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The xi-function is the function
 |  |  |
(1)
|
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(2)
|
where 
is the Riemann zeta function and 
is the gamma function
(Gradshteyn and Ryzhik 2000, p. 1076; Hardy 1999, p. 41; Edwards 2001,
p. 16). This is a variant of the function originally defined by Riemann in his
landmark paper (Riemann 1859), where the above now standard notation follows Landau
(Edwards 2001, p. 16).
It is an entire function (Edwards 2001, p. 16).
It is implemented in the Wolfram Language as RiemannXi[s].
The zeros of 
and of its derivatives are all located on the critical
strip 
,
where 
.
Therefore, the nontrivial zeros of the Riemann
zeta function exactly correspond to those of 
(i.e., the roots of 
are the same as those of 
for real 
), with the additional benefit that 
is purely real.
The first few zeros occur at the values summarized 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 zeros less than 10, 
, 
, ... are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS
A072080; Odlyzko).
 | OEIS |  |
| 1 | A058303 | 14.134725 |
| 2 | 21.022040 | |
| 3 | 25.010858 | |
| 4 | 30.424876 | |
| 5 | 32.935062 | |
| 6 | 37.586178 |
Special values include
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(3)
|
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(4)
|
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(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
The 
function satisfies the functional
equation
 |
(9)
|
(Edwards 2001, p. 16).
The xi-function has the Taylor series about 1/2 of
 |
(10)
|
where
![a_(2n)=4int_1^infty(d[x^(3/2)psi^'(x)])/(dx)((1/2lnx)^(2n))/((2n)!)x^(-1/4)dx](/images/equations/Xi-Function/NumberedEquation3.svg) |
(11)
|
and
 |  |  |
(12)
|
 |  | ![1/2[theta_3(0,e^(-pix))-1]](/images/equations/Xi-Function/Inline45.svg) |
(13)
|
(Edwards 2001, p. 15), with 
a Jacobi theta
function. The coefficient 
has the simple analytic form
 |  |  |
(14)
|
 |  |  |
(15)
|
(OEIS A114720).
As stated by Riemann (1859) and first rigorously proved by Hadamard (1893), the xi-function can be written as
 |
(16)
|
where the product runs over the roots 
of 
(Edwards 2001, pp. 17-21).
 |
The xi-function extended into the complex plane is illustrated above.
The function 
is related to
 |
(17)
|
where 
(Gradshteyn and Ryzhik 2000, p. 1074; Edwards 2001, p. 16), which is the
function originally considered and actually denoted 
by Riemann (Edwards 2001, p. 16). This function can
also be defined as
 |
(18)
|
giving
 |
(19)
|
The de Bruijn-Newman constant is defined in terms of the 
function.
Hardy (1914) proved that 
has infinitely many real roots (Hardy's theorem),
Hardy and Littlewood (1921) proves that the number of real roots between 0 and 
is at least 
for some positive constant 
and all sufficiently large 
, and Selberg (1942) proved that this number is in fact at
least 
for some positive 
and all large 
(Edwards 2001, p. 19).
Coffey (2004) gives a number of formulas of derivatives of 
.
Function." J. Comput. Appl. Math. 166,
525-534, 2004.Conrey, J. B. "The Riemann Hypothesis."
Not. Amer. Math. Soc. 50, 341-353, 2003. 
." §1.8 in 
de Riemann." C. R. Acad. Sci. Paris 158,
1012-1014, 1914.Hardy, G. H. 
Function." Math. Comput. 58,
765-773, 1992.Li, X.-J. "The Positivity of a Sequence of Numbers
and the Riemann Hypothesis." J. Number Th. 65, 325-333, 1997.Odlyzko,
A. M. "The 
th
Zero of the Riemann Zeta Function and 70 Million of Its Neighbors." Preprint.Odlyzko,
A. "Tables of Zeros of the Riemann Zeta Function." 
?" Math. Intel. 8, 57-62, 1986.Wagon,
S.