Particular values of the Riemann zeta function

In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted and is named after the mathematician Bernhard Riemann. When the argument is a real number greater than one, the zeta function satisfies the equation

It can therefore provide the sum of various convergent infinite series, such as Explicit or numerically efficient formulae exist for at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.

The same equation in above also holds when is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at . The complex derivative exists in this more general region, making the zeta function a meromorphic function. The above equation no longer applies for these extended values of , for which the corresponding summation would diverge. For example, the full zeta function exists at (and is therefore finite there), but the corresponding series would be whose partial sums would grow indefinitely large.

The zeta function values listed below include function values at the negative even numbers (s = 2, 4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.

The Riemann zeta function at 0 and 1

At zero, one has

At 1 there is a pole, so ζ(1) is not finite but the left and right limits are:

Since it is a pole of first order, it has a complex residue

Positive integers

Even positive integers

For the even positive integers , one has the relationship to the Bernoulli numbers:

The computation of is known as the Basel problem. The value of is related to the Stefan–Boltzmann law and Wien approximation in physics. The first few values are given by:

Taking the limit , one obtains .

Selected values for even integers
Value Decimal expansion Source
1.6449340668482264364... OEIS: A013661
1.0823232337111381915... OEIS: A013662
1.0173430619844491397... OEIS: A013664
1.0040773561979443393... OEIS: A013666
1.0009945751278180853... OEIS: A013668
1.0002460865533080482... OEIS: A013670
1.0000612481350587048... OEIS: A013672
1.0000152822594086518... OEIS: A013674

The relationship between zeta at the positive even integers and the Bernoulli numbers may be written as

where and are integers for all even . These are given by the integer sequences OEIS: A002432 and OEIS: A046988, respectively, in OEIS. Some of these values are reproduced below:

coefficients
n A B
1 6 1
2 90 1
3 945 1
4 9450 1
5 93555 1
6 638512875 691
7 18243225 2
8 325641566250 3617
9 38979295480125 43867
10 1531329465290625 174611
11 13447856940643125 155366
12 201919571963756521875 236364091
13 11094481976030578125 1315862
14 564653660170076273671875 6785560294
15 5660878804669082674070015625 6892673020804
16 62490220571022341207266406250 7709321041217
17 12130454581433748587292890625 151628697551

If we let be the coefficient of as above,

then we find recursively,

This recurrence relation may be derived from that for the Bernoulli numbers.

Also, there is another recurrence:

which can be proved, using that

The values of the zeta function at non-negative even integers have the generating function:

Since

The formula also shows that for ,

Odd positive integers

The sum of the harmonic series is infinite.

The value ζ(3) is also known as Apéry's constant and has a role in the electron's gyromagnetic ratio. The value ζ(3) also appears in Planck's law. These and additional values are:

Selected values for odd integers
Value Decimal expansion Source
1.2020569031595942853... OEIS: A02117
1.0369277551433699263... OEIS: A013663
1.0083492773819228268... OEIS: A013665
1.0020083928260822144... OEIS: A013667
1.0004941886041194645... OEIS: A013669
1.0001227133475784891... OEIS: A013671
1.0000305882363070204... OEIS: A013673

It is known that ζ(3) is irrational (Apéry's theorem) and that infinitely many of the numbers ζ(2n + 1) : n , are irrational.[1] There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of ζ(5), ζ(7), ζ(9), or ζ(11) is irrational.[2]

The positive odd integers of the zeta function appear in physics, specifically correlation functions of antiferromagnetic XXX spin chain.[3]

Most of the identities following below are provided by Simon Plouffe. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.

Plouffe stated the following identities without proof.[4] Proofs were later given by other authors.[5]

ζ(5)

ζ(7)

Note that the sum is in the form of a Lambert series.

ζ(2n + 1)

By defining the quantities

a series of relationships can be given in the form

where An, Bn, Cn and Dn are positive integers. Plouffe gives a table of values:

coefficients
n A B C D
3 180 7 360 0
5 1470 5 3024 84
7 56700 19 113400 0
9 18523890 625 37122624 74844
11 425675250 1453 851350500 0
13 257432175 89 514926720 62370
15 390769879500 13687 781539759000 0
17 1904417007743250 6758333 3808863131673600 29116187100
19 21438612514068750 7708537 42877225028137500 0
21 1881063815762259253125 68529640373 3762129424572110592000 1793047592085750

These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.

A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.[6][7][8]

Negative integers

In general, for negative integers (and also zero), one has

The so-called "trivial zeros" occur at the negative even integers:

(Ramanujan summation)

The first few values for negative odd integers are

However, just like the Bernoulli numbers, these do not stay small for increasingly negative odd values. For details on the first value, see 1 + 2 + 3 + 4 + · · ·.

So ζ(m) can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.

Derivatives

The derivative of the zeta function at the negative even integers is given by

The first few values of which are

One also has

where A is the Glaisher–Kinkelin constant. The first of these identities implies that the regularized product of the reciprocals of the positive integers is , thus the amusing "equation" .[9]

From the logarithmic derivative of the functional equation,

Selected derivatives
Value Decimal expansion Source
−0.19812624288563685333... OEIS: A244115
−0.93754825431584375370... OEIS: A073002
−0.91893853320467274178... OEIS: A075700
−0.36085433959994760734... OEIS: A271854
−0.16542114370045092921... OEIS: A084448
−0.030448457058393270780... OEIS: A240966
+0.0053785763577743011444... OEIS: A259068
+0.0079838114502686242806... OEIS: A259069
−0.00057298598019863520499... OEIS: A259070
−0.0058997591435159374506... OEIS: A259071
−0.00072864268015924065246... OEIS: A259072
+0.0083161619856022473595... OEIS: A259073

Series involving ζ(n)

The following sums can be derived from the generating function:

where ψ0 is the digamma function.

Series related to the Euler–Mascheroni constant (denoted by γ) are

and using the principal value

which of course affects only the value at 1, these formulae can be stated as

and show that they depend on the principal value of ζ(1) = γ .

Nontrivial zeros

Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The Riemann hypothesis states that the real part of every nontrivial zero must be 1/2. In other words, all known nontrivial zeros of the Riemann zeta are of the form z = 1/2 + yi where y is a real number. The following table contains the decimal expansion of Im(z) for the first few nontrivial zeros:

Selected nontrivial zeros
Decimal expansion of Im(z) Source
14.134725141734693790... OEIS: A058303
21.022039638771554992... OEIS: A065434
25.010857580145688763... OEIS: A065452
30.424876125859513210... OEIS: A065453
32.935061587739189690... OEIS: A192492
37.586178158825671257... OEIS: A305741
40.918719012147495187... OEIS: A305742
43.327073280914999519... OEIS: A305743
48.005150881167159727... OEIS: A305744
49.773832477672302181... OEIS: A306004

Andrew Odlyzko computed the first 2 million nontrivial zeros accurate to within 4×10−9, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.[10][11] A table of about 103 billion zeros with high precision (of ±2-102≈±2·10-31) is available for interactive access and download (although in a very inconvenient compressed format) via LMFDB.[12]

Ratios

Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting particular values of the gamma function into the functional equation

We have simple relations for half-integer arguments

Other examples follow for more complicated evaluations and relations of the gamma function. For example a consequence of the relation

is the zeta ratio relation

where AGM is the arithmetic–geometric mean. In a similar vein, it is possible to form radical relations, such as from

the analogous zeta relation is

References

  1. Rivoal, T. (2000). "La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs". Comptes Rendus de l'Académie des Sciences, Série I. 331 (4): 267–270. arXiv:math/0008051. Bibcode:2000CRASM.331..267R. doi:10.1016/S0764-4442(00)01624-4. S2CID 119678120.
  2. W. Zudilin (2001). "One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational". Russ. Math. Surv. 56 (4): 774–776. Bibcode:2001RuMaS..56..774Z. doi:10.1070/rm2001v056n04abeh000427. S2CID 250734661.
  3. Boos, H.E.; Korepin, V.E.; Nishiyama, Y.; Shiroishi, M. (2002). "Quantum correlations and number theory". J. Phys. A. 35 (20): 4443–4452. arXiv:cond-mat/0202346. Bibcode:2002JPhA...35.4443B. doi:10.1088/0305-4470/35/20/305. S2CID 119143600..
  4. "Identities for Zeta(2*n+1)".
  5. "Formulas for Odd Zeta Values and Powers of Pi".
  6. Karatsuba, E. A. (1995). "Fast calculation of the Riemann zeta function ζ(s) for integer values of the argument s". Probl. Perdachi Inf. 31 (4): 69–80. MR 1367927.
  7. E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).
  8. E. A. Karatsuba: Fast evaluation of ζ(3). Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).
  9. Muñoz García, E.; Pérez Marco, R. (2008), "The Product Over All Primes is ", Commun. Math. Phys. (277): 69–81.
  10. Odlyzko, Andrew. "Tables of zeros of the Riemann zeta function". Retrieved 7 September 2022.
  11. Odlyzko, Andrew. "Papers on Zeros of the Riemann Zeta Function and Related Topics". Retrieved 7 September 2022.
  12. LMFDB: Zeros of ζ(s)

Further reading

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