On the error in

Goldston and Yildirim's ``Small gaps between consecutive primes''

Goldston and Yildirim define

$$\Lambda_R(n,\vec{h}):=\Lambda_R(n;h_1,\dots ,h_k):=\sum_{ d_1\mid... ...n+h_k\atop D_k \le R } \prod_{j=1}^k \mu(d_j) \log \frac {R}{d_j/(D_{j-1},d_j)}$$

where $\mu$ is the Möbius function, $D_j=[d_1,\dots ,d_j]$ is the least common multiple of the $d_i$ for $1\le i \le j$ and $(a,b)$ is the greatest common divisor of $a$ and $b$. Their main theorems involve the sums

$$\sum_{n=1}^N \Lambda_R(n,\vec {h_1})$$

and

$$\sum_{n=1}^N \Lambda_R(n,\vec {h_1})\Lambda_R(n,\vec{h_2})$$

as well as two more sums like these but over prime numbers for arbitrary integer vectors $\vec{h_1}$ and $\vec{h_2}$. The asymptotic evaluations of these sums are correct in the two cases when there is just one $\Lambda_R$ factor but incorrect for certain $\vec{h_1}$ and $\vec{h_2}$ when there are two $\Lambda_R$ factors.

We give a (simplified) version of how the error occurred. Consider the sum:

$$S:=\frac{1}{N} \sum_{n=1}^N \Lambda_R(n; h)^2$$

for some $h$. Here the basic parameter is $N$ and $R$ is about $N^{1/4}$; the magnitude of the expression is $\log R$. Then Goldston and Yildirim express this as a multiple complex integral:

$$\frac{1}{(2\pi i)^4}\int\limits_{(2)}\int\limits_{(2)}\int\limits... ...s_1 dz_0  dz_1}{\zeta(1+s_0+s_1)\zeta(1+z_0+z_1)s_0(s_0+1)z_0(z_0+1)s_1^2z_1^2}$$

where $\zeta(s)$ is the Riemann zeta-function and where all of the paths are $(2)$ which stands for the vertical line from $2-i\infty$ to $2+i\infty$, and where things have been simplified so that the dependency of the sum on $h$ has been suppressed for simplicity of exposition.

Now the idea is to move the path of each integral slightly to the left of the imaginary axis $(0)$ and determine all of the residues, employing Cauchy's residue theorem together with known estimates for the $\zeta$ function.

We move the paths to $\mathcal C$ which is the path $s=\sigma+it$ with $\sigma=-\frac{C}{\log(2+\vert t\vert)}$ for a certain $C>0$; it is known that $\zeta(1+s)$ has no zeros to the right of this path and that $\vert\zeta(s)\vert, 1/\vert\zeta(s)\vert$ are bounded by $\log (2+\vert t\vert)$ on this path.

If we move the paths of integration one at a time to the left to $\mathcal C$ and collect the residues in order from the poles at $s_0=0$, $z_0=0$ $s_1=0$, and $z_1=0$ then it is easy to see that one has a total contribution $\log R$ from these residues. Moreover, the integrals on the new paths are much smaller, actually of size $\exp(-c_1\sqrt{\log R})$, as can be seen by arguments similar to what one uses in the proof of the prime number theorem; the point is that the exponent of $R$ is sufficiently negative on these paths so that the overall integrals are small.

Consequently, it was believed that the integral was asymptotic to $\log R$. However, notice that if we move the paths for $s_0$ and $z_0$ and consider the new integrals (not the residues) then we have a quadruple integral over $\int_{(2)}\int_{\mathcal (2)}\int_{\mathcal C}\int_{\mathcal C}$ that must also be estimated. Now we move the $s_1$ integral and use the residue term $\log R$ from the double pole at $s_1=0$ and then we move the $z_1$ term and use the residue $z_1=-s_0-z_0$ that occurs from the pole of the zeta-function in the numerator; the net result is a term

$$\frac{\log R}{(2\pi i)^2} \int_{\mathcal C} \int_{\mathcal C} \frac{ds_0 dz_0}{\zeta(1+s_0)\zeta(1-s_0) s_0(s_0+1)z_0(z_0+1)(s_0+z_0)^2}.$$

This term was unaccounted for in the manuscript of Goldston and Yildirim and is apparently of the same size as the main term, i.e. it is a constant times $\log R$. Moreover, there doesn't seem to be any way to identify the constant in any reasonable way other than as this numerical integral. It doesn't seem to simplify. Note: the integral shown above actually can be evaluated, because the sum we considered is a bit simpler than that required by Goldston and Yildirim. The integrals in the general case are somewhat more complicated and cannot be evaluated exactly.

The consensus is that the definition of $\Lambda_R$ needs to be changed so that terms like this one do not appear. However, it is not obvious how to do this change. Work is continuing by Goldston and Yildirim and others to rectify the problem. It does seem reasonable to believe that an improvement on the current world record for small gaps between primes will be achieved by these methods; however, the more dramatic result $p_{n+1}-p_n < (\log n)^\alpha$ for some $\alpha <1$, for infinitely many n, seems less likely.