Riemann Prime Counting Function
Riemann defined the function 
by
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
(Hardy 1999, p. 30; Borwein et al. 2000; Havil 2003, pp. 189-191 and 196-197; Derbyshire 2004, p. 299), sometimes denoted 
, 
(Edwards 2001,
pp. 22 and 33; Derbyshire 2004, p. 298), or 
(Havil 2003,
p. 189). Note that this is not an infinite series since the terms become zero
(Derbyshire 2004, p. 299) starting at the 
th term, where 
and 
is the floor function. For 
, 2, ..., the
first few values are 0, 1, 2, 5/2, 7/2, 7/2, 9/2, 29/6, 16/3, 16/3, ... (OEIS A096624 and A096625).
As can be seen, when 
is a prime, 
jumps by 1; when it is the square
of a prime, it jumps by 1/2; when it is a cube of a prime, it jumps by 1/3; and so
on (Derbyshire 2004, pp. 300-301), as illustrated above.
Amazingly, the prime counting function 
is related to 
by the Möbius
transform
 |
(5)
|
where 
is the Möbius
function (Riesel 1994, p. 49; Havil 2003, p. 196; Derbyshire 2004,
p. 302). More amazingly still, 
is connected
with the Riemann zeta function 
by
![(ln[zeta(s)])/s=int_0^inftyf(x)x^(-s-1)dx](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/RiemannPrimeCountingFunction/NumberedEquation2.gif) |
(6)
|
(Riesel 1994, p. 47; Edwards 2001, p. 23; Derbyshire 2004, p. 309). 
is also given by
 |
(7)
|
where 
is the Riemann
zeta function, and (6) and (7) form a
Mellin transform pair.
Riemann (1859) proposed that
 |
(8)
|
where 
is the logarithmic
integral and the sum is over all nontrivial zeros 
of the Riemann
zeta function 
(Mathews
1961, Ch. 10; Landau 1974, Ch. 19; Ingham 1990, Ch. 4; Hardy 1999,
p. 40; Borwein et al. 2000; Edwards 2001, pp. 33-34; Havil 2003,
p. 196; Derbyshire 2004, p. 328). Actually, since the sum of roots is only
conditionally convergent, it must be summed in order of increasing ![I[rho]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/RiemannPrimeCountingFunction/Inline33.gif)
even when
pairing terms 
with their "twins"
, so
![sum_(rho)li(x^rho)=sum_(I[rho]>0)[Li(x^rho)+Li(x^(1-rho))]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/RiemannPrimeCountingFunction/NumberedEquation5.gif) |
(9)
|
(Edwards 2001, pp. 30 and 33).
This formula was subsequently proved by Mangoldt (1895; Riesel 1994, p. 47; Edwards 2001, pp. 48 and 62-65). The integral on the right-hand side converges
only for 
, but since there are no primes
less than 2, the only values of interest are for 
. Since it
is monotonic decreasing, the maximum therefore occurs at 
, which has value
 |
(10)
|
(OEIS A096623; Derbyshire 2004, p. 329).
Riemann also considered the function
 |
(11)
|
sometimes also denoted 
(Borwein et
al. 2000), obtained by replacing 
in the
Riemann function with the logarithmic integral 
, where 
is the Riemann zeta function and 
is the Möbius function (Hardy 1999, pp. 16 and
23; Borwein et al. 2000; Havil 2003, p. 198). 
is plotted
above, including on a semilogarithmic scale (bottom two plots), which illustrate
the fact that 
has a series
of zeros near the origin. These occur at 
for 
(OEIS A143530),
15300.7, 21381.5, 25461.7, 32711.9, 40219.6, 50689.8, 62979.8, 78890.2, 98357.8,
..., corresponding to 
(OEIS A143531), 
,
, 
,
, 
,
, 
,
, 
,
....
The quantity 
is plotted
above.
This function is implemented in the Wolfram Language as RiemannR[x].
Ramanujan independently derived the formula for 
, but nonrigorously
(Berndt 1994, p. 123; Hardy 1999, p. 23). The following table compares
and 
for small
. Riemann conjectured that 
(Knuth
1998, p. 382), but this was disproved by Littlewood in 1914 (Hardy and Littlewood
1918).
The Riemann prime counting function is identical to the Gram series
 |
(12)
|
where 
is the Riemann
zeta function (Hardy 1999, pp. 24-25), but the Gram
series is much more tractable for numeric computations. For example, the plots
above show the difference 
where
is computed using the Wolfram
Language's built-in NSum command (black) and approximated using the
first 
(blue), 
(green), 
(yellow), 
(orange), and
(red) points.
In the table, ![[x]](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/RiemannPrimeCountingFunction/Inline72.gif)
denotes the
nearest integer function. Note that the
values given by Hardy (1999, p. 26) for 
are incorrect.
 |  |  |
| Sloane | A057793 | A057794 |
| 1 | 5 | 1 |
| 2 | 26 | 1 |
| 3 | 168 | 0 |
| 4 | 1227 |  |
| 5 | 9587 |  |
| 6 | 78527 | 29 |
| 7 | 664667 | 88 |
| 8 | 5761552 | 97 |
| 9 | 50847455 |  |
| 10 | 455050683 |  |
| 11 | 4118052495 |  |
| 12 | 37607910542 |  |
Riemann's function is related to the prime counting function by
 |
(13)
|
where the sum is over all complex (nontrivial) zeros 
of 
(Ribenboim
1996), i.e., those in the critical strip so ![0<R[rho]<1](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/RiemannPrimeCountingFunction/Inline85.gif)
, interpreted to mean
 |
(14)
|
However, no proof of the equality of (13) appears to exist in the literature (Borwein et al. 2000).
Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math. 121, 247-296, 2000.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.
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.
Hardy, G. H. and Littlewood, J. E. Acta Math. 41, 119-196, 1918.
Hardy, G. H. "The Series 
." §2.3
in Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, 1999.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, p. 42, 2003.
Ingham, A. E. The Distribution of Prime Numbers. London: Cambridge University Press, p. 83, 1990.
Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.
Landau, E. Handbuch der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, 1974.
Mangoldt, H. von. "Zu Riemann's Abhandlung 'Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.' " J. reine angew. Math. 114, 255-305, 1895.
Mathews, G. B. Ch. 10 in Theory of Numbers. New York: Chelsea, 1961.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 224-225, 1996.
Riemann, G. F. B. "Über die Anzahl der Primzahlen unter einer gegebenen Grösse." Monatsber. Königl. Preuss. Akad. Wiss. Berlin, 671-680, Nov. 1859. Reprinted in Das Kontinuum und Andere Monographen (Ed. H. Weyl). New York: Chelsea, 1972. Also reprinted in English translation in Edwards, H. M. Appendix. Riemann's Zeta Function. New York: Dover, pp. 299-305, 2001.
Riemann, B. "Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe." Reprinted in Gesammelte math. Abhandlungen. New York: Dover, pp. 227-264, 1957.
Riesel, H. and Göhl, G. "Some Calculations Related to Riemann's Prime Number Formula." Math. Comput. 24, 969-983, 1970.
Riesel, H. "The Riemann Prime Number Formula." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 50-52, 1994.
Sloane, N. J. A. Sequences A057793, A057794, A096623, A096624, A096625, A143530, A143531 in "The On-Line Encyclopedia of Integer Sequences."
Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 28-29 and 362-372, 1991.
Weisstein, Eric W. "Riemann Prime Counting Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html
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