Using Zeta Zeros to Tally the Euler Phi Function

The Euler phi () function (also called the totient function) is important in number theory; is the number of positive integers less than or equal to that have no factor in common with . For example, and .
The sum of for positive integers is a function of that is usually denoted by . is an increasing, but irregular, step function.
This Demonstration illustrates the remarkable fact that we can approximate the jumps of this step function by using a sum that involves zeros of the Riemann zeta function .


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Snapshot 1: the graphs of the step function and the formula using no zeta zeros
Snapshot 2: the graphs of and the formula using 30 pairs of zeta zeros
Snapshot 3: for larger , we need more and more zeta zeros in order to closely match the step function
This Demonstration uses the following formula to calculate :
where is the complex zero of the Riemann zeta function.
The first three complex zeros of the zeta function are approximately , , and . The zeros occur in conjugate pairs, so if is a zero, then so is . The important Riemann hypothesis is the unproven conjecture that all these complex zeros have real part 1/2. So far, it has been verified that the first complex zeros do, indeed, have real part 1/2 (see [1]).
If you use the slider to choose, say, one pair of zeta zeros, then the first sum in the above formula, in effect, combines two terms corresponding to the first conjugate pair of zeta zeros and . So, when these terms are added, their imaginary parts cancel while their real parts add. The applied to the first sum is simply an efficient way to combine the two terms for each pair of zeta zeros.
Notice that the second sum has the same form as the first, except that the second sum extends over the so-called "trivial" zeros of the zeta function, namely, . The second sum is small when is large.
The dominant terms in the formula are . This expression is fairly close to , but the accuracy of the formula improves as we include more terms with zeta zeros.
All known zeta zeros have multiplicity 1, which is assumed in the formula. At those values of where jumps from to , the formula converges to the midpoint, , as approaches infinity. You can see this in the graphs.
Also, you can use the tooltip to read off individual values from the graph. The definition of means that . So, for example, at , the graph jumps from 28 to 32; this increase is simply .
To prove formulas like the one for , see [3]. The key is to apply Perron's formula to the following identity, which holds for :
This identity is in [2], or it may be derived (in Mathematica Version 7 or higher) from
DirichletTransform[EulerPhi[n], n, s].
[2] G. H. Hardy and E. M. Wright, The Theory of Numbers, 4th ed., Oxford: Oxford University Press, 1965 p. 250.
[3] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory: I. Classical Theory, Cambridge: Cambridge University Press, 2007 p. 397.
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