Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter $ζ$ (zeta), is a mathematical function of a complex variable defined as $\zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots$ for $\operatorname {Re} (s)>1$ , and its analytic continuation elsewhere.^[2]

The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics.

Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics.^[3]

The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, $ζ (2)$ , provides a solution to the Basel problem. In 1979 Roger Apéry proved the irrationality of $ζ (3)$ . The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet $L$ -functions and $L$ -functions, are known.

Representations[edit]

Dirichlet series^[29][edit]

An extension of the area of convergence can be obtained by rearranging the original series. The series

$\sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1$

1 + 2 + 3 + 4 + ···

Arithmetic zeta function

Generalized Riemann hypothesis

Lehmer pair

Prime zeta function

Riemann Xi function

Renormalization

Riemann–Siegel theta function

ZetaGrid

Apostol, T. M. (2010). . In Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.). NIST Handbook of Mathematical Functions. Cambridge University Press. ISBN 978-0-521-19225-5. MR 2723248..

"Zeta and Related Functions"

; Bradley, David M.; Crandall, Richard (2000). "Computational Strategies for the Riemann Zeta Function" (PDF). J. Comput. Appl. Math. 121 (1–2): 247–296. Bibcode:2000JCoAM.121..247B. doi:10.1016/S0377-0427(00)00336-8. Archived from the original (PDF) on 13 December 2013.

Borwein, Jonathan

Cvijović, Djurdje; Klinowski, Jacek (2002). . J. Comput. Appl. Math. 142 (2): 435–439. Bibcode:2002JCoAM.142..435C. doi:10.1016/S0377-0427(02)00358-8. MR 1906742.

"Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments"

Cvijović, Djurdje; Klinowski, Jacek (1997). . Proc. Amer. Math. Soc. 125 (9): 2543–2550. doi:10.1090/S0002-9939-97-04102-6.

"Continued-fraction expansions for the Riemann zeta function and polylogarithms"

(1974). Riemann's Zeta Function. Academic Press. ISBN 0-486-41740-9. Has an English translation of Riemann's paper.

Edwards, H. M.

(1896). "Sur la distribution des zéros de la fonction $ζ (s)$ et ses conséquences arithmétiques". Bulletin de la Société Mathématique de France. 14: 199–220. doi:10.24033/bsmf.545.

Hadamard, Jacques

(1949). Divergent Series. Clarendon Press, Oxford.

Hardy, G. H.

(1930). "Ein Summierungsverfahren für die Riemannsche $ζ$ -Reihe". Math. Z. 32: 458–464. doi:10.1007/BF01194645. MR 1545177. S2CID 120392534. (Globally convergent series expression.)

Hasse, Helmut

Ivic, A. (1985). The Riemann Zeta Function. John Wiley & Sons. 0-471-80634-X.

ISBN

Motohashi, Y. (1997). Spectral Theory of the Riemann Zeta-Function. Cambridge University Press. 0521445205.

ISBN

; Voronin, S. M. (1992). The Riemann Zeta-Function. Berlin: W. de Gruyter.

Karatsuba, A. A.

Mező, István; Dil, Ayhan (2010). "Hyperharmonic series involving Hurwitz zeta function". Journal of Number Theory. 130 (2): 360–369. :10.1016/j.jnt.2009.08.005. hdl:2437/90539. MR 2564902. S2CID 122707401.

doi

; Vaughan, Robert C. (2007). Multiplicative number theory. I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge University Press. Ch. 10. ISBN 978-0-521-84903-6.

Montgomery, Hugh L.

(1998). Analytic number theory. Graduate Texts in Mathematics. Vol. 177. Springer-Verlag. Ch. 6. ISBN 0-387-98308-2.

Newman, Donald J.

Raoh, Guo (1996). "The Distribution of the Logarithmic Derivative of the Riemann Zeta Function". Proceedings of the London Mathematical Society. s3–72: 1–27. :10.1112/plms/s3-72.1.1.

doi

Riemann, Bernhard (1859). . Monatsberichte der Berliner Akademie.. In Gesammelte Werke, Teubner, Leipzig (1892), Reprinted by Dover, New York (1953).

"Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

Sondow, Jonathan (1994). (PDF). Proc. Amer. Math. Soc. 120 (2): 421–424. doi:10.1090/S0002-9939-1994-1172954-7.

"Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series"

(1986). Heath-Brown (ed.). The Theory of the Riemann Zeta Function (2nd rev. ed.). Oxford University Press.

Titchmarsh, E. C.

; Watson, G. N. (1927). A Course in Modern Analysis (4th ed.). Cambridge University Press. Ch. 13.

Whittaker, E. T.

Zhao, Jianqiang (1999). . Proc. Amer. Math. Soc. 128 (5): 1275–1283. doi:10.1090/S0002-9939-99-05398-8. MR 1670846.

"Analytic continuation of multiple zeta functions"

Media related to Riemann zeta function at Wikimedia Commons

. Encyclopedia of Mathematics. EMS Press. 2001 [1994].

"Zeta-function"

— an explanation with a more mathematical approach

Riemann Zeta Function, in Wolfram Mathworld

Archived 17 May 2009 at the Wayback Machine

Tables of selected zeros

A general, non-technical description of the significance of the zeta function in relation to prime numbers.

Prime Numbers Get Hitched

Visually oriented investigation of where zeta is real or purely imaginary.

X-Ray of the Zeta Function

functions.wolfram.com

Formulas and identities for the Riemann Zeta function

section 23.2 of Abramowitz and Stegun

Riemann Zeta Function and Other Sums of Reciprocal Powers

. "Million Dollar Math Problem" (video). Brady Haran. Archived from the original on 11 December 2021. Retrieved 11 March 2014.

Frenkel, Edward

—Computational examples of Mellin transform methods involving the Riemann Zeta Function

Mellin transform and the functional equation of the Riemann Zeta function

a video from 3Blue1Brown

Riemann zeta function

Representations[edit]

Dirichlet series[29][edit]

1 + 2 + 3 + 4 + ···

Arithmetic zeta function

Generalized Riemann hypothesis

Lehmer pair

Prime zeta function

Riemann Xi function

Renormalization

Riemann–Siegel theta function

ZetaGrid

"Zeta and Related Functions"

Borwein, Jonathan

"Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments"

"Continued-fraction expansions for the Riemann zeta function and polylogarithms"

Edwards, H. M.

Hadamard, Jacques

Hardy, G. H.

Hasse, Helmut

ISBN

ISBN

Karatsuba, A. A.

doi

Montgomery, Hugh L.

Newman, Donald J.

doi

"Über die Anzahl der Primzahlen unter einer gegebenen Grösse"

"Analytic continuation of Riemann's zeta function and values at negative integers via Euler's transformation of series"

Titchmarsh, E. C.

Whittaker, E. T.

"Analytic continuation of multiple zeta functions"

"Zeta-function"

Riemann Zeta Function, in Wolfram Mathworld

Tables of selected zeros

Prime Numbers Get Hitched

X-Ray of the Zeta Function

Formulas and identities for the Riemann Zeta function

Riemann Zeta Function and Other Sums of Reciprocal Powers

Frenkel, Edward

Mellin transform and the functional equation of the Riemann Zeta function

Visualizing the Riemann zeta function and analytic continuation

Dirichlet series^[29][edit]