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MATHEMATICS
Yu. V. KASHIRSKII
ON THE QUESTION OF THE LOCATION OF THE ZEROS OF DIRICHLET \(L\)-SERIES
(Presented by Academician I. M. Vinogradov on 11 X 1960)
The present work is devoted to estimating sums of characters
\[ \left|\sum_{k=T+1}^{T+N}\chi_D(k)\right| \]
(where \(\chi_D(k)\) is a primitive character modulo \(D\)) and to their application to estimating the growth and the location of the zeros of \(L\)-series in the critical strip. The results of the present investigation develop the works of A. G. Postnikov \((^{1})\) and S. M. Rozin \((^{2})\).
§ 1. Estimate of the sum of characters.
Theorem 1. Let
\[ S=\sum_{k=T+1}^{T+N}\chi_D(k), \]
where \(\chi_D(k)\) is a primitive character modulo
\[ D=p_1^{\alpha_1}\cdots p_l^{\alpha_l} \]
with \(\min(p_1,\ldots,p_l)>2\), and let \(\alpha_\nu=\min(\alpha_1,\ldots,\alpha_l)\) be such that
\[ (\alpha_\nu/\log^3\alpha_\nu)^{1/4}\ge 13 \quad (\alpha_\nu>10^8,\ \log\alpha_\nu\ge 18.8), \]
and
\[ \prod_{i=1}^{l} p_i^{3\alpha_i(\log^3\alpha_\nu/\alpha_\nu)^{1/4}}<N. \]
Then the estimate
\[ |S|<N\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}(\alpha_\nu\log\alpha_\nu)^{1/4}}. \]
is valid.
Proof. Choose \(r\) from the equality
\[ r=\left[\frac{\alpha_\nu}{(\alpha_\nu\log\alpha_\nu)^{3/4}-(\alpha_\nu\log\alpha_\nu)^{1/4}}\right]. \]
It is clear that
\[ r>\frac{\alpha_\nu}{(\alpha_\nu\log\alpha_\nu)^{3/4}-(\alpha_\nu\log\alpha_\nu)^{1/4}}-1> \left[\frac{\alpha_\nu}{\log^3\alpha_\nu}\right]^{1/4}-1\ge 12. \]
Let
\[ s_i=\left[\frac{\alpha_i+\delta}{r}\right]+1, \]
where \(\delta=-1\), if \(\alpha_i\ne \alpha p_i^f-\nu\), \((\alpha,p_i)=1\), \(f=1,2,\ldots\), \(0\le \nu\le f-1\), and \(\delta=\nu\), if \(\alpha_i=\alpha p_i^f-\nu\). Then
\[ s_i=\frac{\alpha_i-1}{r}+\frac{s_{1i}}{r},\qquad s_{1i}=d+\delta+1\quad (1\le d\le r). \]
Denote by \(N_1\) such a number that
\[ N=\prod_{i=1}^{l}p_i^{\,2s_i-s_{1i}}N_1+R. \]
\[ s_i-s_{1i} =\alpha_i-1-s_i(r-1) =\alpha_i-1-\left[\frac{\alpha_i+\delta}{r}\right](r-1)-(r-1) \ge \frac{\alpha_i}{r}-\nu-r\ge 0, \]
since
\[ r<\frac{2\alpha_\nu}{(\alpha_\nu\log\alpha_\nu)^{1/4}}<2\alpha_\nu^{3/4}; \qquad \nu\le f\le \log_{p_i}\alpha_i<\log_3\alpha_i, \]
\[ R\le \prod_{i=1}^{l}p_i^{\,2s_i-s_{1i}}\le \prod_{i=1}^{l}p_i^{\,2s_i-1}. \]
Then
\[ \left|\sum_{k=T+1}^{T+N}\chi_D(k)\right| \le N_1|S_1|+\prod_{i=1}^{l}p_i^{\,2s_i-1}. \]
Here
\[ |S_1|=\max_Q\left|\sum_{k=Q+T+1}^{Q+T+\prod_{i=1}^{l}p_i^{\,2s_i-s_{1i}}}\chi_D(k)\right|. \]
But
\[ \sum \chi_D(k)=\sum \chi_{p_1^{\alpha_1}}(k)\ldots \chi_{p_l^{\alpha_l}}(k) =\sum \exp\left\{2\pi i\sum_{i=1}^l \frac{m_i\operatorname{ind}_{g_i} k}{\varphi(p_i^{\alpha_i})}\right\}, \]
where \(g_i\) is a primitive root modulo \(p_i^{\alpha_i}\), and \((m_i,p_i)=1\).
Let, for \(Q=Q_1\), the maximum of the sum be attained
\[ S_1=\sum_{k=1}^{\prod_{i=1}^l p_i^{2s_i-s_{1i}}} \exp\left\{2\pi i\sum_{i=1}^l \frac{m_i\operatorname{ind}_{g_i}(k+T+Q_1)}{\varphi(p_i^{\alpha_i})}\right\}. \]
We make the substitution \(k=x_1+\prod_{i=1}^l p_i^{s_i}x_2\).
\[ |S_1|\le \sum_{x_1=1}^{\prod_{i=1}^l p_i^{s_i}} \left| \sum_{x_2=0}^{\prod_{i=1}^l p_i^{s-s_{1i}}} \exp\left\{2\pi i\sum_{i=1}^l \frac{m_i\operatorname{ind}_{g_i}\left(x_1+T+Q_1+\prod_{i=1}^l p_i^{s_i}x_2\right)} {\varphi(p_i^{\alpha_i})}\right\} \right|. \]
But it is known \((^1)\) that if \((a,p)=1\), and \(aa'\equiv 1\pmod{p^\nu}\), then
\[ \operatorname{ind}_g(a+p^\mu u)\equiv \operatorname{ind}_g a+\lambda(p-1)f(a'u)\pmod{\varphi(p^n)}, \]
where
\[ f(u)\equiv u-\frac{p^\mu u^2}{2}+\ldots+ \frac{(-1)^s p^{\mu s}u^{s+1}}{s+1}\pmod{p^{n-1}}, \qquad s=\left[\frac{n+\delta}{\mu}\right] \]
(\(\delta\) was defined above).
Then
\[ |S_1|\le \sum_{x_1=1}^{\prod p_i^{s_i}} \left| \sum_{x_2=0}^{\prod p_i^{s_i-s_{1i}}} \exp\left\{2\pi i\left( \sum_{i=1}^l \frac{m_i\operatorname{ind}_{g_i}(x_1+Q+T)}{\varphi(p_i^{\alpha_i})} +\right.\right.\right. \]
\[ \left.\left.\left. +\sum_{i=1}^l \frac{m_i\lambda_i\left(c_i x_2-\ldots+ \frac{(-1)^{r-1}}{r}x_1^{\,r-1}(c_i x_2)^r p_i^{s_i(r-1)}\right)} {p_i^{\alpha_i-1}} \right)\right\}\right|, \]
where \(c_i=\prod_{j=1}^l p_j^{s_j}\), \(i\ne j\),
\[ |S_1|\le \prod_{i=1}^l p_i^{s_i}|S_2|, \]
\[ |S_2|= \left| \sum_{x_2=0}^{\prod_{i=1}^l p_i^{s_i-s_{1i}}} \exp\left\{2\pi i\sum_{i=1}^l \frac{ a_{i1}x_2-p_i^{s_{1i}}a_{i2}x_2^2\frac{1}{2}+\ldots+ \frac{(-1)^{r-1}}{r}p_i^{s_i(r-1)}a_{ir}x_2^r } {p_i^{s_i(r-1)+s_i-s_{1i}}} \right\} \right|, \]
where \((a_{ij},p_i)=1\), \(j=1,\ldots,l\); \(i=1,\ldots,l\).
Consider the coefficient of \(x_2^r\). It is equal to
\[ \frac{(-1)^{r-1}}{r}\sum_{i=1}^l \frac{a_{ir}}{p_i^{s_i-s_{1i}}} = \frac{(-1)^{r-1}A_r}{r\prod_{i=1}^l p_i^{s_i-s_{1i}}}, \qquad (A_r,p_i)=1,\quad i=1,\ldots,l; \]
\[ r\prod_{i=1}^l p_i^{s_i-s_{1i}} < \left(\prod_{i=1}^l p_i^{s_i-s_{1i}}\right)^{r-1}. \]
Consequently, \(S_2\) can be estimated by the estimate of I. M. Vinogradov \((^3)\)
\[ |S_2|\le r^{3r\log r}\prod_{i=1}^l p_i^{(s_i-s_{1i})\left(1-\frac{1}{9r^2\log r}\right)} = e^{3r\log^2 r}\prod_{i=1}^l p_i^{(s_i-s_{1i})\left(1-\frac{1}{9r^2\log r}\right)}. \]
Then
\[ |S| \ll \prod_{i=1}^{l} p_i^{2s_i-1} +Ne^{3r\log^2 r}\prod_{i=1}^{l}p_i^{-(s_i-s_{1i})}\frac{1}{q^{r^2\log r}}, \]
\[ \prod_{i=1}^{l}p_i^{2s_i-1} <\frac13\prod_{i=1}^{l}p_i^{2s_i} <\frac13\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}\sqrt[4]{\alpha_\nu\log\alpha_\nu}}\,N, \]
since
\[ 2s_i\leq 2\left(\frac{\alpha_i+\delta}{r}+1\right) \leq 2\left\{ \frac{\alpha_i+\delta} {\displaystyle \frac{\alpha_\nu}{(\alpha_\nu\log\alpha_\nu)^{3/4}-(\alpha_\nu\log\alpha_\nu)^{1/4}-1}} +1 \right\} \]
\[ \ll 3\left\{ \frac{\alpha_i}{\alpha_\nu} \left((\alpha_\nu\log\alpha_\nu)^{3/4} -(\alpha_\nu\log\alpha_\nu)^{1/4}+1\right) \right\} \leq \frac{\alpha_i}{\alpha_\nu} \left(3(\alpha_\nu\log\alpha_\nu)^{3/4} -(\alpha_\nu\log\alpha_\nu)^{1/4}\right). \]
Now let us estimate the second term. It is easy to see that
\[ \prod_{i=1}^{l}p_i^{-4r\log^2 r\,\frac{\alpha_i}{\alpha_\nu}} < \prod_{i=1}^{l}p_i^{-\sqrt[4]{\alpha_\nu\log\alpha_\nu}\,\frac{\alpha_i}{\alpha_\nu}-1} \ll \frac13\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}\sqrt[4]{\alpha_\nu\log\alpha_\nu}} . \]
Estimating (1) by means of this inequality, we obtain
\[ |S|\leq \frac13\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}\sqrt[4]{\alpha_\nu\log\alpha_\nu}}\,N + \frac13\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}\sqrt[4]{\alpha_\nu\log\alpha_\nu}} < N\prod_{i=1}^{l}p_i^{-\frac{\alpha_i}{\alpha_\nu}\sqrt[4]{\alpha_\nu\log\alpha_\nu}}, \]
which was required to be proved.
§ 2. Growth and zeros of \(L\)-functions.
Theorem 2. Let \(Q>3\) be a constant;
\[ \log D\leq \left(\frac{\alpha_\nu}{\log^3\alpha_\nu}\right)^{\frac{Q+1}{4}}; \quad D=p_1^{\alpha_1}\cdots p_l^{\alpha_l}; \quad s=\sigma+it;\quad |s|<C_1; \quad \sigma\geq 1-\frac{1}{(\log D)^{\frac{Q}{Q+1}}}, \]
and let \(\chi_D(k)\) be a primitive character modulo \(D\). Then
\[ |L(s,\chi)|<C(\log D)^{\frac{Q}{Q+1}}. \]
Lemma. If
\[ \left|\sum_{x=1}^{k}\chi_D(x)\right| \ll \begin{cases} k, & \text{for } k\leq D^\omega,\\[4pt] k\displaystyle\prod_{i=1}^{l}p_i^{\beta_i}, & \text{for } k>D^\omega, \end{cases} \qquad \sigma\geq 1-\gamma,\quad |s|<C_1, \]
and the conditions
\[ \begin{aligned} &1)\quad \omega\gamma\leq C_0/\log D;\\ &2)\quad \alpha_i\gamma\leq \beta_i,\qquad i=1,2,\ldots,l, \end{aligned} \]
are satisfied, then
\[ |L(s,\chi)|<C_2/\gamma. \]
Proof of the lemma. It is easy to see that
\[ \sum_{k=1}^{N}a_kb_k = \sum_{k=1}^{N-1}S_k(b_k-b_{k+1}), \qquad \text{where } S_k=\sum_{\nu=1}^{k}a_\nu,\quad S_0=S_N=0. \]
Choose \(a_k=\chi_D(k)\), \(b_k=1/k^s\), \(N=\lambda D\) (\(\lambda\) an integer). Then
\[ \left| \sum_{k=1}^{\lambda D}\frac{\chi_D(k)}{k^s} \right| = \left| \sum_{k=1}^{\lambda D-1}S_k \left(\frac1{k^s}-\frac1{(k+1)^s}\right) \right| < c_1\sum_{k=1}^{\lambda D}\frac{|S_k|}{k^{2-\gamma}}; \]
\[ \sum_{k=1}^{\lambda D}\frac{|S_k|}{k^{2-\gamma}} = \sum_{k=1}^{D}\frac{|S_k|}{k^{2-\gamma}} + \sum_{k_1=1}^{\lambda-1}\sum_{k_2=1}^{D} \frac{|S_{k_2}|}{(k_1D+k_2)^{2-\gamma}} < \]
\[ < \sum_{k=1}^{D} \frac{|S_k|}{k^{2-\gamma}}+ \sum_{k_2=1}^{D}\frac{|S_{k_2}|}{D^{2-\gamma}} \left(\sum_{k_1=1}^{\lambda-1}\frac{1}{k_1^{2-\gamma}}\right) < c_3 \sum_{k=1}^{D}\frac{|S_k|}{k^{2-\gamma}} . \]
Hence
\[ \left|\sum_{k=1}^{\lambda D}\frac{\chi_D(k)}{k^s}\right| < c_4 \sum_{k=1}^{D}\frac{|S_k|}{k^{2-\gamma}} . \]
Let \(\lambda\) be taken so large that
\[ \left|\sum_{k=\lambda D+1}^{\infty}\frac{\chi_D(k)}{k^s}\right|<1 . \]
In this case
\[ |L(s,\chi)|<c_5 \sum_{k=1}^{D}\frac{|S_k|}{k^{2-\gamma}} . \]
But
\[ \sum_{k=1}^{D}\frac{|S_k|}{k^{2-\gamma}} \leq \sum_{k=1}^{D^\omega}\frac{1}{k^{1-\gamma}} +\prod_{i=1}^{l}p_i^{-\beta_i}\sum_{k>D^\omega}^{D}\frac{1}{k^{1-\gamma}} \leq c_6\frac{D^{\omega\gamma}+D^\gamma\prod_{i=1}^{l}p_i^{-\beta_i}}{\gamma}. \]
Consequently,
\[ |L(s,\chi)|<c_7\frac{D^{\omega\gamma}+D^\gamma\prod_{i=1}^{l}p_i^{-\beta_i}}{\gamma}, \]
and, taking into account the conditions of the lemma, we obtain
\[ |L(s,\chi)|<c_2/\gamma . \]
Proof of the theorem. For the proof it is enough to show that, if
\[ \gamma=(\log D)^{-\frac{Q}{Q+1}}, \qquad \log D \leq \left(\alpha_\nu/\log^3\alpha_\nu\right)^{\frac{Q+1}{4}}, \]
then the conditions of the lemma are satisfied, i.e. \(\omega\gamma\leq c_0/\log D\) and \(\alpha_i\gamma\leq \beta_i\), \(i=1,\ldots,l\). But, applying Theorem 1, we have
\(\omega=3(\log^3\alpha_\nu/\alpha_\nu)^{1/4}\),
\(\beta_i=\dfrac{\alpha_i}{\alpha_\nu}(\alpha_\nu\log\alpha_\nu)^{1/4}\).
The conditions of the lemma take the form:
a) \(3(\log^3\alpha_\nu/\alpha_\nu)^{1/4}(\log D)^{Q+1}\leq c_0\);
b) \(\alpha_\nu^{3/4}/(\log\alpha_\nu)^{1/4}<(\log D)^{\frac{Q}{Q+1}}\).
Then a) follows from the fact that
\(\log D\leq(\alpha_\nu/\log^3\alpha_\nu)^{1/4}\), while b) is obvious for \(Q>3\). Thus the theorem is proved.
Theorem 3. If \(\chi_D(k)\) is a primitive character modulo \(D\) and
\[ \log D\leq \left(\alpha_\nu/\log^3\alpha_\nu\right)^{\frac{Q+1}{4}}, \qquad Q>3 \text{ is a constant}, \qquad (\alpha_\nu/\log^3\alpha_\nu)^{1/4}\geq 13, \]
then \(L(s,\chi)\) has no zeros in the region
\[ \sigma>1-\frac{c}{\log^{\frac{Q}{Q+1}}D\cdot \log\log D}. \]
The proof is not given here, since it is entirely analogous to the proof of the theorem for \(D=p^n\) (see (1), Theorem 3).
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
3 X 1960
CITED LITERATURE
- A. G. Postnikov, J. Indian Math. Soc., 20, No. 1, 2, 3 (1956).
- S. M. Rozin, Izv. AN SSSR, ser. matem., 23, No. 4, 503 (1959).
- I. M. Vinogradov, Selected Works, 1952.