Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros

Question

Very recently, Yitang Zhang just gave a (virtual) talk about his work on Landau-Siegel zeros at Shandong University on the 5th of November's morning in China. He will also give a talk on 8th November at Peking University.

The 111-page preprint now can be found on the internet, and it seems this version will be published on arXiv soon. (UPDATE: now it's on arXiv.)

This paper shows that for a real primitive character $\chi$ to the modulus $D$, $$ L(1, \chi) > c_{1}(\log D)^{-2022} $$ where $c_{1} > 0$ is an absolute, effectively computable constant.

Assuming this result is correct, what are some significant number theoretical consequences that would follow?

For example, what would be the impact on PNT error estimates, arithmetic progressions, and other related problems?

Answer 1

It has significant implications on the error term of the PNT for arithmetic progressions.

PNT and Siegel-Walfisz theorem

Let $\psi(x;q,a)$ be the sum of $\Lambda(n)$ over $n\le x$ and $n\equiv a\pmod q$. Then the PNT states that for fixed $q$ there is

$$ \psi(x;q,a)\sim{x\over\varphi(q)}.\tag1 $$

When $q$ is not fixed, Page (1935) proved the following general result:

Theorem 1 (Page): There exists some absolute and effective $c_0>0$ such that for all $(a,q)=1$:

$$ \psi(x;q,a)={x\over\varphi(q)}-\color{blue}{{\chi(a)x^\beta}\over\varphi(q)\beta}+O\{xe^{-c_0\sqrt{\log x}}\},\tag2 $$

where $\chi$ denotes the exceptional character and $\beta$ denotes the Siegel zero. The blue term would be dropped if there are no exceptional characters modulo $q$.

To unify the error terms, we require a result due to Siegel (1935):

Theorem 2 (Siegel): For all $\varepsilon>0$ there exists some $A_\varepsilon>0$ such that $1-\beta>A_\varepsilon q^{-\varepsilon}$.

Plugging this result into the blue term of (2), we have

$$ x^\beta\ll x e^{-A_\varepsilon q^{-\varepsilon}\log x}. $$

If $q\le(\log x)^{2/\varepsilon}$, then the right hand side becomes $\ll xe^{-A_\varepsilon\sqrt{\log x}}$. Combining this with (2) gives us the result of Walfisz (1936):

Theorem 3 (Siegel-Walfisz): For any $M>0$ there exists some $C_M$ such that for all $q\le(\log x)^M$ and $(a,q)=1$ there is

$$ \psi(x;q,a)={x\over\varphi(q)}+O\{e^{-C_M\sqrt{\log x}}\},\tag3 $$

where the O-constant is absolute.

Due to the drawback in the proof of Siegel's theorem, $A_\varepsilon$ and $C_M$ are not effectively computable.

Improvements due to Zhang

However, we can significantly obtain a stronger and effective improvement of Siegel-Walfisz theorem if Zhang's result is used. That is

Theorem 4 (Zhang): There exists $A>0$ and effective $C_1>0$ such that $L(1,\chi)>C_1(\log q)^{-A}$.

Zhang proved this result for $A=2022$, but I choose not to plug it in for generality.

Let $\beta$ be the rightmost real zero of $L(s,\chi)$ for some real $\chi$ modulo $q$ such that $1-\beta\gg(\log q)^{-1}$. Then it follows from the mean value theorem that there exists some $1-\beta<\sigma<1$ such that $1-\beta=L(1,\chi)/L'(\sigma,\chi)$. Applying the classical bound $L'(\sigma,\chi)=O(\log^2q)$ and Zhang's result gives us the zero-free region that

$$ 1-\beta>C_2(\log q)^{-A-2}, $$

where $C_2>0$ is effectively computable, which indicates that the blue term in (2) is dominated by

$$ x^\beta\ll xe^{-C_2(\log x)(\log q)^{-A-2}}. $$

If $(\log q)^{A+2}\le\sqrt{\log x}$, then the right hand side becomes $\ll xe^{-C_2\sqrt{\log x}}$, which allows Theorem 1 to be improved significantly:

Theorem 5: Let $A$ be as in Theorem 4. There exists some absolute $c_0>0$ such that for all $q\le e^{(\log x)^{1/(2A+4)}}$ and $(a,q)=1$, we have

$$ \psi(x;q,a)={x\over\varphi(q)}+O\{e^{-c_0\sqrt{\log x}}\}.\tag4 $$

Asymptotic formulas valid for all $q\ge1$.

Although Theorem 5 is much stronger than Theorem 3, it is difficult to compare them to Theorem 1 without the blue term, so this section is dedicated to deduce asymptotic formula valid for all $q\ge1$ and $(a,q)=1$ so that a better comparison can be made.

Since $\Lambda(n)\le\log n$, we know trivially that

$$ \psi(x;q,a)\le\sum_{\substack{n\le x\\n\equiv a(q)}}\log x\ll{x\log x\over q}. $$

Combining this with (3), we see that Theorem 3 indicates that

$$ \psi(x;q,a)={x\over\varphi(q)}+O_N\{x(\log x)^{-N}\}\quad(N>0).\tag5 $$

If the trivial upper bound is juxtaposed with (4), then we see that there exists some absolute and effective $c_0>0$ such that

$$ \psi(x;q,a)={x\over\varphi(q)}+O\{xe^{-c_0(\log x)^{1/(2A+4)}}\}, $$

which has a substantially better error term than (5).

Answer 2

There will be many important consequences of Zhang's result, if correct. One specific result is that it will reduce one of the last open problem from the era of Gauss and Euler to a finite amount of computation, namely the classification of discriminants of binary quadratic forms with one class per genus. The congruence class of a prime number $p$ modulo $d$ determines which form of discriminant $-d<0$ represents $p$ if and only if there is one class per genus.

Such discriminants which are congruent to $0$ modulo $4$ are Euler's numeri idonei or idoneal numbers. Euler expected there would be infinitely many such discriminants. [Edit: Apparently I'm mis-remembering this. See the remarks of KConrad below.] It was Gauss who conjectured that the only such discriminants are the 65 examples (not necessarily fundamental) known to Euler. There are also 65 known fundamental discriminants (not necessarily even) with one class per genus. The existence of a 66th is still an open problem. By genus theory we know that for discriminants with one class per genus, the class group satisfies $$ C(-d)\cong \left(\mathbb Z/2\right)^{g-1}, $$ where $g$ is the number of prime divisors of $d$. Obviously $d$ is bigger than the absolute value of the smallest fundamental discriminant with $g$ prime divisors, $$ d_g\overset{\text{def.}}=3\cdot4\cdot5\cdot7\cdots p_g. $$ From lower bounds on the size of $p_g$, the $g$-th prime, and on $\theta(x)=\sum_{p\leq x}\log(p)$, one can show that $$ d_g>g^g. $$ Since $ 2^{g-1} \ll \sqrt{g^g}, $ lower bounds for the class number which we expect to be true rule out the possibility for one class per genus for large $g$.

In 1973, Peter Weinberger showed that on GRH, no fundamental discriminant $-d<-5460$ has one class per genus, and unconditionally there is at most one more such $d$.

In contrast, Oesterle explicitly observed that the lower bound due to Goldfeld-Gross-Zagier is not strong enough to finish the classification of discriminants with one class per genus: $ \log(g^g) $ is $\ll 2^{g-1} $. Iwaniec and Kowalski observed that even the full strength of the Birch Swinnerton-Dyer conjecture, "the best effective lower bounds which current technology allows us to hope for" would not suffice, as $ \log(g^g)^r$ is $\ll 2^{g-1}$ for any $r$. In fact, the outlook is still more bleak: Watkins observed that if the discriminant $-d$ is divisible by all the primes up to $(\log\log d)^3$ (as $d_g$ certainly is), the product over primes dividing $d$ in the Goldfeld-Gross-Zagier lower bound is so small the resulting bound is worse than the trivial bound.

If the implied constant is made explicit, Zhang's result would eliminate the possibility of one class per genus for discriminants above some bound. For example, neglecting the constant, the lower bound is discriminants with more than 6007 prime divisors. This works out to $d>3\cdot 10^{25734}$.