List of non-trivial zeros of the Riemann zeta function

The following table lists the first 50 non-trivial zeros of the Riemann zeta function .

Explanation

The Riemann zeta function in the complex plane : The zero, i.e. the origin of the complex plane, is exactly in the middle of the diagram. The so-called non-trivial zeros of the zeta function visible in the picture lie on the vertical line through 0.5, which is not shown. They are recognizable as black points on this imaginary line and are arranged mirror-symmetrically to the real axis, i.e. to the horizontal line through the origin. The so-called trivial zeros lie on the left part of the real axis, namely in −2, −4, −6, −8, ...

The coloring of the complex function values used in the picture above.

The set of all complex zeros of the Riemann zeta function is divided into two subsets: the subset of the so-called trivial zeros, which the Riemann zeta function assumes at the negative even numbers (−2, −4, −6, −8 etc.), and into the Subset of the so-called non-trivial zeros, the real part of which lies between 0 and 1. The Riemann hypothesis from 1859, which has not been proven or disproved to this day, states that all non-trivial zeros of the Riemann zeta functions have the real part 1/2.

The overwhelming number of non-trivial zeros of the Riemann zeta function, which are known to actually have the real part 1/2, include the zeros given in the table below. Therefore, only the imaginary parts of the non-trivial zeros are given in the second column of the table . The associated real part is always 1/2.

The infinite number of non-trivial zeros are arranged mirror-symmetrically to the real axis. So if a non-trivial zero in the following table has the imaginary part , then the too complex conjugate number is also a non-trivial zero of the Riemann zeta function (denotes the imaginary unit ). For this reason, no non-trivial zeros with a negative imaginary part are listed in the following table. For the imaginary parts in the second column, 30 decimal places are given. The last specified decimal place is not rounded. ${\ displaystyle z = 1/2 + it}$ ${\ displaystyle t}$ ${\ displaystyle z}$ ${\ displaystyle {\ overline {z}} = 1/2-it}$ ${\ displaystyle \ textstyle i}$

The numbering of the non-trivial zeros in the first column follows increasing values of the imaginary parts of the zeros. It is therefore the non-trivial zero of the Riemann zeta function with the smallest, positive imaginary part. The non-trivial zero has the second smallest, positive imaginary part, etc. ${\ displaystyle 1/2 + i \ cdot 14 {,} 1347 \ ldots}$ ${\ displaystyle 1/2 + i \ cdot 21 {,} 0220 \ ldots}$

The simplest method for the numerical calculation of non-trivial zeros of the zeta function uses the Euler-Maclaurin formula . With their help, the Danish mathematician Gram was able to calculate the first 15 non-trivial zeros with an accuracy of a few decimal places by 1903.

Advanced methods are based on the Riemann-Siegel Z function . The zeros of this real-valued function of real argument agree with the imaginary parts of the zeros of the zeta function with real part 1/2. The Riemann-Siegel formula and the asymptotic development of the Riemann-Siegel theta function are used to calculate the zeros of the Z function . Together with the knowledge of the number of non-trivial zeros in the considered interval of the imaginary part, it can then be checked whether the calculated zeros of the zeta function with a real part of 1/2 already cover all non-trivial zeros in this interval. The method by Odlyzko and Schönhage is also based on the Z function. Compared to older methods that use the Z function, it increases its speed e.g. B. by using fast Fourier transforms .

table

Riemann's zeta function : This picture shows the contour lines real part (zeta (s)) = 0, blue, and imaginary part (zeta (s)) = 0, lilac, for −5 <Re (s) <3 and −25 <Im (s ) <65, as well as the "critical straight line" Re (s) = 1/2, brown. The intersections of the blue and lilac-colored contour lines in the "critical stripe" 0 <Re (s) <1 are non-trivial zeros of the Riemann zeta function.

{\ displaystyle \ zeta (s)}

n	Imaginary part
1	14.134725141734693790457251983562 ...
2	21.022039638771554992628479593896 ...
3	25.010857580145688763213790992562 ...
4th	30.424876125859513210311897530584 ...
5	32.935061587739189690662368964074 ...
6th	37.586178158825671257217763480705 ...
7th	40.918719012147495187398126914633 ...
8th	43.327073280914999519496122165406 ...
9	48.00 5150881167159727942472749427 ...
10	49.773832477672302181916784678563 ...
11	52.970321477714460644147296608880 ...
12	56.446247697063394804367759476706 ...
13	59.347044002602353079653648674992 ...
14th	60.831778524609809844259901824524 ...
15th	65.112544048081606660875054253183 ...
16	67.079810529494173714478828896522 ...
17th	69.546401711173979252926857526554 ...
18th	72.067157674481907582522107969826 ...
19th	75.704690699083933168326916762030 ...
20th	77.144840068874805372682664856304 ...
21st	79.337375020249367922763592877116 ...
22nd	82.910380854086030183164837494770 ...
23	84.735492980517050105735311206827 ...
24	87.425274613125229406531667850919 ...
25th	88.809111207634465423682348079509 ...
26th	92.491899270558484296259725241810 ...
27	94.651344040519886966597925815208 ...
28	95.870634228245309758741029219246 ...
29	98.831194218193692233324420138622 ...
30th	101.317851005731391228785447940292 ...
31	103.725538040478339416398408108695 ...
32	105.446623052326094493670832414111 ...
33	107.168611184276407515123351963086 ...
34	111.029535543169674524656450309944 ...
35	111.874659176992637085612078716770 ...
36	114.320220915452712765890937276191 ...
37	116.226680320857554382160804312064 ...
38	118.790782865976217322979139702699 ...
39	121.370125002420645918945532970499 ...
40	122.946829293552588200817460330770 ...
41	124.256818554345767184732007966129 ...
42	127.516683879596495124279323766906 ...
43	129.578704199956050985768033906179 ...
44	131.087688530932656723566372461501 ...
45	133.497737202997586450130492042640 ...
46	134.756509753373871331326064157169 ...
47	138.116042054533443200191555190282 ...
48	139.736208952121388950450046523382 ...
49	141.123707404021123761940353818475 ...
50	143.111845807620632739405123868913 ...

literature

Most of the technical literature on the mathematics of the Riemann zeta function and its zeros was written in English. There is comparatively little literature in German on this topic.

Tom M. Apostol : Introduction to Analytic Number Theory . Springer, New York 1976, ISBN 0-387-90163-9 (especially chapters 11, 12 and 13).
Peter Borwein , Stephen Choi, Brendan Rooney, Andrea Weirathmueller: The Riemann Hypothesis . Springer, New York 2008, ISBN 978-0-387-72125-5 (especially Chapters 2 and 3).
Jörg Brüdern : Introduction to analytical number theory . Springer, Berlin, Heidelberg 1995, ISBN 3-540-58821-3 .
John Brian Conrey : More than two fifths or the zeros of the Riemann zeta function are on the critical line . In: Journal for Pure and Applied Mathematics (Crelles Journal) . tape 1989 , no. 399 . Walter de Gruyter, Berlin, New York 1989, pp. 1-26 .
Harold Edwards : Riemann's Zeta Function . Dover, 2001, ISBN 0-486-41740-9 (This book explains in detail the mathematics in Bernhard Riemann's famous original work "About the number of prime numbers under a given size" from 1859. It contains an English translation of this original work in the appendix. ).
Aleksandar Ivić : The Riemann Zeta-Function: theory and applications . Dover, Mineola 2003, ISBN 0-486-42813-3 .
Henryk Iwaniec , Emmanuel Kowalski : Analytic Number Theory . American Mathematical Society , Providence 2004, ISBN 0-8218-3633-1 (especially Chapters 1 and 5).
Eugen Jahnke : boards of higher functions . Teubner, Stuttgart 1966.
Anatoly A. Karatsuba , SM Voronin : The Riemann Zeta-Function . Walter de Gruyter, Berlin 1992, ISBN 3-11-013170-6 .
Peter Meier, Jörn Steuding: Anyone who knows the zeta function knows the world! In: Spectrum of Science Dossier 6/2009: “The greatest riddles in mathematics” . ISBN 978-3-941205-34-5 , pp. 12-19 .
Jürgen Neukirch : Algebraic number theory . Springer, Berlin 1992, ISBN 3-540-54273-6 (especially chapter 7).
Samuel Patterson : An introduction to the theory of the Riemann Zeta-Function . Cambridge University Press, New York 1995, ISBN 0-521-49905-4 .
Paulo Ribenboim : The world of prime numbers . 2nd Edition. Springer, Berlin, Heidelberg 2011, ISBN 978-3-642-18078-1 (especially Chapter 4, Section I.).
Bernhard Riemann : About the number of prime numbers under a given size . In: Monthly reports of the Royal Prussian Academy of Sciences in Berlin . Berlin 1859, p. 671-680 ( Wikisource ).
Atle Selberg : On the zeros of the Riemann zeta-function . In: Skr. Norske Vid. Akad. Oslo . tape 10 , 1942, pp. 1-59 .
Edward Charles Titchmarsh : The Theory of the Riemann Zeta-Function . 1951.
Sergei Michailowitsch Voronin : Theorem on the 'universality' of the Riemann zeta-function . In: Mathematics of the USSR-Izvestiya . tape 9 , no. 3 , 1975, p. 443-445 .

Numerical calculation of non-trivial zeros of the Riemann zeta function:

Richard P. Brent : On the zeros of the Riemann zeta function in the critical strip . In: Mathematics of Computation . tape 33 , no. 148 , 1979, ISSN 0025-5718 , pp. 1361–1372 , doi : 10.1090 / S0025-5718-1979-0537983-2 (English, ams.org ).
RP Brent , J. van de Lune, HJJ te Riele, DT Winter: On the zeros of the Riemann zeta function in the critical strip. II . In: Mathematics of Computation . tape 39 , no. 160 , 1982, ISSN 0025-5718 , pp. 681-688 , doi : 10.1090 / S0025-5718-1982-0669660-1 (English, ams.org ).
J. van de Lune, HJJ te Riele: On the zeros of the Riemann zeta function in the critical strip. III . In: Mathematics of Computation . tape 41 , no. 164 , 1983, ISSN 0025-5718 , pp. 759-767 , doi : 10.1090 / S0025-5718-1983-0717719-3 (English, ams.org ).
J. van de Lune, HJJ te Riele, DT Winter: On the zeros of the Riemann zeta function in the critical strip. IV . In: Mathematics of Computation . tape 46 , no. 174 , 1986, ISSN 0025-5718 , pp. 667-681 , doi : 10.1090 / S0025-5718-1986-0829637-3 (English, ams.org ).

Web links

The LMFDB Collaboration (L-functions and Modular Forms Database): Zeros of zeta (s). Retrieved June 8, 2018 (English, online calculation of non-trivial zeros of the Riemann zeta function).
Andrew Odlyzko : Table of zeros of the Riemann zeta function. Retrieved June 8, 2018 .
Gleb Beliakov, Yuri Matiyasevich (2013): Zeroes of Riemann's zeta function on the critical line with 40000 decimal digits accuracy (1.48 GB)

Individual evidence

↑ Harold Edwards : Riemann's Zeta Function , Dover, Mineola, 2001, ISBN 0-486-41740-9 , sections 6.1–6.4, pp. 96–118 (English).
^ Andrew Odlyzko : The 10 ²² -nd zero of the Riemann zeta function (PDF; 178 kB). In: M. van Frankenhuysen, ML Lapidus (Ed.): Dynamical, Spectral, and Arithmetic Zeta Functions. (= Contemporary Mathematics. No. 290). American Mathematical Society, 2001, ISBN 0-8218-5626-X , pp. 139-144 (English).
↑ Andrew Odlyzko : Analytic Computations in Number Theory (PDF; 188 kB). In: W. Gautschi (Ed.): Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (= Proceedings of symposia in applied mathematics. No. 48). American Mathematical Society, 1994, ISBN 0-8218-0291-7 , pp. 451-463 (English).
↑ Andrew Odlyzko , Arnold Schönhage : Fast Algorithms for multiple evaluations of the Riemann zeta function (PDF; 1.2 MB). In: Transactions of the American Mathematical Society. Volume 309, No. 2, 1988, pp. 797-809 (English).

[1] Harold Edwards : Riemann's Zeta Function , Dover, Mineola, 2001, ISBN 0-486-41740-9 , sections 6.1–6.4, pp. 96–118 (English).

[2] Andrew Odlyzko : The 10 ²² -nd zero of the Riemann zeta function (PDF; 178 kB). In: M. van Frankenhuysen, ML Lapidus (Ed.): Dynamical, Spectral, and Arithmetic Zeta Functions. (= Contemporary Mathematics. No. 290). American Mathematical Society, 2001, ISBN 0-8218-5626-X , pp. 139-144 (English).

[3] Andrew Odlyzko : Analytic Computations in Number Theory (PDF; 188 kB). In: W. Gautschi (Ed.): Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics (= Proceedings of symposia in applied mathematics. No. 48). American Mathematical Society, 1994, ISBN 0-8218-0291-7 , pp. 451-463 (English).

[4] Andrew Odlyzko , Arnold Schönhage : Fast Algorithms for multiple evaluations of the Riemann zeta function (PDF; 1.2 MB). In: Transactions of the American Mathematical Society. Volume 309, No. 2, 1988, pp. 797-809 (English).