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Totient Summatory Function

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TotientSummatoryFunction

The totient summatory function Phi(n) is the summatory function of the totient function phi(n), defined for a nonnegative integer n 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, where mu(d) is the Möbius function and |_x_| denotes the floor function. For n=1, 2, ..., the first values of Phi(n) are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (OEIS A002088).

For real x>=0, set Phi(x)=Phi(|_x_|). As x->infty,

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 (Perott 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

For a positive integer N, 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 (see also Bordellès and Cloitre 2013)

S(N)=A(gamma+lnN)-B+O((lnN)/N),
(9)

where gamma is the Euler-Mascheroni constant. Writing mu(k) for the Möbius function, the constants are

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; 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).

Writing p_k for the kth prime, 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=Asum_(k=1)^(infty)(lnp_k)/(p_k^2-p_k+1)
(18)
=(315)/(2pi^4)zeta(3)sum_(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)
=2.20385...
(22)

(OEIS A065484) is sometimes known as the totient constant (Niklasch 2002), 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

Bordellès, O. and Cloitre, B. "An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions." J. Integer Seq. 16, Art. 13.6.3, 2013. https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.pdf.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." Aug. 23, 2002. https://oeis.org/A001692/a001692.html.Perott, J. "Sur la sommation des nombres phi." Bull. Sci. Math. Astron., Sér. 2 5, 37-40, 1881. https://www.numdam.org/item/BSMA_1881_2_5_1_37_0/.Sloane, N. J. A. Sequences A002088, A028415, A048049, A065483, A065484, A082695, A085609, and A098468 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, Germany: 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 Resource. https://mathworld.wolfram.com/TotientSummatoryFunction.html

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