Search: a305744 -id:a305744
|
|
A002410
|
|
Nearest integer to imaginary part of n-th zero of Riemann zeta function.
(Formerly M4924 N2113)
|
|
+10
59
|
|
|
14, 21, 25, 30, 33, 38, 41, 43, 48, 50, 53, 56, 59, 61, 65, 67, 70, 72, 76, 77, 79, 83, 85, 87, 89, 92, 95, 96, 99, 101, 104, 105, 107, 111, 112, 114, 116, 119, 121, 123, 124, 128, 130, 131, 133, 135, 138, 140, 141, 143, 146, 147, 150, 151, 153, 156, 158, 159, 161
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
"All these zeros of the form s + it have real part s = 1/2 and are simple. Thus the Riemann hypothesis is true at least for t < 3330657430697." - Wedeniwski
All nontrivial zeros on the critical line, of the form 1/2 + i*t, have an associated conjugate nontrivial zero of the form 1/2 - i*t.
Any nontrivial zeros off the critical line, if ever found, would come in pairs (1/2 +- delta) + i*t, 0 < delta < 1/2. Each of these pairs, again if ever found, would then have their associated conjugate pair (1/2 +- delta) - i*t, 0 < delta < 1/2. (End)
The sequence is not strictly increasing. - Joerg Arndt, Jan 17 2015
The fraction of numbers n such that a(n) = a(n-1) has density 1. There are only finitely many numbers n with a(n) > a(n-1) + 1, see A208436. - Charles R Greathouse IV, Mar 07 2018
Conjecture: Noninteger rationals of the form m/2^bigomega(m) that can be used to approximate this sequence, i.e. a(n) ~~ 2*Pi*A374074(n)/2^bigomega(A374074(n)) - n/2 +- (...), where '~~' means 'close to'. - Friedjof Tellkamp, Jul 04 2024
|
|
REFERENCES
|
Gregory Benford, Gravity's whispers, Futures Column, Nature, 446 (Jul 15 2010), p. 406. [Gravity waves are detected on Earth that turn out to contain a list of the zeros of the Riemann zeta function, essentially this sequence]
E. Bombieri, "The Riemann Hypothesis" in 'The Millennium Prize Problems' Chap. 7 pp. 107-128 Eds: J. Carlson, A. Jaffe & A. Wiles, Amer. Math. Soc. Providence RI 2006.
P. Borwein et al., The Riemann Hypothesis, Can. Math. Soc. (CMS) Ottawa ON 2007.
S. Chowla, Riemann Hypothesis and Hilbert's Tenth Problem, Mathematics and Its Application Series Vol. 4, Taylor & Francis NY 1965.
J. Derbyshire, Prime Obsession, Penguin Books 2004.
K. Devlin, The Millennium Problems, Chapter 1 (pp. 19-62) Basic Books NY 2002.
M. du Sautoy, The Music of the Primes, Fourth Estate/HarperCollins NY 2003.
H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.
A. Ivic, The Riemann Zeta-Function: Theory and Applications, Dover NY 2003.
D. S. Jandu, Riemann Hypothesis and Prime Number Theorem, Infinite Bandwidth Publishing, N. Hollywood CA 2006.
A. A. Karatsuba & S. M. Voronin, The Riemann Zeta-Function, Walter de Gruyter, Berlin 1992.
G. Lachaud, "L'hypothèse de Riemann" in La Recherche No.346 October 2001 pp. 24-30 (or Les Dossiers de La Recherche No. Aug 20 2005 pp. 26-35) Paris.
M. L. Lapidus, In Search of the Riemann Zeros, Amer. Math. Soc. (AMS) Providence RI 2008.
P. Meier & J. Steuding, "L'hypothèse de Riemann" in 'Pour la Science' (French Edition of 'Scientific American') pp 22-9, March 2009, Issue No. 377, Paris.
P. Odifreddi, The Mathematical Century, Chapter 5.2, p. 168, Princeton Univ. Press NJ 2004.
S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge Univ. Press, UK 1995.
D. N. Rockmore, Stalking the Riemann Hypothesis, Jonathan Cape UK 2005.
K. Sabbagh, The Riemann Hypothesis, Farrar Straus Giroux NY 2003.
K. Sabbagh, Dr. Riemann's Zeros, Atlantic Books London 2003.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press NY 1986.
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ 2*Pi*(n - 11/8)/ProductLog((n - 11/8)/exp(1)). This is the asymptotic by Guilherme França and André LeClair. - Mats Granvik, Mar 10 2015; corrected May 16 2016
|
|
EXAMPLE
|
The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453).
|
|
MATHEMATICA
|
|
|
PROG
|
(Sage)
Z = lcalc.zeros(n)
return [round(z) for z in Z]
|
|
CROSSREFS
|
Cf. A013629 (floor), A092783 (ceiling), A057641, A057640, A058209, A058210, A120401, A122526, A072080, A124288 ("unstable" zeta zeros), A124289 ("unstable twins"), A236212, A177885, A374074 (approximation).
|
|
KEYWORD
|
nonn,easy,nice
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 08 2004
|
|
STATUS
|
approved
|
|
|
|
|
A013629
|
|
Floor of imaginary parts of nontrivial zeros of Riemann zeta function.
|
|
+10
33
|
|
|
14, 21, 25, 30, 32, 37, 40, 43, 48, 49, 52, 56, 59, 60, 65, 67, 69, 72, 75, 77, 79, 82, 84, 87, 88, 92, 94, 95, 98, 101, 103, 105, 107, 111, 111, 114, 116, 118, 121, 122, 124, 127, 129, 131, 133, 134, 138, 139, 141, 143, 146, 147, 150, 150, 153, 156, 157, 158, 161
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
REFERENCES
|
H. M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974, p. 96.
C. B. Haselgrove and J. C. P. Miller, Tables of the Riemann Zeta Function. Royal Society Mathematical Tables, Vol. 6, Cambridge Univ. Press, 1960, p. 58.
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ (2*Pi*e) * e^(W0(n/e)), where W0 is the principal branch of Lambert's W function. - Hal M. Switkay, Oct 04 2021
|
|
EXAMPLE
|
The imaginary parts of the first 4 zeros are 14.134725... (A058303), 21.0220396... (A065434), 25.01085758... (A065452), 30.424876... (A065453). Therefore the sequence starts: 14, 21, 25, 30, ..., as does A002410 (rounded values; main entry). But the 5th, 6th and 7th values are 32.935... (A192492), 37.586... (A305741), 40.9187... (A305742), whence a(n) = A002410(n)-1 and A002410 = A092783 (ceiling) for these. - M. F. Hasler, Nov 23 2018
|
|
MATHEMATICA
|
|
|
PROG
|
(Sage)
Z = lcalc.zeros(n)
return [floor(z) for z in Z]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
John Morrison (John.Morrison(AT)armltd.co.uk)
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|
|
A058303
|
|
Decimal expansion of the imaginary part of the first nontrivial zero of the Riemann zeta function.
|
|
+10
26
|
|
|
1, 4, 1, 3, 4, 7, 2, 5, 1, 4, 1, 7, 3, 4, 6, 9, 3, 7, 9, 0, 4, 5, 7, 2, 5, 1, 9, 8, 3, 5, 6, 2, 4, 7, 0, 2, 7, 0, 7, 8, 4, 2, 5, 7, 1, 1, 5, 6, 9, 9, 2, 4, 3, 1, 7, 5, 6, 8, 5, 5, 6, 7, 4, 6, 0, 1, 4, 9, 9, 6, 3, 4, 2, 9, 8, 0, 9, 2, 5, 6, 7, 6, 4, 9, 4, 9, 0, 1, 0, 3, 9, 3, 1, 7, 1, 5, 6, 1, 0, 1, 2, 7, 7, 9, 2
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,2
|
|
COMMENTS
|
"The Riemann Hypothesis, considered by many to be the most important unsolved problem of mathematics, is the assertion that all of zeta's nontrivial zeros line up with the first two all of which lie on the line 1/2 + sqrt(-1)*t, which is called the critical line. It is known that the hypothesis is obeyed for the first billion and a half zeros." (Wagon)
We can compute 105 digits of this zeta zero as the numerical integral: gamma = Integral_{t=0..gamma+15} (1/2)*(1 - sign((RiemannSiegelTheta(t) + Im(log(zeta(1/2 + i*t))))/Pi - n + 3/2)) where n=1 and where the initial value of gamma = 1. The upper integration limit is arbitrary as long as it is greater than the zeta zero computed recursively. The recursive formula fails at zeta zeros with indices n equal to sequence A153815. - Mats Granvik, Feb 15 2017
|
|
REFERENCES
|
S. Wagon, "Mathematica In Action," W. H. Freeman and Company, NY, 1991, page 361.
|
|
LINKS
|
|
|
FORMULA
|
zeta(1/2 + i*14.1347251417346937904572519836...) = 0.
|
|
EXAMPLE
|
14.1347251417346937904572519835624702707842571156992...
|
|
MAPLE
|
|
|
MATHEMATICA
|
FindRoot[ Zeta[1/2 + I*t], {t, 14 + {-.3, +.3}}, AccuracyGoal -> 100, WorkingPrecision -> 120]
(* The following numerical integral takes about 9 minutes to compute *)Clear[n, t, gamma]; gamma = 1; numberofzetazeros = 1; Quiet[Do[gamma = N[NIntegrate[(1/2)*(1 - Sign[(RiemannSiegelTheta[t] + Im[Log[Zeta[I*t + 1/2]]])/Pi - n + 3/2]), {t, 0, gamma + 15}, PrecisionGoal -> 110, MaxRecursion -> 350, WorkingPrecision -> 120], 105]; Print[gamma], {n, 1, numberofzetazeros}]]; RealDigits[gamma][[1]] (* Mats Granvik, Feb 15 2017 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A065434
|
|
Decimal expansion of imaginary part of 2nd nontrivial zero of Riemann zeta function.
|
|
+10
19
|
|
|
2, 1, 0, 2, 2, 0, 3, 9, 6, 3, 8, 7, 7, 1, 5, 5, 4, 9, 9, 2, 6, 2, 8, 4, 7, 9, 5, 9, 3, 8, 9, 6, 9, 0, 2, 7, 7, 7, 3, 3, 4, 3, 4, 0, 5, 2, 4, 9, 0, 2, 7, 8, 1, 7, 5, 4, 6, 2, 9, 5, 2, 0, 4, 0, 3, 5, 8, 7, 5, 9, 8, 5, 8, 6, 0, 6, 8, 8, 9, 0, 7, 9, 9, 7, 1, 3, 6, 5, 8, 5, 1, 4, 1, 8, 0, 1, 5, 1, 4
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i*21.0220396387715549926284795938969...
|
|
MAPLE
|
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) lfunzeros(1, [21, 22])[1] \\ M. F. Hasler, Nov 23 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A065452
|
|
Decimal expansion of imaginary part of 3rd nontrivial zero of Riemann zeta function.
|
|
+10
18
|
|
|
2, 5, 0, 1, 0, 8, 5, 7, 5, 8, 0, 1, 4, 5, 6, 8, 8, 7, 6, 3, 2, 1, 3, 7, 9, 0, 9, 9, 2, 5, 6, 2, 8, 2, 1, 8, 1, 8, 6, 5, 9, 5, 4, 9, 6, 7, 2, 5, 5, 7, 9, 9, 6, 6, 7, 2, 4, 9, 6, 5, 4, 2, 0, 0, 6, 7, 4, 5, 0, 9, 2, 0, 9, 8, 4, 4, 1, 6, 4, 4, 2, 7, 7, 8, 4, 0, 2, 3, 8, 2, 2, 4, 5, 5, 8, 0, 6, 2, 4
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
COMMENTS
|
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i * 25.01085758014568876321379099256282181865954967...
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) lfunzeros(1, [25, 26])[1] \\ or: lfunzeros(1, 26)[3]. - M. F. Hasler, Nov 23 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|
|
A065453
|
|
Decimal expansion of imaginary part of 4th nontrivial zero of Riemann zeta function.
|
|
+10
17
|
|
|
3, 0, 4, 2, 4, 8, 7, 6, 1, 2, 5, 8, 5, 9, 5, 1, 3, 2, 1, 0, 3, 1, 1, 8, 9, 7, 5, 3, 0, 5, 8, 4, 0, 9, 1, 3, 2, 0, 1, 8, 1, 5, 6, 0, 0, 2, 3, 7, 1, 5, 4, 4, 0, 1, 8, 0, 9, 6, 2, 1, 4, 6, 0, 3, 6, 9, 9, 3, 3, 2, 9, 3, 8, 9, 3, 3, 3, 2, 7, 7, 9, 2, 0, 2, 9, 0, 5, 8, 4, 2, 9, 3, 9, 0, 2, 0, 8, 9, 1
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
COMMENTS
|
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i * 30.42487612585951321031189753058409132...
|
|
MATHEMATICA
|
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A192492
|
|
Decimal expansion of imaginary part of 5th nontrivial zero of Riemann zeta function.
|
|
+10
13
|
|
|
3, 2, 9, 3, 5, 0, 6, 1, 5, 8, 7, 7, 3, 9, 1, 8, 9, 6, 9, 0, 6, 6, 2, 3, 6, 8, 9, 6, 4, 0, 7, 4, 9, 0, 3, 4, 8, 8, 8, 1, 2, 7, 1, 5, 6, 0, 3, 5, 1, 7, 0, 3, 9, 0, 0, 9, 2, 8, 0, 0, 0, 3, 4, 4, 0, 7, 8, 4, 8, 1, 5, 6, 0, 8, 6, 3, 0, 5, 5, 1, 0, 0, 5, 9, 3, 8, 8, 4, 8, 4, 9, 6, 1, 3, 5, 3
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
COMMENTS
|
The real part of the 5th nontrivial zero is of course 1/2 (A020761; the Riemann hypothesis is here assumed to be true).
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i * 32.9350615877391896906623689640749...
|
|
MATHEMATICA
|
(* ZetaZero was introduced in Version 6.0 *) RealDigits[ZetaZero[5], 10, 100][[1]]
|
|
PROG
|
|
|
CROSSREFS
|
Cf. A002410: nearest integer to imaginary part of n-th zero of Riemann zeta function (main entry); also A013629 (floor) and A092783 (ceiling).
The real parts of the trivial zeros are given by A005843 multiplied by -1 (and ignoring the initial 0 of that sequence).
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Example and cross-references edited by M. F. Hasler, Nov 23 2018
|
|
STATUS
|
approved
|
|
|
|
|
A092783
|
|
Ceiling of imaginary parts of zeros of Riemann zeta function.
|
|
+10
12
|
|
|
15, 22, 26, 31, 33, 38, 41, 44, 49, 50, 53, 57, 60, 61, 66, 68, 70, 73, 76, 78, 80, 83, 85, 88, 89, 93, 95, 96, 99, 102, 104, 106, 108, 112, 112, 115, 117, 119, 122, 123, 125, 128, 130, 132, 134, 135, 139, 140, 142, 144, 147, 148, 151, 151, 154, 157, 158, 159, 162, 164, 166, 168, 170, 170, 174
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
LINKS
|
|
|
FORMULA
|
|
|
MATHEMATICA
|
|
|
PROG
|
(Sage)
Z = lcalc.zeros(n)
return [ceil(z) for z in Z]
(PARI) apply(ceil, lfunzeros(1, [1, 180])) \\ M. F. Hasler, Nov 23 2018
|
|
CROSSREFS
|
Cf. A002410: nearest integer to imaginary part of n-th zero of Riemann zeta function (main entry).
Cf. A013629: floor of imaginary parts of zeros of Riemann zeta function.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
More terms, link and cross-references from M. F. Hasler, Nov 23 2018
|
|
STATUS
|
approved
|
|
|
|
|
A305741
|
|
Decimal expansion of imaginary part of 6th nontrivial zero of Riemann zeta function.
|
|
+10
12
|
|
|
3, 7, 5, 8, 6, 1, 7, 8, 1, 5, 8, 8, 2, 5, 6, 7, 1, 2, 5, 7, 2, 1, 7, 7, 6, 3, 4, 8, 0, 7, 0, 5, 3, 3, 2, 8, 2, 1, 4, 0, 5, 5, 9, 7, 3, 5, 0, 8, 3, 0, 7, 9, 3, 2, 1, 8, 3, 3, 3, 0, 0, 1, 1, 1, 3, 6, 2, 2, 1, 4, 9, 0, 8, 9, 6, 1, 8, 5, 3, 7, 2, 6, 4, 7, 3, 0, 3, 2, 9, 1, 0
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i * 37.58617815882567125721776348070533282140559735083...
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) lfunzeros(1, [37, 38])[1] \\ M. F. Hasler, Nov 23 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|
|
A305742
|
|
Decimal expansion of imaginary part of 7th nontrivial zero of Riemann zeta function.
|
|
+10
12
|
|
|
4, 0, 9, 1, 8, 7, 1, 9, 0, 1, 2, 1, 4, 7, 4, 9, 5, 1, 8, 7, 3, 9, 8, 1, 2, 6, 9, 1, 4, 6, 3, 3, 2, 5, 4, 3, 9, 5, 7, 2, 6, 1, 6, 5, 9, 6, 2, 7, 7, 7, 2, 7, 9, 5, 3, 6, 1, 6, 1, 3, 0, 3, 6, 6, 7, 2, 5, 3, 2, 8, 0, 5, 2, 8, 7, 2, 0, 0, 7, 1, 2, 8, 2, 9, 9, 6, 0, 0, 3, 7, 1, 9
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
2,1
|
|
LINKS
|
|
|
EXAMPLE
|
The zero is at 1/2 + i * 40.918719012147495187398126914633254395726165962777...
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) lfunzeros(1, [40, 41])[1] \\ M. F. Hasler, Nov 23 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
Search completed in 0.018 seconds
|