I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University.

Today I found a detail in the proof that seems outright wrong, and I am starting to worry that it is a serious problem (in particular since I promised to give a talk about the result next week).

Can anyone explain the last three lines on page 22, or if there indeed is a mistake on those lines give an alternative argument for obtaining the error term that we want to have? Zhang has the following sum $$ \mathcal E_i= \sum_{ d < D^2, d | \mathcal P } \tau_3(d) \rho_2(d) \sum_{ c \in \mathcal C_i ( d ) } | \Delta( \theta,d,c) | . $$ Then he says: By Cauchy's inequality and Theorem 2 we have $$ \mathcal E_i \ll x \mathcal L^{-A}. $$ Here $\mathcal L$ denotes $\log x$. I do not see this. The reason is that Theorem 2 which is given in the following way: For $ 1 \leq i \leq k_0 $ we have $$ \sum_{ d < D^2 , d | \mathcal P } \sum_{ c \in \mathcal C_i(d) } | \Delta (\theta,d,c)|\ll x \mathcal L^{-A}, $$

is given in $L^1$-norm, Cauchy's inequality would need something in $L^2$-norm. The natural inequality to use would be $$ \| f g \|_1 \leq \| f \| _\infty \| g \| _1, $$ where the first function would be the divisor function $\tau_3(d) \rho_2(d)$ and the second would be the sum in $c$. Thus it seems that Theorem 2 in its current form should not really give anything better than

$$ \mathcal E_i \ll \left( \max_{ d < D^2, d | \mathcal P} \tau_3(d) \rho_2(d) \right) \sum_{ d < D^2, d | \mathcal P } \sum_{ c \in \mathcal C_i(d) } | \Delta( \theta,d,c ) | \ll \left( \max_{ d < D^2, d | \mathcal P } \tau_3(d) \rho_2(d) \right) x \mathcal L^{-B} $$ for any $B>0$. Here $ D^2 $ is a little more than $\sqrt x$ (to be precise $ x^{ 1 / 2 + 1 / 584 } $), and $\mathcal P$ is the product of all primes up to a small power of $x$.

Now the sup norm for the divisor function $\tau_3(n)$ certainly grows faster than any power of $\log x$ (even on square free numbers, e.g on primorials), even if it on the average grows like a power of $\log x$). The function $\rho_2(d)$ is defined on page 7 is multiplicative, has support on square free numbers and defined to be $v_p-1$ on the primes, where $v_p$ are the number of residue classes of $\mathcal H$ mod p. With the exception of finitely many $p$ this will be $k_0-1 =3.5\cdot 10^6-1$, i.e really large and contribute much more than the divisor function. Thus this should not get what we need, i.e. a bound of the form $x \mathcal L^{-B}$.

If my concerns are correct, I guess that you can start looking closely at the proof of theorem 2 and see if the same proof holds with the divisor functions thrown in.

If I have made som simple error in the above reasoning I would appreciate your help to understand it, since I would like to understand at least all details on how Theorem 1 implies Theorem 2 before my talk next week (Of course as much as possible of the proof of Theorem 2 also).

Reference: Yitang Zhang: Bounded gaps between primes http://annals.math.princeton.edu/articles/7954