MATHEMATICS

M. B. BARBAN, Corresponding Member of the Academy of Sciences of the USSR Yu. V. LINNIK, N. G. CHUDAKOV

ON THE DISTRIBUTION OF PRIME NUMBERS IN SHORT PROGRESSIONS mod \(p^n\)

It is known that the study of the problem of the distribution of prime numbers in arithmetic progressions requires considerable information about the “critical” zeros of \(L\)-functions. For example, obtaining an asymptotic law for primes in the progression

\[ q \equiv l(\bmod D), \qquad (l,D)=1,\qquad q \ll D^{2+\varepsilon}, \quad D \to \infty, \]

requires the validity of the extended Riemann hypothesis.* Therefore the investigation of the distribution of primes in progressions of general type encounters difficulties. However, there is every reason to believe that for partial sets of values of \(D\) the problem will prove easier. In this note we show that for \(D=p^n\) (\(p\) a fixed prime, \(n=1,2,\ldots\)) one can obtain estimates close to hypothetical ones. The following is proved.

Theorem. Let \(D=p^n\), \(x \gg D^{8/3+\varepsilon}\). Then

\[ \pi(x,D,l)=h^{-1}\operatorname{li}x+O(xh^{-1}\eta), \tag{1} \]

where \(h=\varphi(D)\), \(\eta=O((\lg D)^{-A})\), \(A\) is any positive number.

Moduli of the indicated type have already been considered in the literature (see \((^1)\)). For them it naturally turns out to be useful to employ the theory of functions of a \(p\)-adic variable.

The proof of (1) is based on several lemmas. Let

\[ \psi_0(x)=\psi(x,D,l)=h^{-1}\sum_{\chi}\overline{\chi}(l)\sum_{n\le x}\Lambda(n)\chi(n), \]

\[ \psi_m(x)=\int_0^x \psi_{m-1}(x)\,dx \qquad (m=0,1,2,\ldots), \]

\[ \psi_m(x)=x^{m+1}h^{-1}\left(\frac{1}{(m+1)!}+\delta_m(x)\right). \]

The classical theory shows that the derivation of (1) reduces to obtaining analogous estimates for \(\delta_0(x)\); moreover, one can show that

\[ \delta_0(x)\ll \delta_m^{\frac{1}{2m+1}}, \qquad x>x_0(D), \]

where \(\delta_m=\sup |\delta_m(x)|\) for \(x_0 \le x<\infty\), if there exists such an \(x_1\) that for \(x>x_1\) the estimate \(\delta_m\ll(\lg D)^{-A}\) is valid. We shall show that \(x_1=D^{8/3+\varepsilon}\).

Lemma 1.

\[ \delta_m(x)=-\sum_{\chi}\overline{\chi}(l)\sum_{\rho(x)} \frac{x^{\rho-1}}{\rho(\rho+1)\cdots(\rho+m)} +O(x^{-1}h\lg^2 x), \]

where \(\rho(x)\) is the aggregate of critical zeros of \(L(s,\chi)\), \(m\ge 1\).

This lemma for \(m=1\) is an analogue of the well-known identity for \(\psi_1(x)\) in the case \(D=1\) (see, for example, \((^2)\), p. 43). Termwise \(m\)-fold integration and shifting the contour to the line \(\sigma=-\frac12\) prove the lemma.

\[ \text{* Or a very strong density hypothesis.} \]

Lemma 2. If, for some set \(\{D\}\), the following facts hold:

1) \(N(\sigma,T)\ll T^A D^{B(1-\sigma)}\lg^c D\), where \(N(\sigma,T)\) is the total number of zeros of all \(L(s,\chi)\bmod D\) in the rectangle \(\sigma\leq \sigma'\leq 1;\ |t|\leq T\);

2) \(L(s,\chi)\) in the region
\[ \sigma\geq \sigma_0=1-c_1(\lg D)^{-\alpha},\quad \alpha<1,\quad |t|\leq \tau=(\lg D)^2, \]
then, for \(x\geq D^{B+\varepsilon}\), \(\varepsilon>0\), the estimate
\[ \delta_m(x)\ll \tau^{-1} \]
holds.

For the proof we choose \(m\) so large that the series \(\sum_{\nu=1}^{\infty}\nu^{A-m}\) converges rapidly. Then we select in the identity for \(\delta_m(x)\) all those \(\rho\) for which \(|\operatorname{Im}\rho|\leq (\lg D)^L=\tau\). The estimate of the sum over such \(\rho\) gives:
\[ \sum_{\chi}\bar\chi(l)\sum_{\rho}'\frac{x^{\rho-1}}{\rho(\rho+1)\ldots(\rho+m)} \ll \int_0^{\sigma_0} x^{\sigma-1}N(\sigma,\tau)\,d\sigma+ \]
\[ +\,O\bigl(x,\tau^{-1}D\lg \tau D\bigr)\ll \exp\{-c_1(\lg D)^{1-\alpha}\},\quad c_1>0. \]

We distribute the remaining part of the sum into strips \(\nu\leq |t|<\nu+1\) and again apply within these strips the estimate for \(N(\sigma,\tau)\). The choice of the number \(m\) is such that the total contribution to \(\delta_m(x)\) coming from all strips is of order \(O(\tau^{-1})\), which proves the lemma.

Lemma 3. If
\[ |L(1/2+it,\chi)|\leq M(D)(|t|+2)^{c_0}, \]
then
\[ N(\sigma,T)\ll T^{1+2c_0}(M^2D)^{2(1-\sigma)}\lg^7 D \]
in the rectangle \(\sigma\leq \sigma'\leq 1,\ |t|\leq T\), for any \(\sigma\geq 0,\ T\geq 2\).

For the proof of the lemma see \((^3)\), p. 422.

Lemma 4. If \(D=p^n\), then \(L(s,\chi)\ne 0\) in the region
\[ \sigma\geq 1-\lg^{0.9}D,\quad |t|\leq \lg^L D \]
(\(L>0\) may be arbitrary).

The proof of this lemma is based on estimates for sums of values of the characters \(\chi(\nu)\bmod D\), obtained in the work \((^4)\).

Lemma 5. If \(D=p^n\), then
\[ L(1/2+it,\chi)\ll D^{1/6}\lg^2D\,(|t|+1). \]

This lemma follows directly from the identity
\[ L(1/2+it,\chi)=(1/2+it)\int_0^\infty S(x)x^{-3/2-it}\,dx, \]
where
\[ S(x)=\sum_{\nu\leq x}\chi(\nu), \]
provided one shows that
\[ S(x)\ll D^{1/6}\sqrt{x\lg D} \]
in the interval \((D^{1/3},D^{2/3})\). The estimate in this interval is essentially reduced to the investigation of the sum
\[ S=\sum_{\nu=N}^{N'}\chi(\nu), \]
where \(p^s\ll N\ll N'<2N\ll D^{2/3}\), and \(s\) is the least natural number \(\geq (n+2)/3\). Putting
\[ \nu=l+p^s u,\quad (l,p)=1,\quad ll^*\equiv 1\pmod {p^n}, \]
we obtain
\[ S\ll p^{s/2}S_1^{1/2}, \]

where

\[ S_1=\sum_{(l,p)=1}^{p^s}\left|\sum_{N_1\leqslant u\leqslant N_2}\chi(1+l^*up^s)\right|^2 . \]

But, by a theorem of A. G. Postnikov ((1), p. 217) and by virtue of the choice of the number \(s\):

\[ \chi(1+l^*up^s)=\exp 2\pi i(\alpha u^2+\beta u),\qquad \alpha=\Lambda a_2' l^{*2}p^{-r}, \]

\[ \beta=\Lambda l^*p^{s-n},\qquad r=n-2s,\qquad a_2=a_2'p,\qquad p\nmid a_2', \]

where \(\Lambda\) and \(a_2\) have the same meaning as in \((1)\).

Thus the estimation of \(S_1\) is reduced to the investigation of the sum

\[ \left|\sum_{N_1}^{N_2}\exp 2\pi i(\alpha u^2+\beta u)\right|^2, \]

whose estimation is carried out by well-known methods. After simple computations we have:

\[ S_1\ll \sum_{|u-u'|\leqslant Np^{-s}} \sum_{l=0}^{p^s-1} \min\left(\frac{N}{p^s},\frac{1}{\{\alpha(u-u')\}}\right) \ll \left(\frac{N^2}{p^{2s}}+N\right)\lg D . \]

Here it must be taken into account that if \(l\) runs through a reduced system of residues \(\bmod\, p^s\), then \(l^{*2}\) runs through the same system, but with possible repetitions of multiplicity \(\ll 2p^\delta\). Since \(N^2p^{-2s}\ll N\) for \(N\leqslant D^{2/3}\), we obtain the required estimate of \(S(x)\).

Comparing all the lemmas listed above, we obtain the proof of the main theorem.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
4 XI 1963

REFERENCES

1. A. G. Postnikov, J. Indian Math. Soc., 20, 1—3, 217 (1956).
2. A. E. Ingham, The Distribution of Prime Numbers, 1936.
3. M. B. Barban, Matem. sborn., 61 (103), No. 4, 418 (1963).
4. S. M. Rozin, Izv. AN SSSR, ser. matem., 23, 503 (1959).