A simple note on the number-theoretic function $\Omega (n)$ and Dirichlet L-functions

Abstract

We give a relation between the orders of potential non-trivial zeros of Dirichlet L-functions and analytic properties of certain Dirichlet series first introduced by Selberg.

1 Introduction

For a positive integer n we denote as usual by $\omega (n)$ the number of distinct prime factors of n and by $\Omega (n)$ the total number of prime factors of n counting multiplicities. These classical prime factor counting functions have many relations in classical and analytic number theory, in particular to the Riemann Hypothesis (RH) as well as the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions $L(s,\chi )$, where $\chi$ is a primitive Dirichlet character modulo q. For example, Wolke in [9] showed that RH is true if and only if in the classical asymptotic expansion of the summatory function $\sum _{n\leqslant x}\omega (n)$, the error term is of order $O_\epsilon (x^{1/2+\epsilon })$ for every $\epsilon >0$. The same is true with $\omega$ replaced by $\Omega$. More generally, under GRH one expects a similar formula for the summatory function of $\omega$ over arithmetic progressions, see Meng [1].

For and $z\in \textbf{C}$ with $|z|<2$ we set

(1)

Elementary estimates show that this Dirichlet series is absolutely convergent. (One could also refer to the more elaborate formula given in [7, Theorem II.6.2].) Moreover, the coefficients are completely multiplicative, hence one has an Euler product

$$\begin{aligned} D_z(s,\chi )=\prod _p\,\bigl (1-z\chi (p) p^{-s}\bigr )^{-1} \end{aligned}$$

(2)

which is easily seen to be absolutely convergent in the given range. Note that the Euler factors do not vanish. We also observe that $D_1(s,\chi )=L(s,\chi )$.

The series (1) (with $\chi =\chi _0$ the principal character) was first introduced by Selberg [6] with the aim of studying asymptotic properties of $\Omega (n)$ and $\omega (n)$, generalizing previous work of Sathe [2,3,4,5]. The latter were also studied in the book by Tenenbaum [7, Chapter II 4, Example 181 and Chapter II 6].

In this short note we will give a relationship between the orders of potential zeros of $L(s,\chi )$ in $\sigma >{1/2}$ and the analytic properties of the series $D_{1/N}(s,\chi )$ where N is a positive integer. As far as we know such a connection has not been given explicitly in the literature before.

The precise statement of our result will be given in Sect. 2 and the proof in Sect. 3. The proof uses only elementary facts from complex analysis and roughly speaking rests on the observation that for a non-principle character $\chi$ one has

$$\begin{aligned} L(s,\chi )=D_{1/N}(s,\chi )^NH(s),\quad \sigma >1, \end{aligned}$$

where H(s) is analytic and non-zero in the half-plane $\sigma > {1/ 2}$. The latter fact indeed seems to have been known (at least to experts) for a long time.

2 Statement of results

For a primitive Dirichlet character $\chi$ modulo q we denote by

(3)

the associated L-series. Recall the Euler product expansion

Note that $\zeta (s)=L(s,\chi _0)$ has meromorphic continuation to $\textbf{C}$ and is holomorphic except for a simple pole at $s=1$. If $\chi$ is not the principal character, then the series in (3) is convergent for $\sigma >0$ and has holomorphic continuation to $\textbf{C}$. By the Euler product expansion $L(s,\chi )\ne 0$ for $\sigma > 1$, while GRH states that $L(s,\chi )\ne 0$ for $\sigma >{1/ 2}$.

Theorem 2.1

Suppose that $\chi$ is non-principal. Fix a positive integer N and $\delta > {1/ 2}$. Then the following assertions are equivalent:

• (i) Every zero of $L(s,\chi )$ in $\sigma >\delta$ has order divisible by N.

• (ii) The series $D_{1/N}(s,\chi )$ defined by (1) has holomorphic continuation to $\sigma >\delta$.

As an immediate consequence we obtain

Corollary 2.2

Suppose that $D_{1/N}(s,\chi )$ has holomorphic continuation to $\sigma >\delta$. Then $D_{1/M}(s,\chi )$ has holomorphic continuation to $\sigma >\delta$ for any positive divisor $M\,{|}\,N$.

We also have

Corollary 2.3

Let $\lbrace N_\nu \rbrace _{\nu \geqslant 1}$ be an infinite set of positive integers and suppose that $D_{1/N_\nu }(s,\chi )$ has holomorphic continuation to $\sigma >{1/ 2}$ for all $\nu$. Then GRH is true for $L(s,\chi )$.

Indeed, a positive integer can have only finitely many positive divisors.

Remark 2.4

Similar statements as above hold for $\zeta (s)$. However, to take care of the pole at $s=1$ one has to replace $\zeta (s)$ by and $D_{1/N}(s,\chi )$ by $(s-1)^{1/N}D_{1/N}(s,\chi )$ in the statement of Theorem 2.1. Here and throughout the paper real powers of complex numbers are defined by means of the principal branch of the complex logarithm.

3 Proof of Theorem 2.1

We need some preparations (compare [8]). First recall the binomial series expansion

where

$$\begin{aligned} {\alpha \atopwithdelims ()\nu }={{\alpha (\alpha -1)\cdots (\alpha -\nu +1)}\over \nu !} \end{aligned}$$

is a generalized binomial coefficient.

We put $z=-\chi (p) p^{-s}$ where p is a prime and $\sigma >0$, and put $\alpha =-{1/ N}$ to obtain

$$\begin{aligned} (1-\chi (p) p^{-s})^{-1/N}=P_{N,p}(s,\chi ), \quad \sigma >0, \end{aligned}$$

where

Observe that for $\nu >0$ one has

hence

$$\begin{aligned} 0<(-1)^\nu {-1/N\atopwithdelims ()\nu }\leqslant 1. \end{aligned}$$

Therefore, $|P_{N,p}(s,\chi )|\leqslant (1-p^{-\sigma })^{-1}$ which implies that the infinite product

is absolutely convergent for $\sigma >1$ and defines a holomorphic function in this half-plane. Furthermore, by construction

(4)

Next we note that the p-Euler factor $P_{N,p}(s,\chi )$ starts with

$$\begin{aligned} 1+\chi (p)\,{1\over N}\,p^{-s} \end{aligned}$$

and this expression is non-zero for $\sigma >0$. Therefore, for $\sigma >0$ we can write

$$\begin{aligned} P_{N,p}(s,\chi )=\biggl (1+\chi (p)\,{1\over N}\,p^{-s}\biggr )\biggl (1+{{Q_{N,p}(s,\chi )}\over {1+\chi (p){1\over N}p^{-s}}}\biggr ) \end{aligned}$$

(5)

where

We claim that the product over all p of the second factors in (5) is unconditionally convergent in compact subsets of the half-plane $\sigma > {1/ 2}$ and hence defines a holomorphic function in this domain.

Indeed, let K be a compact subset of this domain. Since $1+\chi (p){1\over N}p^{-s}$ is bounded from below on K by a positive constant depending only on K, it will be sufficient to show that

$$\begin{aligned} \sum _p |Q_{N,p}(s,\chi )|\ll _K1 \end{aligned}$$

for all $s\in K$. To see this we note that

$$\begin{aligned}|Q_{N,p}(s,\chi )|\leqslant \sum _{\nu \geqslant 2}p^{-\nu \sigma } ={1\over {p^\sigma (p^{\sigma }-1)}} \leqslant {1\over {p^{1/2+\epsilon }(p^{1/2+\epsilon }-1)}}\end{aligned}$$

where in the last inequality we have assumed that $\sigma \geqslant {1/ 2}+\epsilon$ with $\epsilon >0$ depending only on K.

Choose an integer $M\geqslant 2$ such that

$$\begin{aligned} {M\over {M-1}}\leqslant p^{1/2+\epsilon } \end{aligned}$$

for all p. Then

whence

$$\begin{aligned}\sum _p |Q_{N,p}(s,\chi )|\ll \sum _p {1 \over {p^{1+2\epsilon }}} \ll \sum _{n\geqslant 1} {1\over {n^{1+2\epsilon }}} \ll 1.\end{aligned}$$

This proves our claim.

We therefore obtain that

$$\begin{aligned} P_N(s,\chi )=\prod _p\,\biggl (1+\chi (p)\,{1\over N}\,p^{-s}\biggr )\cdot R_N(s,\chi ), \quad \sigma >1, \end{aligned}$$

where $R_N(s,\chi )$ is holomorphic for $\sigma >{1/ 2}$. Furthermore, one has

$$\begin{aligned}\prod _p\,\biggl (1+\chi (p)\,{1\over N}\,p^{-s}\biggr ) =\prod _p {1\over {1-\chi (p){1\over N}p^{-s}}}\cdot \prod _p\,\biggl (1-\chi ^2(p)\,{1\over N^2}\,p^{-2s}\biggr ), \quad \sigma > 1.\end{aligned}$$

The second factor on the right is holomorphic in $\sigma >{1/ 2}$ and the first factor equals $D_{1/N}(s,\chi )$ by (2). Therefore

where $S_N(s,\chi )$ is holomorphic for $\sigma > {1/ 2}$. Since all Euler factors of $S_N(s,\chi )$ for $\sigma > {1/ 2}$ are non-zero (as follows from tracing back the definitions), we finally conclude that $P_N(s,\chi )$ has analytic continuation to any given subdomain of $\sigma >{1/ 2}$ if and only if this holds for $D_{1/N}(s,\chi )$.

Now suppose that assertion (i) of the theorem holds. Note that by assumption $L(s,\chi )$ has analytic continuation to $\textbf{C}$. We have to show that $D_{1/N}(s,\chi )$ has analytic continuation to $\sigma>\delta >{1/ 2}$. For this it is sufficient to show that $D_{1/N}(s,\chi )$ continues analytically to (where ), for any $c>0$,.by the uniqueness of analytic continuation. The function $L(s,\chi )$ has only finitely many zeros on $A_c$. Indeed, $L(s,\chi )$ is zero-free on $\sigma >1$ and $\lbrace s\,{\in }\, \textbf{C}\, {|} 1+\epsilon _0\,{\geqslant }\, \sigma \,{\geqslant }\,\delta , |t|\,{\leqslant }\, c\rbrace$, $\epsilon _0>0$, is compact. Let $s_1,\dots , s_r$, $r\geqslant 0$, be those zeros and $\nu _1,\dots , \nu _r$, $r\geqslant 0$, be their multiplicities. Note that the latter are divisible by N, by hypothesis. Write

$$\begin{aligned} L(s,\chi )=(s-s_1)^{\nu _1}\cdots (s-s_r)^{\nu _r}{\cdot }\, g(s),\quad s\in A_c, \end{aligned}$$

where g(s) is holomorphic and free of zeros on $A_c$. Since $A_c$ is simply connected, we have $g(s)=h(s)^N$ for some function h(s) holomorphic on $A_c$. Hence

$$\begin{aligned} L(s,\chi )=\ell _N(s,\chi )^N \end{aligned}$$

where $\ell _N(s,\chi )$ is holomorphic on $A_c$.

Using (4) it now follows that

$$\begin{aligned} \ell _N(s,\chi )=\kappa _N P_N(s,\chi ), \quad \sigma >1, \end{aligned}$$

(6)

where $\kappa _N$ is an N-th root of unity. Equation (6) now gives the analytic continuation of $P_N(s,\chi )$ to $A_c$. From what was proved before, it follows that $D_{1/N}(s,\chi )$ has analytic continuation to $A_c$. This proves our assertion.

The converse statement, namely that (ii) implies (i) is trivial. Indeed, if $D_{1/N}(s,\chi )$ has holomorphic continuation to $\sigma >\delta$ then the same is true for $P_N(s,\chi )$ as was said before, hence (4) holds for $\sigma >\delta$. In particular the orders of all zeros of $L(s,\chi )$, $\sigma >\delta$, are divisible by N.