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# Greek alphabet

The Greek alphabet is used to write the Greek language, and at one point was even used for a numeral system.[1] Mathematicians the world over, however, use certain letters of the Greek alphabet as mathematical symbols.

The Greek alphabet and its usage for mathematical symbols
Letter Name Constants Functions Miscellaneous
${\displaystyle \mathrm {A} \,}$ Alpha
${\displaystyle \alpha \,}$ alpha Fine structure constant ${\displaystyle \scriptstyle \alpha \,\approx \,0.007297\,}$ (A003673)
${\displaystyle \mathrm {B} \,}$ Beta Euler Beta function ${\displaystyle \scriptstyle \mathrm {B} (p,q),\,(p,q)\,\in \,\mathbb {C} ^{2}\,}$
${\displaystyle \beta \,}$ beta
${\displaystyle \Gamma \,}$ Gamma Euler Gamma function ${\displaystyle \scriptstyle \Gamma (z),\,z\,\in \,\mathbb {C} \,}$
${\displaystyle \gamma \,}$ gamma Euler-Mascheroni constant ${\displaystyle \scriptstyle \gamma \,\approx \,0.5772\,}$ (A001620)

${\displaystyle \scriptstyle \gamma _{n}\,}$, the ${\displaystyle \scriptstyle n\,}$th Stieltjes constant

${\displaystyle \Delta \,}$ Delta Nabla ${\displaystyle \scriptstyle \nabla \,}$ (del in vector calculus)

Laplacian or Laplace operator ${\displaystyle \scriptstyle \Delta \,\equiv \,\nabla ^{2}\,}$

${\displaystyle \delta \,}$ delta Silver ratio ${\displaystyle \scriptstyle \delta \,=\,1+{\sqrt {2}}\,\approx \,2.4142\,}$ (A014176) Dirac delta function ${\displaystyle \scriptstyle \delta (x)\,}$

Kronecker delta function ${\displaystyle \scriptstyle \delta _{ij}\,}$

Wieirstraas epsilon-delta notation ${\displaystyle \scriptstyle \delta \,}$
${\displaystyle \mathrm {E} \,}$ psilon
${\displaystyle \epsilon \,}$, ${\displaystyle \varepsilon \,}$ epsilon Wieirstraas epsilon-delta notation ${\displaystyle \scriptstyle \epsilon \,}$

Levi-Civita permutation symbol ${\displaystyle \scriptstyle \epsilon _{ijk}\,}$[2]

${\displaystyle \mathrm {Z} \,}$ Zeta
${\displaystyle \zeta \,}$ zeta Riemann zeta function ${\displaystyle \scriptstyle \zeta (z),\,z\,\in \,\mathbb {C} \,}$ (the classic "zeta function")[3]

Dedekind zeta function[4]
Weierstrass zeta function[5]
● Complex ${\displaystyle \scriptstyle n\,}$th root of unity ${\displaystyle \scriptstyle \zeta _{n}\,=\,\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}\,}$[6]

${\displaystyle \mathrm {H} \,}$ Eta
${\displaystyle \eta \,}$ eta
${\displaystyle \Theta \,}$ Theta
${\displaystyle \theta \,}$, ${\displaystyle \vartheta \,}$ theta Chebyshev or Tschebycheff function ${\displaystyle \scriptstyle \theta (x)\,=\,\sum _{i=1}^{\pi (x)}\log p_{i}\,}$[7][8] ● Angle symbol ${\displaystyle \scriptstyle \theta \,}$
${\displaystyle \mathrm {I} \,}$ Iota
${\displaystyle \iota \,}$ iota
${\displaystyle \mathrm {K} \,}$ Kappa
${\displaystyle \kappa \,}$ kappa
${\displaystyle \Lambda \,}$ Lambda Mangoldt lambda function ${\displaystyle \scriptstyle \Lambda (n)\,}$ ${\displaystyle \scriptstyle n\,}$-dimensional lattice ${\displaystyle \scriptstyle \Lambda _{n}\,}$[9]
${\displaystyle \lambda \,}$ lambda Carmichael lambda function ${\displaystyle \scriptstyle \lambda (n)\,}$ (A002322)[10]

Liouville lambda function ${\displaystyle \scriptstyle \lambda (n)\,}$ (A008836)

${\displaystyle \mathrm {M} \,}$ Mu
${\displaystyle \mu \,}$ mu ● Madachy's constant ${\displaystyle \scriptstyle \mu \,\approx \,1.39418655\,}$ (A130701)[11] Möbius function ${\displaystyle \scriptstyle \mu (n)\,}$ (A008683) Fundamental unit in ${\displaystyle \scriptstyle \mathbb {A} (m)\,}$ when ${\displaystyle \scriptstyle m\,>\,0\,}$ [12]
${\displaystyle \mathrm {N} \,}$ Nu
${\displaystyle \nu \,}$ nu ● Exponent ${\displaystyle \scriptstyle \nu (n)\,}$ of the highest power of the prime ${\displaystyle \scriptstyle p\,}$ that divides ${\displaystyle \scriptstyle n\,}$[13]

● Alternate notation for number of distinct prime factors function ${\displaystyle \scriptstyle \nu (n)\,}$ (also ${\displaystyle \scriptstyle \omega (n)\,}$) (A001221)[14]

${\displaystyle \Xi \,}$ Xi
${\displaystyle \xi \,}$ xi
${\displaystyle \mathrm {O} \,}$ Omicron
${\displaystyle \mathrm {o} \,}$ omicron
${\displaystyle \Pi \,}$ Pi Product operator
${\displaystyle \pi \,}$, ${\displaystyle \varpi \,}$ pi Archimedes' constant ${\displaystyle \scriptstyle \pi \,\approx \,3.1416\,}$ (A000796) Prime counting function ${\displaystyle \scriptstyle \pi (x)\,}$ (A000720) gives ${\displaystyle \scriptstyle \pi (n)\,}$)

● Euler's original notation ${\displaystyle \scriptstyle \pi (n)\,}$ for the totient function ${\displaystyle \scriptstyle \varphi (n)\,}$[15]

${\displaystyle \mathrm {P} \,}$ Rho
${\displaystyle \rho \,}$, ${\displaystyle \varrho \,}$ rho Prime constant ${\displaystyle \scriptstyle \rho \,\approx \,0.4146\,}$ (A051006) Digital root function ${\displaystyle \scriptstyle \rho (n)\,}$[16] (A010888) Pollard's rho method

${\displaystyle \scriptstyle n\,}$th zero of ${\displaystyle \scriptstyle \zeta (s)\,}$ in the upper half of the critical strip[17]

${\displaystyle \Sigma \,}$ Sigma Summation operator
${\displaystyle \sigma \,}$, ${\displaystyle \varsigma \,}$ sigma Number of divisors function ${\displaystyle \scriptstyle \sigma (n)\,=\,\sigma _{0}(n)\,}$ (A000005)

Sum of divisors function ${\displaystyle \scriptstyle \sigma _{1}(n)\,}$
Sum of ${\displaystyle \scriptstyle k\,}$th powers of divisors function ${\displaystyle \scriptstyle \sigma _{k}(n)\,}$
● The real part of a complex number, by Riemann's notation[18]

${\displaystyle \mathrm {T} \,}$ Tau
${\displaystyle \tau \,}$ tau ● Alternative symbol ${\displaystyle \scriptstyle \tau \,}$ for the golden ratio ${\displaystyle \scriptstyle \phi \,}$[19] ● Alternative notation ${\displaystyle \scriptstyle \tau (n)\,}$ for ${\displaystyle \scriptstyle \sigma _{0}(n)\,}$[20][21]

Ramanujan tau function ${\displaystyle \scriptstyle \tau (n)\,}$ (A000594)

${\displaystyle \Upsilon \,}$ Upsilon
${\displaystyle \upsilon \,}$ upsilon
${\displaystyle \Phi \,}$ Phi Totient summatory function ${\displaystyle \scriptstyle \Phi (n)\,}$ ${\displaystyle \scriptstyle n\,}$th cyclotomic polynomial ${\displaystyle \scriptstyle \Phi _{n}(X)\,}$[22]

● The Frattini subgroup ${\displaystyle \scriptstyle \Phi (G)\,}$ of a group ${\displaystyle \scriptstyle G\,}$
● The group of units of ${\displaystyle \scriptstyle \mathbb {Z} _{n}\,}$[23]

${\displaystyle \phi \,}$, ${\displaystyle \varphi \,}$ phi Golden ratio ${\displaystyle \scriptstyle \phi \,=\,{\tfrac {1+{\sqrt {5}}}{2}}\,\approx \,1.6180\,}$ (A001622) Euler's totient function ${\displaystyle \scriptstyle \varphi (n)\,}$
${\displaystyle \mathrm {X} \,}$ Chi
${\displaystyle \chi \,}$ chi ● A characteristic function, like the characteristic function of the primes ${\displaystyle \chi _{p}(n)}$ Modular character
${\displaystyle \Psi \,}$ Psi
${\displaystyle \psi \,}$ psi ● Alternative notation ${\displaystyle \scriptstyle \psi (n)\,}$ for the Liouville lambda function ${\displaystyle \scriptstyle \lambda (n)\,}$

● Alternative notation ${\displaystyle \scriptstyle \psi (n)\,}$ for the Carmichael reduced totient function ${\displaystyle \scriptstyle \lambda (n)\,}$
Dedekind psi function ${\displaystyle \scriptstyle \psi (n)\,}$

${\displaystyle \Omega \,}$ Omega Chaitin's constant ${\displaystyle \scriptstyle \Omega \,}$ Number of prime factors function ${\displaystyle \scriptstyle \Omega (n)\,}$ (A001222)[24] ● The ${\displaystyle \scriptstyle i\,}$th omega subgroup ${\displaystyle \scriptstyle \Omega _{i}(G)\,}$ of a ${\displaystyle \scriptstyle p\,}$-group ${\displaystyle \scriptstyle G\,}$

● The ${\displaystyle \scriptstyle i\,}$th agemo subgroup ${\displaystyle \scriptstyle \mho ^{i}(G)\,}$ of a ${\displaystyle \scriptstyle p\,}$-group ${\displaystyle \scriptstyle G\,}$

${\displaystyle \omega \,}$ omega ● Smallest countably infinite ordinal number ${\displaystyle \scriptstyle \omega \,}$ Number of distinct prime factors function ${\displaystyle \scriptstyle \omega (n)\,}$[25][26] ● Complex cubic root of unity ${\displaystyle \scriptstyle \omega \,\equiv \,\zeta _{3}\,}$[27]

## Notes

1. Kühner, p. 121
2. PlanetMath, Levi-Civita permutation symbol
3. Dan Rockmore, Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers. New York: Pantheon Books (2005) p. 143
4. Rockmore, ibid.
5. Weisstein, Eric W. "Weierstrass Zeta Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/WeierstrassZetaFunction.html
6. Paulo Ribenboim, The New Book of Prime Number Records New York: Springer-Verlag (1996): p. xxi
7. Or as in Paulo Ribenboim, The New Book of Prime Number Records New York: springer-Verlag (1996): p. xx
8. The variant ${\displaystyle \scriptstyle \vartheta (x)\,}$ is also used. Weisstein, Eric W. "Chebyshev Functions." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChebyshevFunctions.html
9. N. J. A. Sloane & Simon Plouffe, The Encyclopedia of Integer Sequences San Diego: Academic Press (1995): p. xii
10. Paulo Ribenboim, The New Book of Prime Number Records New York: springer-Verlag (1996): p. xviii
11. Alfred S. Posamentier & Ingmar Lehmann, The (Fabulous) Fibonacci Numbers, New York: Prometheus Books, 2007, page 171.
12. Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach Mineola, New York: Dover Publications (1969, reprinted 2007): 175
13. L. Carlitz, "The Highest Power of a Prime Dividing Certain Quotients" Archiv der Mathematik 18 2 (1967) p. 157
14. In Wolfram Mathematica, this seems to be more about keeping the function PrimeOmega[n] unambiguously reserved for ${\displaystyle \Omega (n)}$. http://reference.wolfram.com/mathematica/ref/PrimeNu.html
15. D. N. Lehmer, "Dickson's History of the Theory of Numbers" Bull. Amer. Math. Soc. 26 3 (1919), 128.
16. Thomas Koshy, Elementary Number Theory with Applications. Harcourt Academic Press (2002): p. 283, Supplementary Exercises 1 - 3. This is the only place I've seen this meaning assigned to this notation.
17. Paulo Ribenboim, The New Book of Prime Number Records New York: springer-Verlag (1996): p. xx
18. Władysław Narkiewicz, The Development of Prime Number Theory: From Euclid to Hardy and Littlewood Berlin: Springer-Verlag (2000): p. xi
19. As for example, in N. J. A. Sloane & Simon Plouffe, The Encyclopedia of Integer Sequences. San Diego: Academic Press (1995): p. xii
20. Thomas Koshy, Elementary Number Theory with Applications. Harcourt Academic Press (2002): p. 353, Section 8.2
21. Or as in Paulo Ribenboim, The New Book of Prime Number Records New York: Springer-Verlag (1996): p. xix
22. Paulo Ribenboim, The New Book of Prime Number Records New York: Springer-Verlag (1996): p. xxi
23. Ethan D. Bolker, Elementary Number Theory: An Algebraic Approach Mineola, New York: Dover Publications (1969, reprinted 2007): 175
24. Manfred R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity 5th Ed. Springer (2009) p. 407
25. As for example, in Paulo Ribenboim, The New Book of Prime Number Records New York: springer-Verlag (1996): p. xvii
26. Or as in Manfred R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity 5th Ed. Springer (2009) p. 407
27. Manfred R. Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing and Self-Similarity 5th Ed. Springer (2009) p. 407

## References

• R. Kühner, Grammar of the Greek Language, for the Use of High Schools and Colleges. Andover: Allen, Morrill and Wardwell (1844) p. 15