it) dengan nilai t berkisar antara −30 dan 30.

Persamaan zeta Riemann ζ(s) adalah sebuah fungsi dengan 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 $\zeta (s)=0$ ketika $s$ bernilai −2, −4, −6, .... Akar-akar ini disebut akar-akar sederhana (trivial). Fungsi zeta juga memiliki akar pada nilai-nilai $s$ yang lain, yang disebut dengan akar-akar tak-sederhana (nontrivial). Hipotesis Riemann memperhatikan lokasi dari akar-akar tak-sederhana ini, dan menyatakan bahwa:

Bagian real dari setiap akar tak-sederhana dari fungsi zeta Riemann adalah 12.

Akibatnya, jika hipotesis ini benar, semua akar tak-sederhana akan terletak pada garis kritis ${\frac {1}{2}}+it$ , dengan $t$ merupakan bilangan real dan $i$ adalah unit imajiner.

Fungsi zeta Riemann

Fungsi zeta Riemann terdefinisi pada bilangan kompleks $s$ dengan bagian real lebih besar dari 1, lewat deret takhingga yang konvergen absolut $\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots .$ Leonhard Euler telah mempelajari deret ini pada tahun 1730-an untuk nilai real $s$ , bersamaan dengan solusi mengenai masalah Basel. Ia juga membuktikan deret itu sama dengan darab (perkalian) Euler

\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

dengan darab takhingga dilakukan atas semua bilangan prima $p$ .^[2]

Hipotesis Riemann membahas akar-akar diluar daerah konvergensi dari deret itu dan darab Euler. Untuk dapat memahami maksud dari hipotesis, diperlukan kontinuasi (perluasan) analitik dari fungsi untuk mendapatkan bentuk yang valid untuk semua bilangan kompleks $s$ . Karena fungsi zeta termasuk 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

\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 ,

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 $s$ dengan bagian real positif. Akibatnya, fungsi zeta dapat didefinisikan ulang sebagai $\eta (s)/(1-2/2^{s})$ , memperluas domain dari ${\text{Re}}(s)>1$ menjadi ${\text{Re}}(s)>0$ , kecuali untuk titik-titik yang menyebabkan $1-2/2^{s}$ bernilai nol. Titik-titik tersebut memiliki bentuk $s=1+2\pi in/\log 2$ dengan $n$ 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 $s$ dengan bagian real positif; kecuali untuk kutub sederhana $s=1$ .

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

\zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s).

Fungsi $\zeta (s)$ juga dapat didefinisikan untuk bilangan kompleks $s$ yang tersisa (yakni ${\text{Re}}(s)\leq 0$ dan $s\neq 0$ ) dengan menggunakan fungsi ini di luar pita, lalu membuat $\zeta (s)$ bernilai sama dengan ruas kanan kapanpun $s$ memenuhi ${\text{Re}}(s)\leq 0$ (dan $s\neq 0$ ).

Jika $s$ merupakan bilangan genap negatif, maka $\zeta (s)=0$ karena faktor ${\textstyle \sin({\frac {\pi s}{2}})}$ bernilai nol; ini adalah akar-akar sederhana dari fungsi zeta. Argumen ini tidak berlaku ketika $s$ berupa bilangan genap positif karena akar dari fungsi sinus tercoret dengan kutub-kutub dari fungsi gamma. Nilai ${\textstyle \zeta (s)=-{\frac {1}{2}}}$ tidak terdefinisi lewat persamaan fungsional, namun lewat nilai limit $\zeta (s)$ ketika $s$ 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 $0<{\text{Re}}(s)<1$ .

Catatan

^ Bombieri (2000).
^ Leonhard Euler. 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 $s=1$ , and in Theorem 8 he proves it more generally. In the first corollary to his Theorem 7 he notes that $\zeta (1)=\log \infty$ , 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 $\log \log \infty$ .

Referensi

Terdapat beberapa buku nonteknis mengenai hipotesis Riemann, seperti (Derbyshire 2003), (Rockmore 2005), Sabbagh (2003a, 2003b), (du Sautoy 2003), dan (Watkins 2015). Buku-buku seperti (Edwards 1974), (Patterson 1988), (Borwein et al. 2008), (Mazur & Stein 2015) dan (Broughan 2017) memberikan pengenalan secara matematis, sedangkan (Titchmarsh 1986), (Ivić 1985) dan (Karatsuba & Voronin 1992) berupa monografi yang lebih lanjut.

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