The Order of Euler’s Totient Function

The Mobius function is commonly used to define Euler’s totient function and the Mangoldt function. Similarly, the summatory Mobius function (the Mertens function) is used to define the summatory totient function and the summatory Mangoldt function (the second Chebyshev function). Analogues of the product formula for the totient function are introduced. An analogue of the summatory totient function with many additive properties is introduced.


Introduction
If n ≥ 1 the Euler totient function ϕ(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n. If n > 1, then n = p a1 1 p a2 2 · · · p a k k where p 1 , p 2 , . . . , p k are primes. The Möbius function µ(n) is defined as follows: µ(1) = 1, µ(n) = (−1) k if a 1 = a 2 = · · · = a k = 1, or µ(n) = 0 otherwise. Note that µ(n) = 0 if and only if n has a square factor > 1. The Euler totient function is related to the Möbius function through the following formula: (1) ϕ(n) = d|n µ(d) n d See Theorem 2.3 of Apostol's [1] book. The Mangoldt function Λ(n) is defined to be log(p) if n = p m for some prime p and some m ≥ 1, or 0 otherwise. Λ(n) is expressed in terms of the logarithm as follows:
Mikolás [2] proved that M ( x/(in) ) = 1, n = 1, 2, 3, ..., x (since x/n /i = x/(in) ). Let R x denote a square matrix where element (i, j) equals 1 if j divides i or 0 otherwise. (In a Redheffer matrix, element (i, j) equals 1 if i divides j or if j = 1. Redheffer [3] proved that the determinant of such a x by x matrix equals M (x).) Let T x denote the matrix obtained from R x by element-by-element multiplication of the columns by Let U x denote the matrix obtained from T x by element-by-element multiplication of the columns by ϕ(j). The sum of the sums of the columns of U x then equals Φ(x). i = d|i ϕ(d), so x i=1 M ( x/i )i (the sum of the sums of the rows of U x ) equals Φ(x). This relationship was proved by Cox [4] (and previously by Mikolás).
A plot of the sums of the rows for x = 8000 is given in Figure 1.
The multiple for a squarefree number q is M (q − 1). Each term in the series is the sum of an arithmetic progression and is thus quadratic in terms of x. About 0.81 times the first term equals Φ(x).

The Second Chebyshev Function
Let U x denote the matrix obtained from T x by element-by-element multiplication of the columns by Λ(j). A plot of the sums of the columns for x = 30 is is given in Figure 5. The sum of the sums of the columns is then ψ(x). Let r n , n = 1, 2, 3, . . . , x, denote the sums of the rows. A plot of r n for n = 1, 2, 3, . . . 200 is given in Figure 6.  As with Φ(x), the process continues with larger squarefree divisors. Again, the multiple for a squarefree number q is M (q − 1). A plot of the partial sum of the Chebyshev function corresponding to the squarefree numbers 2, 3, 5, 6, 7, and 10 is given in Figure 10.

The Function H(x)
Let S x denote the matrix obtained from T x by element-by-element multiplication of the columns by log(j). Let r n , n = 1,2,3,. . .,x denote the sums of the rows of S x . A plot of r n for x = 500 is given in Figure 11. The negative curves are derived from the logarithms of quadratic curves. A plot of the exponential of the negatives of one of the curves (the third one) is given in Figure 12. For a quadratic least-squares fit of the data, p 1 = 1 with a 95% confidence interval of (0.9999, 1), p 2 = 0.002261 with a 95% confidence interval of (−0.007718, 0.01224), p 3 = −0.03387 with a 95% confidence interval of (−0.2574, 0.1897), SSE=0.2631, R-squared=1, and RMSE=0.1244. The positive curves are also quadratic. A plot of one of the curves for x = 100 is given in Figure 13. For a quadratic least-squares fit of the data, p 1 = −0.0001926 with a 95% confidence interval of (−0.0002054, −0.0001798), p 2 = 0.05611 with a 95% confidence interval of (0.05421, 0.05801), p 3 = 5.511 with a 95% confidence interval of (5.5443, 5.579), SSE=0.0002127, R-squared=0.9999, and RMSE=0.003898. The number of negative or positive curves is equal to the number of distinct non-zero Mertens function values less than or equal to x. Sums involving step functions of certain types can be expressed as integrals by means of the following theorem.

The Selberg Identity
Let R x denote the matrix obtained from T x by element-by-element multiplication of the columns by Λ(j) log j + d|j Λ(d)Λ( j d ). Let r n , n = 1,2,3,. . .,x denote the sums of the rows of R x . A plot of r n for x = 500 is given in Figure 14. A plot of the r n for n = x/2 + 1 for x = 100 is given in Figure 15. For a quadratic least-squares fit of the data, p 1 = −0.0006047 with a 95% confidence interval of (−0.0006221, −0.0005874), p 2 = 0.2067 with a 95% confidence interval of (0.2049, 0.2102), p 3 = 6.475 with a 95% confidence interval of (6.379, 6.572), SSE=0.006063, R-squared=1, and RMSE=0.01136.

A Product Formula for ϕ(n)
The sum for ϕ(n) in Theorem 2.3 of Apostol's book can also be expressed as a product extended over the distint prime divisors of n.  The same intervals corresponding to the squarefree numbers occur and the same multiples of the Mertens function are applicable. In the following, the r n values are "normalized", that is, divided by the corresponding Mertens function value (except when that value is 0). Let r n denote the normalized values. r n values of zero are neglected. A product formula similar to Euler's is applicable, but p + 1 factors occur instead of p − 1 factors.
Let σ(n) denote the sum of positive divisors function. Let W x denote the matrix obtained from T x by element-by-element multiplication of the columns by σ(j). Let r n , n = 1, 2, 3,. . . , x, denote the sums of the rows. A plot of r n for x = 200 is given in Figure 18. Let r n , n = 1, 2, 3, . . . , x denote the sums of the rows of Y x . A plot of r n for x = 100 is given in Figure 20. Compared to the sums of the rows of U x , the sums of the rows of Y x have been "normalized". An empirical result is (17) r n is usually non-positive for n < x/3 + 1, positive for n > x/2 , and 0 in between.
There do not appear to be analogous properties for r n values at n = p 3 , n = p 4 , n = p 5 , etc.
These and similar relationships appear to account for all n values.

Ramanujan and Robin's Theorems
Let γ denote Euler's constant.  [5] book. Ramanujan [6] proved that the Riemann hypothesis implies σ(n) < e γ n log log n for sufficiently large n. Robin [7] proved that this inequality is true for all n ≥ 5041 if and only if the Riemann hypothesis is true. Robin's theorems are (26) If the Riemann hypothesis is true, then for each n ≥ 5041, d|n d ≤ e γ n log log n where γ is Euler's constant.
(27) If the Riemann hypothesis is false, then there exist constant 0 < β < 1 2 and C > 0 such that d|n e γ n log log n + Cn log log n (log n) β holds for infinitely many n.
The constant β can be chosen to take any value 1 − b < β < 1 2 where b = (ρ) for some zero ρ of ξ(s) with R(ρ) > 1 2 and C > 0 must be chosen sufficiently small, depending on ρ.
A relationship proved by Cox is

Conclusion
The modified Redheffer matrix technique is applicable to any formula containing the Möbius function. The Möbius function arises in many different places in number theory. The above is the basis for a more rigorous investigation (currently everything is empirically derived). Abel's identity may result in a proof.