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Riemann Zeta Function Zeros


Zeros of the Riemann zeta function zeta(s) come in two different types. So-called "trivial zeros" occur at all negative even integers s=-2, -4, -6, ..., and "nontrivial zeros" occur at certain values of t satisfying

 s=sigma+it
(1)

for s in the "critical strip" 0<sigma<1. In general, a nontrivial zero of zeta(s) is denoted rho, and the nth nontrivial zero with t>0 is commonly denoted rho_n (Brent 1979; Edwards 2001, p. 43), with the corresponding value of t being called t_n.

Wiener (1951) showed that the prime number theorem is literally equivalent to the assertion that zeta(s) has no zeros on sigma=1 (Hardy 1999, p. 34; Havil 2003, p. 195). The Riemann hypothesis asserts that the nontrivial zeros of zeta(s) all have real part sigma=R[s]=1/2, a line called the "critical line." This is known to be true for the first 10^(13) zeros.

Wolfram Riemann Zeta Zeros Poster

An attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).

RiemannZetaZerosReImAbs
Min Max
Re
Im Powered by webMathematica

The plots above show the real and imaginary parts of zeta(s) plotted in the complex plane together with the complex modulus of zeta(z). As can be seen, in right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of zeta(s) lie.

RiemannZetaZerosContoursReIm

The position of the complex zeros can be seen slightly more easily by plotting the contours of zero real (red) and imaginary (blue) parts, as illustrated above. The zeros (indicated as black dots) occur where the curves intersect.

RiemannZetaSurfaces

The figures above highlight the zeros in the complex plane by plotting |zeta(z)| (where the zeros are dips) and 1/|zeta(z)| (where the zeros are peaks).

RiemannZetaAbs

The above plot shows |zeta(1/2+it)| for t between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros less than 10, 10^2, 10^3, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).

nOEISt_n
1A05830314.134725
221.022040
325.010858
430.424876
532.935062
637.586178
XiFunctionRoots

The so-called xi-function xi(z) defined by Riemann has precisely the same zeros as the nontrivial zeros of zeta(z) with the additional benefit that xi(z) is entire and xi(1/2+it) is purely real and so are simpler to locate.

ZetaGrid is a distributed computing project attempting to calculate as many zeros as possible. It had reached 1029.9 billion zeros as of Feb. 18, 2005. Gourdon (2004) used an algorithm of Odlyzko and Schönhage to calculate the first 10×10^(12) zeros (Pegg 2004, Pegg and Weisstein 2004). The following table lists historical benchmarks in the number of computed zeros (Gourdon 2004).

yearnauthor
190315J. P. Gram
191479R. J. Backlund
1925138J. I. Hutchinson
19351041E. C. Titchmarsh
19531104A. M. Turing
195615000D. H. Lehmer
195625000D. H. Lehmer
195835337N. A. Meller
1966250000R. S. Lehman
19683500000J. B. Rosser, J. M. Yohe, L. Schoenfeld
197740000000R. P. Brent
197981000001R. P. Brent
1982200000001R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter
1983300000001J. van de Lune, H. J. J. te Riele
19861500000001J. van de Lune, H. J. J. te Riele, D. T. Winter
200110000000000J. van de Lune (unpublished)
2004900000000000S. Wedeniwski
200410000000000000X. Gourdon and P. Demichel

Numerical evidence suggests that all values of t corresponding to nontrivial zeros are irrational (e.g., Havil 2003, p. 195; Derbyshire 2004, p. 384).

No known zeros with order greater than one are known. While the existence of such zeros would not disprove the Riemann hypothesis, it would cause serious problems for many current computational techniques (Derbyshire 2004, p. 385).

Some nontrivial zeros lie extremely close together, a property known as Lehmer's phenomenon.

The Riemann zeta function can be factored over its nontrivial zeros rho as the Hadamard product

 zeta(s)=(e^([ln(2pi)-1-gamma/2]s))/(2(s-1)Gamma(1+1/2s))product_(rho)(1-s/rho)e^(s/rho)
(2)

(Titchmarsh 1987, Voros 1987).

Let rho_k denote the kth nontrivial zero of zeta(s), and write the sums of the negative integer powers of such zeros as

 Z(n)=sum_(k)rho_k^(-n)
(3)

(Lehmer 1988, Keiper 1992, Finch 2003, p. 168), sometimes also denoted sigma_n (e.g., Finch 2003, p. 168). But by the functional equation, the nontrivial zeros are paired as rho and 1-rho, so if the zeros with positive imaginary part are written as sigma_k+it_k, then the sums become

 Z(n)=sum_(k)[(sigma_k+it_k)^(-n)+(1-sigma_k-it_k)^(-n)].
(4)

Such sums can be computed analytically, and the first few are

Z(1)=1/2[2+gamma-ln(4pi)]
(5)
Z(2)=1+gamma^2-1/8pi^2+2gamma_1
(6)
Z(3)=1+gamma^3+3gammagamma_1+3/2gamma_2-7/8zeta(3)
(7)
Z(4)=1+gamma^4-1/(96)pi^4+4gamma^2gamma_1+2gamma_1^2+2gammagamma_2+2/3gamma_3
(8)
Z(5)=1+gamma^5+5gamma^3gamma_1+5/2gamma^2gamma_2+5/2gamma_1gamma_2+5gammagamma_1^2+5/6gammagamma_3+5/(24)gamma_4-(31)/(32)zeta(5)
(9)
Z(6)=1+gamma^6-1/(960)pi^6+6gamma^4gamma_1+2gamma_1^3+3gamma^3gamma_2+3/4gamma_2^2+gamma_1gamma_3+9gamma^2gamma_1^2+gamma^2gamma_3+6gammagamma_1gamma_2+1/4gammagamma_4+1/(20)gamma_5,
(10)

where gamma is the Euler-Mascheroni constant, gamma_i are Stieltjes constants, zeta(n) is the Riemann zeta function, and zeta(3) is Apéry's constant. These values can also be written in terms of the Li constants (Bombieri and Lagarias 1999).

The case

 Z(1)=0.0230957...
(11)

(OEIS A074760; Edwards 2001, p. 160) is classical and was known to Riemann, who used it in his computation of the roots of zeta(s) (Davenport 1980, pp. 83-84; Edwards 2001, pp. 67 and 159). It is also equal to the constant lambda_1 from Li's criterion.

Assuming the truth of the Riemann hypothesis (so that sigma=1/2), equation (◇) can be written for the first few values of n in the simple forms

Z(1)=sum_(k)4/(1+4t_k^2)
(12)
Z(2)=-sum_(k)(8(4t_k^2-1))/((4t_k^2+1)^2)
(13)
Z(3)=-sum_(k)(16(12t_k^2-1))/((4t_k^2+1)^3)
(14)
Z(4)=-sum_(k)(32(16t_k^4-24t_k^2+1))/((4t_k^2+1)^4)
(15)

and so on.


See also

Critical Line, Critical Strip, Explicit Formula, Gram's Law, Gram Point, Hadamard Product, Hardy's Theorem, Hilbert-Pólya Conjecture, Landau's Formula, Lehmer's Phenomenon, Li's Criterion, Mangoldt Function, Montgomery-Odlyzko Law, Montgomery's Pair Correlation Conjecture, Riemann Hypothesis, Riemann-von Mangoldt Formula, Riemann Zeta Function, Xi-Function

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References

Bombieri, E. and Lagarias, J. C. "Complements to Li's Criterion for the Riemann Hypothesis." J. Number Th. 77, 274-287, 1999.Brent, R. P. "On the Zeros of the Riemann Zeta Function in the Critical Strip." Math. Comput. 33, 1361-1372, 1979.Brent, R. P.; van de Lune, J.; te Riele, H. J. J.; and Winter, D. T. "On the Zeros of the Riemann Zeta Function in the Critical Strip. II." Math. Comput. 39, 681-688, 1982.Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.Derbyshire, J. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, 2004.Edwards, H. M. Riemann's Zeta Function. New York: Dover, 2001.Farmer, D. W. "Counting Distinct Zeros of the Riemann Zeta-Function." Electronic J. Combinatorics 2, No. 1, R1, 1-5, 1995. http://www.combinatorics.org/Volume_2/Abstracts/v2i1r1.html.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 168, 2003.Gourdon, X. "Computation of Zeros of the Zeta Function." http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeroscompute.html.Gourdon, X. "The 10^(13) First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height." Oct. 24, 2004. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf.Gram, J.-P. "Sur les zéros de la fonction zeta(s) de Riemann." Acta Math. 27, 289-304, 1903.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. "The Zeros of Zeta." §16.6 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 193-196, 2003.Hayes, B. "The Spectrum of Riemannium." Amer. Sci. 91, 296-300, 2003.Hutchinson, J. I. "On the Roots of the Riemann Zeta-Function." Trans. Amer. Math. Soc. 27, 49-60, 1925.Keiper, J. B. "Power Series Expansions of Riemann's xi Function." Math. Comput. 58, 765-773, 1992.Landau, E. "Über die Nullstellen der Zetafunction." Math. Ann. 71, 548-564, 1911.Lehmer, D. H. "The Sum of Like Powers of the Zeros of the Riemann Zeta Function." Math. Comput. 50, 265-273, 1988.Odlyzko, A. M. "The 10^(20)th Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Odlyzko, A. "Tables of Zeros of the Riemann Zeta Function." http://www.dtc.umn.edu/~odlyzko/zeta_tables/.Pegg, E. Jr. "Math Games: Ten Trillion Zeta Zeros." Oct. 18, 2004. http://www.maa.org/editorial/mathgames/mathgames_10_18_04.html.Pegg, E. Jr. and Weisstein, E. W. "Seven Mathematical Tidbits." MathWorld Headline News. Nov. 8, 2004. http://mathworld.wolfram.com/news/2004-11-08/seventidbits/#3.Sabbagh, K. Dr. Riemann's Zeros: The Search for the $1 Million Solution to the Greatest Problem in Mathematics. Atlantic Books, 2002.Sloane, N. J. A. Sequences A002410/M4924, A058303, A072080, and A074760 in "The On-Line Encyclopedia of Integer Sequences."Titchmarsh, E. C. The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.Tyagi, S. "Double Exponential Method for Riemann Zeta, Lerch and Dirichlet L-Functions." https://arxiv.org/abs/2203.02509. 7 Mar 2022.Voros, A. "Spectral Functions, Special Functions and the Selberg Zeta Function." Commun. Math. Phys. 110, 439-465, 1987.Wagon, S. "The Evidence: Where Are the Zeros of Zeta of s?" Math. Intel. 8, 57-62, 1986.Wagon, S. Mathematica in Action. New York: W. H. Freeman, 1991.Wiener, N. §19 et seq. in The Fourier Integral and Certain of Its Applications. New York: Dover, 1951.Wolfram Research. "The Riemann Zeta Function on the Critical Line Plotted by Mathematica." 1995. http://mathworld.wolfram.com/pdf/posters/Zeta.pdf.ZetaGrid. http://www.zetagrid.net/.

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Riemann Zeta Function Zeros

Cite this as:

Weisstein, Eric W. "Riemann Zeta Function Zeros." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannZetaFunctionZeros.html

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