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Totient Summatory Function

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TotientSummatoryFunction

The summatory function Phi(n) of the totient function phi(n) is defined by

Phi(n)=sum_(k=1)^(n)phi(k)
(1)
=sum_(m=1)^(n)msum_(d|m)(mu(d))/d
(2)
=sum_(d=1)^(n)mu(d)sum_(d^'=1)^(|_n/d_|)d^'
(3)
=1/2sum_(d=1)^(n)mu(d)|_n/d_|(1+|_n/d_|)
(4)

(Hardy and Wright 1979, p. 268), plotted as the red curve above. The first values of Phi(n) are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (OEIS A002088).

Phi(n) has the asymptotic series

Phi(x)∼1/(2zeta(2))x^2+O(xlnx)
(5)
∼3/(pi^2)x^2+O(xlnx),
(6)

where zeta(z) is the Riemann zeta function (Perrot 1881; Nagell 1951, p. 131; Hardy and Wright 1979, p. 268; blue curve above). An improved asymptotic estimate due to Walfisz (1963) is given by

Phi(x)∼(3x^2)/(pi^2)+O[x(lnx)^(2/3)(lnlnx)^(4/3)].
(7)
TotientInverseSummatory

Consider the summatory function of 1/phi(n),

S(N)=sum_(n=1)^N1/(phi(n)),
(8)

plotted as the red curve above. For N=1, 2, ..., the first few terms are 1, 2, 5/2, 3, 13/4, 15/4, 47/12, 25/6, ... (OEIS A028415 and A048049). The sum diverges as N->infty, but Landau (1900) showed that the asymptotic behavior is given by

S(N)∼A(gamma+lnN)+B+O((lnN)/N),
(9)

where gamma is the Euler-Mascheroni constant,

A=sum_(k=1)^(infty)([mu(k)]^2)/(kphi(k))
(10)
=(zeta(2)zeta(3))/(zeta(6))
(11)
=(315)/(2pi^4)zeta(3)
(12)
=1.9435964368...
(13)
B=sum_(k=1)^(infty)([mu(k)]^2lnk)/(kphi(k))
(14)
=1.18244...
(15)

(OEIS A082695), mu(k) is the Möbius function, zeta(z) is the Riemann zeta function, and p_k is the kth prime (Landau 1900; Halberstam and Richert 1974, pp. 110-111; DeKoninck and Ivić 1980, pp. 1-3; Finch 2003, p. 116; Havil 2003, p. 115; Dickson 2005).

A and B can also be written as

A=product_(k=1)^(infty)(1-p_k^(-6))/((1-p_k^(-2))(1-p_k^(-3)))
(16)
=product_(k=1)^(infty)[1+1/(p_k(p_k-1))]
(17)

and

B=Aproduct_(k=1)^(infty)(lnp_k)/(p_k^2-p_k+1)
(18)
=(315)/(2pi^4)zeta(3)product_(k=1)^(infty)(lnp_k)/(p_k^2-p_k+1),
(19)

respectively, making these constants similar in form to Artin's constant (Finch 2003, pp. 116-117).

The sum

C_(totient)=sum_(n=1)^(infty)1/(nphi(n))
(20)
=zeta(2)product_(p)[1+1/(p^2(p-1))]
(21)
=2.20386...
(22)

(OEIS A118262) is sometimes known as the totient constant (Niklasch), where

product_(p)[1+1/(p^2(p-1))]=1.33978...
(23)

(OEIS A065483) and the products are taken over the primes p.

See also

Prime Products, Totient Function, Totient Valence Function

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References

DeKoninck, J.-M. and Ivić, A. Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 113-158, 2005.Finch, S. R. "Euler Totient Constants." §2.7 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 115-119, 2003.Halberstam, H. and Richert, H.-E. Sieve Methods. New York: Academic Press, 1974.Hardy, G. H. and Wright, E. M. "The Average Order of phi(n)." §18.5 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 268-269, 1979.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Landau, E. "Über die zahlentheoretische Function phi(n) und ihre Beziehung zum Goldbachschen Satz." Nachr. Königlichen Ges. Wiss. Göttingen, Math.-Phys. Klasse, 177-186, 1900. Werke, Vol. 1 (Ed. L. Mirsky, I. J. Schoenberg, W. Schwarz, and H. Wefelscheid). Thales Verlag, pp. 106-115, 1983. Mitrinović, D. S. and Sándor, J. §I.27 in Handbook of Number Theory. Dordrecht, Netherlands: Kluwer, 1995.Nagell, T. "Relatively Prime Numbers. Euler's phi-Function." §8 in Introduction to Number Theory. New York: Wiley, pp. 23-26, 1951.Niklasch, G. "Some Number-Theoretical Constants." http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml.Perrot, J. 1811. Quoted in Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 126, 2005.Sloane, N. J. A. Sequences A028415, A048049, A065483, A082695, A085609, A098468, and A118262 in "The On-Line Encyclopedia of Integer Sequences."Stephens, P. J. "Prime Divisor of Second-Order Linear Recurrences, I." J. Number Th. 8, 313-332, 1976.Walfisz, A. Ch. 5 in Weyl'sche Exponentialsummen in der neueren Zahlentheorie. Berlin: Deutscher Verlag der Wissenschaften, 1963.

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Totient Summatory Function

Cite this as:

Weisstein, Eric W. "Totient Summatory Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TotientSummatoryFunction.html

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