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.