9827

Using Zeta Zeros to Count the Squarefree Integers

A squarefree number is a positive integer that is not divisible by the square of a prime. The first 10 squarefree numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14.
The number of squarefree integers from 1 through is usually denoted by . For example, , because 4, 8, and 9 are divisible by prime squares, so they are not counted. The graph of is an 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 .

SNAPSHOTS

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DETAILS

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: a closeup showing that the formula with no zeta zeros misses some jumps of , but when 40 pairs of zeros are used, the formula closely matches
This Demonstration uses the following formula to calculate :
,
where is the complex zero of the Riemann zeta function. See [1].
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 [3]).
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 term in the formula is , which is the density of the squarefree integers, that is, the limiting proportion of integers up to that are squarefree as approaches .
All known zeta zeros have multiplicity 1, which the formula assumes. At those values of where jumps from to , the formula converges to the midpoint, , as approaches infinity.
As we include more zeta zeros in the first sum, we more closely replicate the jumps in the step function . This means that the zeros of the Riemann zeta function somehow know which integers are divisible by squares of primes.
To prove formulas like the one above, see [3]. Define to be 1 if is squarefree and 0 otherwise. Then this identity holds for (see [2]):

.
When we apply Perron's formula to this identity, we get the above equation for .
References:
[2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford: Oxford University Press, 1965 p. 255.
[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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