Bombieri–Vinogradov Theorem in Shorter Intervals

1. Introduction

Let x denote a sufficiently large integer and p denote prime numbers. Let

$$\pi \left(x\right)=\sum _{p⩽x}1\mathrm{and}\pi (x;q,a)=\sum _{\begin{array}{c}p⩽x\\ p\equiv a(modq)\end{array}}1.$$

Prime Number Theorem tells us that $\pi \left(x\right)\sim \mathrm{Li}\left(x\right)$, the logarithmic integral function. The well-known Bombieri–Vinogradov Theorem, proved independently by Bombieri [1] and Vinogradov [2] in 1965, states that

$$\sum _{q⩽Q}\underset{(a,q)=1}{max}\underset{z⩽x}{max}\left|\pi (z;q,a)-{\displaystyle \frac{\mathrm{Li}\left(z\right)}{\phi \left(q\right)}}\right|\ll {\displaystyle \frac{x}{{(logx)}^A}},$$

where A is a large positive constant, $Q=x^{{\displaystyle \frac{1}{2}}}{(logx)}^{-B}$ and $B=B\left(A\right)>0$.

In 1969, Jutila [3] first considered the analogous result for short intervals. By using zero-density method, he established a result of the following form:

$$\sum _{q⩽Q}\underset{(a,q)=1}{max}\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|\pi (z+h;q,a)-\pi (z;q,a)-{\displaystyle \frac{\mathrm{Li}(z+h)-\mathrm{Li}\left(z\right)}{\phi \left(q\right)}}\right|\ll {\displaystyle \frac{y}{{(logx)}^A}},$$

(1)

where $y=x^{\theta }$ and $\theta ⩽1$. Write $Q=x^{\psi }{(logx)}^{-B}$, Jutila showed that (1) holds for

$$\psi <{\displaystyle \frac{4c\theta +2\theta -1-4c}{6+4c}},\mathrm{if}\zeta \left({\displaystyle \frac{1}{2}}+it\right)\ll t^c.$$

After Jutila, many mathematicians improved this result. In 1971, Motohashi [4] showed that (1) holds for

$$\psi ⩽{\displaystyle \frac{8}{26}}\theta -{\displaystyle \frac{5}{26}},{\displaystyle \frac{5}{8}}<\theta ⩽1.$$

In 1975, Huxley and Iwaniec [5] showed that (1) holds for

$$\psi ⩽\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}<\theta ⩽1;$$

$$\psi <{\left({\displaystyle \frac{1}{5}}+\sqrt{{\displaystyle \frac{3}{5}}\left(\theta -{\displaystyle \frac{3}{5}}\right)}\right)}^2,{\displaystyle \frac{29}{48}}<\theta ⩽{\displaystyle \frac{3}{4}};$$

$$\psi <3\theta -{\displaystyle \frac{7}{4}},{\displaystyle \frac{7}{12}}<\theta ⩽{\displaystyle \frac{29}{48}}.$$

In 1978, Ricci [6] showed that (1) holds for

$$\psi <min\left(\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{5}{2}}\theta -{\displaystyle \frac{3}{2}}\right),{\displaystyle \frac{3}{5}}<\theta ⩽1.$$

In 1984, Perelli, Pintz and Salerno [7] showed that (1) holds for

$$\psi ⩽\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{5}}<\theta ⩽1.$$

In 1985, Perelli, Pintz and Salerno [8] showed that (1) holds for

$$\psi ⩽{\displaystyle \frac{1}{40}},{\displaystyle \frac{7}{12}}<\theta ⩽1.$$

In 1989, Zhan [9] showed that (1) holds for

$$\psi ⩽{\displaystyle \frac{1}{38.5}},{\displaystyle \frac{7}{12}}<\theta ⩽1.$$

In 1988, Timofeev [10] showed that (1) holds for

$$\psi ⩽\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{5}}<\theta ⩽1;$$

$$\psi ⩽\theta -{\displaystyle \frac{11}{20}},{\displaystyle \frac{7}{12}}<\theta ⩽1.$$

The Zero-Density Hypothesis implies that (1) holds for

$$\psi ⩽\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}<\theta ⩽1.$$

In 2012, under the assumption of sixth power large sieve mean-value of Dirichlet L-function, Lao [11] showed that (1) holds for

$$\psi ⩽\theta -{\displaystyle \frac{1}{2}},{\displaystyle \frac{7}{12}}<\theta ⩽1.$$

Lou and Yao [12] and Wu [13] proved that a generalized version of (1) holds under some conditions. The range of $\theta$ is ${\displaystyle \frac{7}{12}}<\theta ⩽1$ in [12] and ${\displaystyle \frac{3}{5}}<\theta ⩽1$ in [13].

Huxley and Iwaniec [5], and several results before, used only zero-density methods. After Perelli, Pintz and Salerno [7], Heath-Brown’s “generalized Vaughan’s identity” [14] was used in the proof of many results on this topic. However, all unconditional results above stop at $\theta ={\displaystyle \frac{7}{12}}$ or larger values. Recently, using the new method of Guth and Maynard [15], Chen [16] announced a better large value estimate for Dirichlet L-functions, which brings the possibility of obtaining new results of type (1). In the persent paper, instead of using Heath-Brown’s identity, we follow the zero-density method used by Huxley and Iwaniec [5] to show that (1) holds for a wider range of $\theta$.

Theorem 1.1.

The estimate (1) holds true for

$$\psi <2\theta -{\displaystyle \frac{8}{7}},{\displaystyle \frac{4}{7}}<\theta ⩽{\displaystyle \frac{7}{12}}.$$

.

Chen’s new zero-density estimate is also applicable for a variant of Bombieri–Vinogradov Theorem whose moduli can be divisible by powers of a given integer. Using similar arguments, one can also show that

$$\sum _{\begin{array}{c}q⩽x^{{\displaystyle \frac{9}{20}}}{l}^{-1}{(logx)}^{-B}\\ (q,l)=1\end{array}}\underset{(a,ql)=1}{max}\underset{z⩽x}{max}\left|\pi (z;ql,a)-{\displaystyle \frac{\mathrm{Li}\left(z\right)}{\phi \left(ql\right)}}\right|\ll {\displaystyle \frac{x}{\phi \left(l\right){(logx)}^A}}$$

(2)

holds for $l⩽x^{{\displaystyle \frac{3}{7}}}exp\left(-{(loglogx)}^3\right)$ that are powers of a given integer. This gives an improvement of Theorem 1.2 of Guo [17]. One can also see another application of Chen’s estimate due to Harm [18].

2. Proof of Theorem 1.1

Now we follow the steps in [5]. Instead of showing (1) directly, we are going to prove an equivalent form of (1):

$$\sum _{q⩽Q}\underset{(a,q)=1}{max}\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|\sum _{\begin{array}{c}z<n⩽z+h\\ n\equiv a(modq)\end{array}}\mathsf{\Lambda }\left(n\right)-{\displaystyle \frac{h}{\phi \left(q\right)}}\right|\ll {\displaystyle \frac{y}{{(logx)}^A}}$$

(3)

holds for ${\displaystyle \frac{4}{7}}<\theta ⩽{\displaystyle \frac{7}{12}}$ and $\psi ={\displaystyle \frac{7}{2}}\theta -2$, where $\mathsf{\Lambda }\left(x\right)$ denote the von Mangoldt function. We have

$$\sum _{\begin{array}{c}z<n⩽z+h\\ n\equiv a(modq)\end{array}}\mathsf{\Lambda }\left(n\right)-{\displaystyle \frac{h}{\phi \left(q\right)}}={\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi (modq)}\overline{\chi }\left(a\right)E(z,h;\chi ),$$

(4)

where

$$\begin{array}{c}\hfill E(z,h;\chi )=\left\{\begin{array}{cc}{\sum }_{z<n⩽z+h}\mathsf{\Lambda }\left(n\right)\chi \left(n\right),\hfill & \chi \mathrm{is}\mathrm{not}\mathrm{principal};\hfill \\ {\sum }_{z<n⩽z+h}\mathsf{\Lambda }\left(n\right)\chi \left(n\right)-h,\hfill & \chi \mathrm{is}\mathrm{principal}.\hfill \end{array}\right.\end{array}$$

(5)

If the character ${\chi }_1$, proper mod f, induces $\chi$ mod q, then

$$E(z,h;{\chi }_1)=E(z,h;\chi )+O(logqlogz).$$

(6)

Since we have

$$\sum _{\begin{array}{c}q⩽Q\\ f\mid q\end{array}}{\displaystyle \frac{1}{\phi \left(q\right)}}\ll {\displaystyle \frac{logQ}{\phi \left(f\right)}},$$

(7)

we can estimate the left-hand side of (3) as

$$\begin{array}{cc}\hfill & \sum _{q⩽Q}\underset{(a,q)=1}{max}\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|\sum _{\begin{array}{c}z<n⩽z+h\\ n\equiv a(modq)\end{array}}\mathsf{\Lambda }\left(n\right)-{\displaystyle \frac{h}{\phi \left(q\right)}}\right|\hfill \\ \hfill \ll & \sum _{f⩽Q}{\displaystyle \frac{logQ}{\phi \left(f\right)}}\sum _{\chi (modf)}^{\ast }\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|E(z,h;\chi )\right|+Q{(logx)}^2,\hfill \end{array}$$

(8)

where ${\sum }^{\ast }$ denote sums over proper characters. In order to deal with the term $Q{(logx)}^2$, we need to assume that $Q\ll y{(logx)}^{-C}$, where C is a large constant that may have different values at different places. We recall the Explicit Formula:

$$E(z,h;\chi )=-\sum _{\begin{array}{c}\rho =\beta +i\gamma \\ \left|\gamma \right|<T\end{array}}{\displaystyle \frac{{(z+h)}^{\rho }-z^{\rho }}{\rho }}+O\left({\displaystyle \frac{x{(logx)}^2}{T}}\right)\mathrm{for}z⩽x,T\ll x\mathrm{and}\mathrm{proper}\chi .$$

(9)

Now, for ${\displaystyle \frac{x}{2}}⩽z⩽x$ we have

$$\begin{array}{c}\hfill {\displaystyle \frac{{(z+h)}^{\rho }-z^{\rho }}{\rho }}\ll \left\{\begin{array}{cc}y{x}^{\beta -1},\hfill & \left|\gamma \right|⩽{\displaystyle \frac{x}{y}};\hfill \\ {\displaystyle \frac{x^{\beta }}{\left|\gamma \right|}},\hfill & \left|\gamma \right|>{\displaystyle \frac{x}{y}}.\hfill \end{array}\right.\end{array}$$

(10)

By a standard dyadic division technique ($F⩽f<2F$), we only need to show that

$$\sum _{F⩽f<2F}\sum _{\chi (modf)}^{\ast }\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|E(z,h;\chi )\right|\ll Fy{(logx)}^{-C}.$$

(11)

By (9) and (10), we have

$$\begin{array}{cc}\hfill & \sum _{F⩽f<2F}\sum _{\chi (modf)}^{\ast }\underset{h⩽y}{max}\underset{{\displaystyle \frac{x}{2}}⩽z⩽x}{max}\left|E(z,h;\chi )\right|\hfill \\ \hfill \ll & \sum _{F⩽f<2F}\sum _{\chi (modf)}^{\ast }\left(\sum _{\begin{array}{c}\rho =\beta +i\gamma \\ \left|\gamma \right|⩽{\displaystyle \frac{x}{y}}\end{array}}y{x}^{\beta -1}+\sum _{\begin{array}{c}\rho =\beta +i\gamma \\ {\displaystyle \frac{x}{y}}<\left|\gamma \right|<T\end{array}}{\displaystyle \frac{x^{\beta }}{\left|\gamma \right|}}\right)+F^2{T}^{-1}x{(logx)}^2\hfill \\ \hfill \ll & logx\underset{{\displaystyle \frac{1}{2}}⩽\alpha ⩽1}{max}\sum _{F⩽f<2F}\sum _{\chi (modf)}^{\ast }y{x}^{\alpha -1}N\left(\alpha ,{\displaystyle \frac{x}{y}},\chi \right)\hfill \\ \hfill & +{(logx)}^2\underset{{\displaystyle \frac{1}{2}}⩽\alpha ⩽1}{max}\underset{{\displaystyle \frac{x}{y}}<U<T}{max}\sum _{F⩽f<2F}\sum _{\chi (modf)}^{\ast }{U}^{-1}{x}^{\alpha }N\left(\alpha ,U,\chi \right)+F^2{T}^{-1}x{(logx)}^2,\hfill \end{array}$$

(12)

where

$$N\left(\sigma ,T,\chi \right)=\#\left\{\mathrm{zeros}\mathrm{of}L(s,\chi ):\beta >\sigma ,\left|\gamma \right|<T\right\}.$$

We can deal with the last term on the right-hand side of (12) by letting $T=Fxy{(logx)}^C$. Clearly this choice satisfies $T\ll x$.

Now, we need several bounds for

$$M\left(F,U\right)=\sum _{f<2F}\sum _{\chi (modf)}^{\ast }N\left(\alpha ,U,\chi \right).$$

We start from (12). If $\alpha ⩾1-c{\left(max\left(logF,{(logx)}^{4/5}\right)\right)}^{-1}$ for some constant c, then $M\left(F,U\right)$ is 0 or 1 and the only possible zero is an exceptional zero. By Siegel’s Theorem, its contribution to (12) can be bounded by $y{(logx)}^{-C}$.

If ${\displaystyle \frac{6}{7}}⩽\alpha <1-c{\left(max\left(logF,{(logx)}^{4/5}\right)\right)}^{-1}$, we can use the zero-density estimate of Montgomery [19], Theorem 12.2, (12.14)]:

$$M\left(F,U\right)\ll {\left(F^{2}U\right)}^{{\displaystyle \frac{2(1-\alpha )}{\alpha }}}{(logx)}^{14}.$$

(13)

Since $y=x^{\theta }$, the terms of (12) is $\ll Fy{(logx)}^{-C}$ if

$$\theta >{\displaystyle \frac{4}{7}}.$$

(14)

If ${\displaystyle \frac{5}{7}}⩽\alpha ⩽{\displaystyle \frac{6}{7}}$, an application of the new result of Chen [16], Theorem 1.3] tells us that

$$M\left(F,U\right)\ll {\left(F^{2}U\right)}^{{\displaystyle \frac{7(1-\alpha )}{3}}+\epsilon }.$$

(15)

In this case, the terms of (12) is $\ll Fy{(logx)}^{-C}$ if

$$\psi <{\displaystyle \frac{7}{2}}\theta -2.$$

(16)

Finally, if ${\displaystyle \frac{1}{2}}⩽\alpha ⩽{\displaystyle \frac{5}{7}}$, the arguments in [5] shows that the terms of (12) is $\ll Fy{(logx)}^{-C}$ if

$$\psi <\underset{{\displaystyle \frac{1}{2}}⩽\alpha ⩽{\displaystyle \frac{5}{7}}}{min}{\displaystyle \frac{(1-\alpha )(3\theta -1-\alpha )}{4-5\alpha }},$$

(17)

and we know that

$$\underset{{\displaystyle \frac{1}{2}}⩽\alpha ⩽{\displaystyle \frac{5}{7}}}{min}{\displaystyle \frac{(1-\alpha )(3\theta -1-\alpha )}{4-5\alpha }}=2\theta -{\displaystyle \frac{8}{7}}$$

(18)

for $\theta ⩽{\displaystyle \frac{30}{49}}$. Now since

$$0<2\theta -{\displaystyle \frac{8}{7}}<{\displaystyle \frac{7}{2}}\theta -2$$

(19)

for $\theta >{\displaystyle \frac{4}{7}}$, the proof of Theorem 1.1 is completed.