To calculate the residues associated with the singularities of the integrand function , we can work with the functions and separately:

has only one singularity: a simple pole at 0 with residue 1, so the product function there has a residue of .

To determine the residues of , we first consider the function

Here denotes Euler's Gamma function.


Note: Riemann would have used different notation, and stated

where . This is now more often denoted by .

We can also write where s = 1/2 + it, which reduces the Riemann Hypothesis to the assertion that all zeros of are real.


Recall that this is the function with the simple functional equation

and whose only zeros are the nontrivial zeros of . The trivial zeros are 'annihilated' by the Gamma function

This is an integral function of order 1, meaning that its sequence of zeros {zn} is such that the sum of |zn| -a converges if and only if a > 1 (see Theorem 18, Ingham).

We further note that (see Theorem 15, Ingham) and may now apply Hadamard's Factorisation Theorem.

This gives us

which then gives

Here bo and b1 are constants which have yet to be determined, and the are the zeros of , or equivalently, the nontrivial zeros of .

Taking logarithms and then deriving gives

Letting s = 0, while observing that , (Euler's constant) and , we see that .


Note: The third term in this equation is expressible in terms of the digamma function (confusingly) notated

.

This is not related to Chebyshev's psi function encountered earlier. It is known to satisfy


Therefore we have

for a constant c. This is easily seen to have singularities at

s = 1 (residue = -1)
s = {-2,-4,-6,...} (trivial zeta zeros, residues = 1)
s = (nontrivial zeta zeros, residues = 1)

Hence has the following singularities and residues:

singularities residues
0 (due to )
1 (due to pole of zeta function) -x1/1 = -x
{-2,-4,-6,...} (due to trivial zeta zeros) {x -2/2, x -4/4, x -6/6,...} which sum to [log(1-1/x2)]/2
(due to nontrivial zeta zeros)


This leads directly to the explicit formula. To summarise, the steps for computing the residues are as follows:

1. Define in terms of

2. Apply Hadamard's factorisation theorem to get an expression for in terms of an infinite product over the nontrivial zeta zeros

3.Use this to produce an expression for in terms of the Gamma function and a sum over nontrivial zeros.

4.Rewrite terms involving the Gamma function to give an expression for in terms of sums over both trivial and nontrivial zeros.

5.Calculate residues directly from definition.

 


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