A. I. VINOGRADOV

ON THE DENSITY AND QUASI-RIEMANN HYPOTHESES

(Presented by Academician I. M. Vinogradov on 29 IV 1964)

To study the properties of zeros of \(L\)-series, Yu. V. Linnik \({}^{(2)}\) applied the principle of constructing sums that take a large value at a zero of an \(L\)-series. Here it will be shown how this principle can be supplemented by the estimates of I. M. Vinogradov \({}^{(1)}\) and a new theorem on the zeros of \(\zeta(s)\) obtained. In addition, if this principle is supplemented by the estimates of Burgess \({}^{(4)}\) or A. G. Postnikov \({}^{(5)}\), then an analogous theorem can be obtained for all \(L\)-series \((\bmod D)\) having zeros near the real axis.

Theorem 1. For a given \(T>1\), all zeros of \(\zeta(s)\) in the critical strip of height \(T\) can be divided into no more than \(\ln T\) classes \(\{C_0,C_1,\ldots,C_r\}\) in such a way that the real part of the zeros of the class \(C_\nu\) satisfies the estimate

\[ \operatorname{Re}\rho_\nu < 1-\eta_\nu, \]

and the total number of zeros of the class \(C_\nu\) having real part \(\geq \sigma\) does not exceed

\[ T^{2(1+\varepsilon_\nu)(1-\sigma)}\ln^5 T, \]

where \(\varepsilon_\nu=\nu/\ln T,\ \eta_\nu=C_0\varepsilon_\nu^3,\ \nu=0,1,\ldots,r\leq \ln T\).

Proof. Let \(\zeta(\rho)=0\) and \(T/2\leq \operatorname{Im}\rho\leq T\); then

\[ \sum_{n\leq T}\frac{1}{n^\rho}=O\left(T^{-\beta}\right),\qquad \beta=\operatorname{Re}\rho. \]

Multiplying the right- and left-hand sides of this equality by the sum \(\sum_{n\leq T}\frac{\mu(n)}{n^\rho}\), we obtain the estimate

\[ \left|\sum_{T\leq n\leq T^2}\frac{a_n}{n^\rho}\right|>\frac12,\qquad a_n=\sum_{\substack{d\mid n,\ n/T\leq d\leq T}}\mu(d),\qquad \operatorname{Re}\rho>\frac12. \tag{1} \]

If we introduce the sum \(S(n)=\sum_{m\leq n} a_m m^{-i\gamma}\), where \(\gamma=\operatorname{Im}\rho\), then by partial summation from (1) we find

\[ \sum_{T\leq n\leq T^2}\frac{|S(n)|}{n^{1+\beta}}+\frac{|S(T)|}{T^\beta}\geq \frac12. \]

It follows from this that, for the given zero \(\rho\) satisfying \(\operatorname{Re}\rho>1/2,\ T/2\leq \operatorname{Im}\rho\leq T\), there exists an \(n_\rho\) from the interval \((T,T^2)\) such that

\[ |S(n_\rho)|>\frac{n_\rho^\beta}{2\ln T}. \tag{2} \]

We assign to the class \(C_\nu\) those zeros for which \(n_\rho\) lies in the interval \((T_\nu,T_{\nu+1})\), where

\[ T_\nu=T^{1+\varepsilon_\nu}. \]

Let \(\rho \in C_\nu\). For it, the corresponding sum from (2) can be written in the form

\[ S_\nu(n_\rho)= \sum_{n_\rho/T\le d\le T}\mu(d)d^{-i\gamma} \sum_{m\le n_\rho/d} m^{-i\gamma}. \tag{3} \]

I. M. Vinogradov’s method gives the estimate

\[ \sum_{m\le x}m^{-i\gamma}\ll x^{1-\eta}, \qquad \rho=2c_0\left(\frac{\ln x}{\ln\gamma}\right)^2 . \tag{4} \]

In our case \(x=n_\rho/d\ge T^{\varepsilon_\nu}\); hence \(\eta\ge 2c_0\varepsilon_\nu^2\). Substituting (4) into (3), we obtain

\[ |S_\nu(n_\rho)|\ll n_\rho^{1-\eta_\nu}, \qquad \eta_\nu\ge c_0\varepsilon_\nu^3. \]

Comparing this inequality with the estimate (2), we find

\[ \operatorname{Re}\rho_\nu\ll 1-\eta_\nu . \]

To study the density of zeros of the class \(C_\nu\), we “smooth” the sums \(S_\nu(n_\rho)\):

\[ S_\nu(n_\rho)=\frac{1}{2\pi i} \int_{\frac{1}{\ln T}-iT_\nu}^{\frac{1}{\ln T}+iT_\nu} \frac{(T_{\nu+1})^s}{s} \left(\sum_{m\le T_{\nu+1}} a_m m^{-i\gamma-s}\right)\,ds+O(1). \]

From this equality we obtain the estimate

\[ |S_\nu(n_\rho)|^2\ll \int_0^{T_\nu}\frac{\ln T}{|s|} \left|\sum_{m\le T_{\nu+1}} a_m m^{-i\gamma-s}\right|^2 dt+1; \qquad s=\frac{1}{\ln T}+it. \tag{5} \]

But the integral on the right-hand side of the resulting inequality is not greater than

\[ \sum_{0\le n\le T_{\nu+1}} \frac{\ln T}{n+\alpha} \int_{\gamma+n}^{\gamma+n+1} \left|\sum_{m\le T_{\nu+1}} a_m m^{-it+\alpha}\right|^2 dt, \qquad \alpha=\frac{1}{\ln T}. \]

Sum the right- and left-hand sides of (5) over all \(\rho_\nu\) satisfying \(\operatorname{Re}\rho_\nu\ge \sigma\):

\[ \sum_{\operatorname{Re}\rho_\nu\ge\sigma}|S_\nu(n_{\rho_\nu})|^2\ll \]

\[ \ll \sum_{0\le n\le T_{\nu+1}} \frac{\ln T}{n+\alpha} \sum_{\gamma_\nu} \int_{\gamma_\nu+n}^{\gamma_\nu+n+1} \left|\sum_{m\le T_{\nu+1}} a_m m^{-it}\right|^2 dt + N_\nu\left(\sigma,T,\frac12 T\right). \]

But in a critical strip of length \(1\) there lie no more than \(O(\ln T)\) zeros; therefore

\[ \sum_{\gamma_\nu} \int_{\gamma_\nu+n}^{\gamma_\nu+n+1} \ll \ln T\cdot \int_0^{2T_{\nu+1}} . \]

Consequently,

\[ \sum_{\operatorname{Re}\rho_\nu\ge\sigma}|S_\nu(n_{\rho_\nu})|^2\ll \ln^3 T \int_0^{2T_{\nu+1}} \left|\sum_{m\le T_{\nu+1}} a_m m^{-it+\alpha}\right|^2 dt + N_\nu\left(\sigma,T,\frac12 T\right); \]

moreover, from (2) there follows the lower estimate

\[ \sum_{\operatorname{Re}\rho_\nu\ge\sigma}|S_\nu(n_{\rho_\nu})|^2> \frac{T_{\nu+1}^{2\sigma}}{4\ln^2 T} N_\nu\left(\sigma,T,\frac12 T\right). \]

Comparing the upper and lower estimates, we obtain

\[ N_\nu(\sigma,T)\ll T^{2(1+\varepsilon_\nu)(1-\sigma)}\ln^5 T; \]

Theorem 1 is proved. It follows from it that the greater the density of zeros in a class, the farther they are from the unit line.

Let us consider what new information about the distribution of zeros of \(L\)-series this method gives. Let \(N(\sigma,D)\) denote the number of zeros of all \(L\)-series \((\bmod\, D)\) whose real part is \(\geq \sigma\), and whose imaginary part is \(\leq (\ln D)^c\). In [3], for almost all \(D\) the estimate

\[ N(\sigma,D)\ll D^{2(1+1/3+\varepsilon)(1-\sigma)}\ln^{c_1}D \]

was obtained.

We shall show how this estimate can be differentiated according to classes of zeros, analogously to how this was done in Theorem 1.

Theorem 2. For a given \(D>1\), the zeros of all \(L\)-series \((\bmod\, D)\) satisfying \(|\operatorname{Im}\rho|<\ln^c D,\ \operatorname{Re}\rho\geq 3/4\) can be divided into no more than \(\ln D\) classes \((C_0,C_1,\ldots,C_r)\) so that the real part of the zeros of the class \(C_\nu\) has the estimate

\[ \operatorname{Re}\rho<1-\eta_\nu, \]

and the total number of zeros of the class \(C_\nu\) having real part \(\geq \sigma>3/4\) does not exceed the quantity

\[ D^{2(1+1/4+\varepsilon_\nu)(1-\sigma)}\ln^{c_1}D, \]

where

\[ \varepsilon_\nu=\frac{\nu}{\ln D},\qquad \eta_\nu=\left(\varepsilon_\nu-\frac{1}{4r}-\varepsilon\right)\frac{1}{r},\qquad \varepsilon>0 \]

is arbitrarily small, and \(r\) is an arbitrarily large quantity.

Proof. If \(L(s,\chi)\) has a zero \(\rho\) satisfying \(\operatorname{Re}\rho>3/4,\ |\operatorname{Im}\rho|<\ln^c D\), then for it, just as in the proof of Theorem 1, we obtain the inequality

\[ |S(\chi,n_\rho)|>\frac{n_\rho^\beta}{\ln^{c_0}D}\qquad (D\leq n_\rho\leq D^2), \tag{6} \]

where

\[ S(\chi,n_\rho)=\sum_{m\leq n_\rho} a_m\chi(m) = \sum_{n_\rho/D\leq d\leq D}\mu(d)\chi(d) \sum_{m\leq n_\rho/d}\chi(m). \]

We divide all \(L\)-series \((\bmod\, D)\) into \(O(\ln D)\) classes. To the class \(C_\nu\) we assign those for which \(n_\rho\) lies in the interval \((D_\nu,D_{\nu+1})\), where \(D_\nu=D^{1+\varepsilon_\nu}\).

If \(\varepsilon_\nu>1/4\), then to the sum \(\sum_{m\leq n_\rho/d}\chi(m)\) we apply Burgess’s estimate [4]; we obtain

\[ |S_\nu(\chi,n_\rho)|\ll n_\rho^{1-\eta_\nu},\qquad \eta_\nu\geq \left(\varepsilon_\nu-\frac{1}{4}-\frac{1}{4r}-\varepsilon\right)\frac{1}{r}. \]

Comparing this inequality with estimate (6), we find

\[ \operatorname{Re}\rho_\nu<1-\eta_\nu. \]

In order to obtain the density of zeros in the class \(C_\nu\), we again “smooth” the sums \(S_\nu(\chi,n_\rho)\), after which we obtain the estimate

\[ |S_\nu(\chi,n_\rho)|^2 \ll \sum_{0\leq n\leq D_{\nu+1}} \frac{\ln D}{n+\alpha} \int_n^{n+1} \left| \sum_{m\leq D_{\nu+1}}\chi(m)a_m m^{-it+\alpha} \right|^2dt, \qquad \alpha=\frac{1}{\ln D}. \]

Summing it over all characters whose \(L\)-series have zeros in this class satisfying \(\operatorname{Re}\rho_\nu>\sigma\), we obtain the estimate:

\[ \sum_{\chi_\nu}|S_\nu(\chi_\nu,n_\rho)|^2\ll D_{\nu+1}^2\ln^c D. \tag{7} \]

\[ \text{* The condition } \operatorname{Re}\rho>3/4 \text{ can in principle be replaced by } \operatorname{Re}\rho>1/2, \text{ at the cost of complicating the proof given above.} \]

But, on the other hand, from inequality (6) it follows that

\[ \sum_{\chi_\nu} \left| S_\nu(\chi_\nu,\eta_\rho) \right|^2 > \frac{D^{2\sigma}_{\nu+1}}{(\ln D)^{2c_0}}\,N_\nu(\sigma,D). \tag{8} \]

Comparing (7) and (8), we obtain

\[ N_\nu(\sigma,D) \ll D^{2(1+\varepsilon_\nu)(1-\sigma)} \ln^{c_1} D . \]

Since Burgess’s estimates are valid only for \(\varepsilon_\nu > 1/4\), by combining all classes with \(\nu \leq \tfrac14 \ln D\) into one class, we obtain Theorem 2.

If one considers only moduli \(D=p^n\), then the estimates of A. G. Postnikov \((^5)\) are applicable to the character sums

\[ \sum_{m \leq x} \chi(m). \]

They are nontrivial beginning with \(x > D^{1/n}\). Accordingly, Theorem 2 takes the following form:

Theorem 3. If \(D=p^n,\ n \geq 4\), then the zeros of all \(L\)-series \((\bmod D)\) satisfying \(|\operatorname{Im}\rho| < \ln D\) can be divided into no more than \(\ln D\) classes \((C_0,C_1,\ldots,C_r)\) such that the real part of the zeros of the class \(C_\nu\) has the estimate

\[ \operatorname{Re}\rho_\nu < 1-\eta_\nu, \]

and the total number of zeros of the class \(C_\nu\), having real part \(\geq \sigma \geq 3/4\), does not exceed the quantity

\[ D^{2(1+1/n+\varepsilon_\nu)(1-\sigma)} \ln^{c_1} D, \]

where \(\varepsilon_\nu=\nu/\ln D,\ \eta_\nu=c_0\varepsilon_\nu^3,\ \nu=0,1,\ldots,r\).

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

Received
20 IV 1964

CITED LITERATURE

\(^{1}\) I. M. Vinogradov, Izv. AN SSSR, Ser. Matem., 22, No. 2, 161 (1958).
\(^{2}\) Yu. V. Linnik, Izv. AN SSSR, Ser. Matem., 14 (1950).
\(^{3}\) M. B. Barban, Matem. sborn., 61 (103), No. 4 (1963).
\(^{4}\) D. A. Burgess, Proc. London Math. Soc., 12, No. 46 (1962); 13, No. 52 (1963).
\(^{5}\) A. G. Postnikov, Izv. AN SSSR, Ser. Matem., 19, No. 1, 11 (1955).