Using Zeta Zeros to Compute a Summatory Liouville Function

An important function in number theory is , the number of integers in the range that have an even number of prime factors minus the number of integers in that range that have an odd number of prime factors. This function is nicely expressible as a sum of values of Liouville's function: , where is the number or prime factors of , via .
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 .


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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 50 pairs of zeta zeros
Snapshot 3: for , showing that , , and . It is surprising that is the last for which until .
In number theory, we define (capital omega) to be the number of prime factors of , counting multiplicity. Therefore, , , , , and . The Liouville lambda function is defined to be . So, , , , , and .
The function is defined to be the sum of for . Whenever has an even number of prime factors (counting multiplicity), that contributes to the sum. Whenever has an odd number of prime factors, that contributes to the sum. This means that is the number of integers in the range that have an even number of prime factors, minus the number that have an odd number of prime factors.
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.
The dominant part of the formula is . As increases, this expression becomes more and more negative. This means that it must be the sum involving zeta zeros that causes the graph of to fluctuate.
For example, at , . But when we include 100 zeta zeros, the above formula gives the value , which is much closer to the exact value, . We could get even closer to this exact value by taking larger values of in the above formula.
You can see from the graphs that for small , is less than or equal to zero. In 1919, the mathematician George Pólya conjectured that for all . However, in 1958, this conjecture was proven to be false. We now know that the smallest counterexample for which is . In fact, .
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.
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" how many prime factors each of the integers has.
To prove formulas like the one above, see [3]. The key is to apply Perron's formula to the following identity, which holds for :
This identity is found in [2], or it may be derived (in Mathematica Version 7 or higher) from
DirichletTransform[LiouvilleLambda[n], n, s].
[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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