Using Zeta Zeros to Compute Sigma Sums

In number theory, the sum of the divisors of an integer is usually denoted by . ( is the lowercase Greek letter sigma.) For example, 4 has three divisors (namely, 1, 2, and 4), so .
Suppose . The sum of for is an irregular step function that jumps up at every integer . For example, for , this sum is . For , this sum is .
Similarly, the sum of for is also an irregular step function. For example, for , this sum is .
This Demonstration shows how we can approximate each of these step functions with sums that involve 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 100 pairs of zeta zeros
Snapshot 3: the graphs of and the formula using 100 pairs of zeta zeros
Here is the step function that is graphed when the left-hand button is pressed:
After you use the slider to choose (the number of pairs of zeta zeros to use), this Demonstration uses the following formula to calculate :
(1) .
In this formula,
In equation (1), is the complex zero of the Riemann zeta function with positive real part. The first three complex zeros of the zeta function are approximately , , and . These zeros occur in conjugate pairs, so if is a zero, then so is .
If you use the slider to choose, say, (one pair of zeta zeros), then the sum in equation (1) adds the two terms that correspond to the first pair of conjugate zeros, and . These terms are conjugates of each other. When these terms are added, their imaginary parts cancel while their real parts add. So, the applied to the sum is merely an efficient way to combine the two terms for each pair of zeta zeros.
If you plot a graph using no zeta zeros, then the graph is computed with only the terms through .
Now consider the step function that is graphed when the right-hand button is pressed:
Here is the formula used to graph this function:
where is Euler's constant.
If you plot a graph using no zeta zeros, then the graph is computed with only the terms through .
Where Do Equations (1) and (2) Come From?
To prove equation (1), we start with the following identity, which holds for (see [3], equation 5.39):
(3) .
Perron's formula (see reference [4]) takes an identity like equation (3) and gives a formula for the sum of the numerators as a function of , in this case, .
When we apply Perron's formula to equation (3), we get equation (1). To apply Perron's formula, we integrate this integrand
around a contour in the complex plane. Each part of equation (1) is the residue at one of the poles of this integrand. The residue at the pole at is . At , has a pole (of order 1), so the integrand has poles of order 1 at , at , and at . The residue at the pole is . The residue at is . The residue at is .
Mathematica can compute these residues. For example, this calculation Residue[Zeta[s] Zeta[s-1] Zeta[s-2]/Zeta[2 s-2] x^s/s,{s,3}] gives , where has the value given above for equation (1).
Finally, the integrand has a pole at each complex zero of . The first sum in equation (1) is just the sum of the residues at these complex zeros of zeta. Each complex zero gives rise to one term in the sum. (The complex zero of is , so is the complex zero of .)
The integrand also has poles at the real zeros of (i.e., at , in other words, at ). However, all these residues are 0 except at , where the residue is .
Equation (2) is proved the same way. We start with this identity, which holds for (see [1], equation D-58, and [2], Theorem 305):
As with equation (1), we apply Perron's formula, this time using the integrand
The Demonstration "Using Zeta Zeros to Tally Squarefree Divisors" gives additional details on Perron's formula.
[1] H. W. Gould and T. Shonhiwa, "A Catalog of Interesting Dirichlet Series".
[2] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford: Oxford University Press, 1965, p. 256.
[3] P. J. McCarthy, Introduction to Arithmetical Functions, New York: Springer-Verlag, 1986, p. 237.
[4] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory: I. Classical Theory, Cambridge: Cambridge University Press, 2007, p. 397.
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