EULER–KRONECKER CONSTANTS FOR CYCLOTOMIC FIELDS

Abstract

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

1 Introduction

For a number field K, the Euler–Kronecker constant $\gamma _K$ is given by

$$ \begin{align*} \gamma_K:=\lim_{s \to 1^+}\bigg(\frac{\zeta_K'(s)}{\zeta_K(s)} + \frac{1}{s-1}\bigg), \end{align*} $$

where $\zeta _K(s)$ is the Dedekind zeta-function for $K.$ The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ case, where $\zeta _{{\mathbb Q}}(s)=\zeta (s)$ is the Riemann zeta-function. We consider the constants $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q)$ , where $q\in {\mathbb Z}^{+}$ and $\zeta _q$ is a primitive qth root of unity.

The recent interest in the distribution of the $\gamma _q$ is inspired by work of Ihara [4, 5]. He proposed, for every $\varepsilon> 0$ , that there is a $Q (\varepsilon )$ for which

$$ \begin{align*} (c_1-\varepsilon)\log q\le \gamma_q\le (c_2+\varepsilon)\log q \end{align*} $$

for every integer $q \ge Q (\epsilon )$ , where $0 < c_1 \le c_2 < 2$ are absolute constants. This conjecture was disproved by Ford et al. in [2] assuming a strong form of the Hardy–Littlewood k-tuple conjecture. However, assuming the Elliott–Halberstam conjecture (see [1]), these same authors also proved that the conjecture holds for almost all primes $q,$ with $c_1 = c_2 = 1.$ We recall the Elliott–Halberstam Conjecture as formulated in terms of the Von Mangoldt function $\Lambda (n),$ the Chebyshev function $\psi (x)$ and Euler’s totient function $\varphi (n).$

Elliott–Halberstam Conjecture (EH).

If we let

$$ \begin{align*}E(x;m,a):=\sum_{\substack{p\equiv a \pmod m\\ p\leq x\ {\mathrm{prime}}}} \Lambda(p) -\frac{\psi(x)}{\varphi(m)},\end{align*} $$

then for every $\varepsilon>0$ and $A>0$ , we have

$$ \begin{align*}\sum_{m\le x^{1-\varepsilon}}\max_{(a,m) = 1}\lvert{E(x;m,a)}\rvert\ll_{A,\varepsilon} \frac{x}{(\log x)^A}.\end{align*} $$

Assuming EH, Ford et al. proved (see [2, Theorem 6(i)]), for every $\varepsilon> 0$ , that

$$ \begin{align*}1-\varepsilon < \frac{\gamma_q}{\log q} <1+\varepsilon \end{align*} $$

for almost all primes q (that is, the number of exceptional $q \le x$ is $o(\pi (x))$ as $x\to \infty $ ). Here we extend and refine this result to all integers $q.$

Theorem 1.1. Under EH, for $Q\rightarrow +\infty $ , we have

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} \lvert{\gamma_q - \log q}\rvert = o(\log Q), \end{align*} $$

where the sum is over integers q.

Remark 1.2. Theorem 1.1 shows that EH implies that the distribution of $\gamma _q /\!\log q$ in $[Q, 2Q]$ converges to the one point distribution supported on $1$ .

To prove Theorem 1.1, we use the work of Fouvry [3] that allowed him to unconditionally prove that

$$ \begin{align*}\frac{1}{Q}\sum_{Q<q\le 2Q}\gamma_q=\log Q+O(\log\log Q).\end{align*} $$

Our conditional result is a point-wise refinement of Fouvry’s asymptotic formula under EH.

2 Proof of Theorem 1.1

For brevity, we shall assume that the reader is familiar with Fouvry’s paper [3]. The key formula is (see (3) of [3]) the following expression for $\gamma _q$ in terms of logarithmic derivatives of Dirichlet L-functions:

(2.1) $$ \begin{align} \gamma_q = \gamma + \sum_{1<q^*|q\ }\sum_{\chi^*\!\bmod q^*}\frac{L'(1,\chi^*)}{L(1,\chi^*)}. \end{align} $$

Here the inner sum runs over the primitive Dirichlet characters $\chi ^*$ modulo $q^*$ .

We follow the strategy and notation in [3], which makes use of the modified Chebyshev function

$$ \begin{align*} \psi(x;q,a):=\sum_{\substack{n\leq x\\ n\equiv a \pmod q}} \Lambda(n), \end{align*} $$

and the integral

$$ \begin{align*} \Phi_{\chi^*}(x):=\frac{1}{x-1}\int_{1}^x \bigg(\sum_{n\leq t}\frac{\Lambda(n)}{n}\chi^*(n)\bigg) \,dt. \end{align*} $$

However, we replace the sums $\Gamma _i(Q)$ and $\Gamma _{1,j}(Q)$ defined in [3] with the pointwise terms $\gamma _i(q)$ and $\gamma _{1,j}(q)$ . Following the approach in [3], which is based on (2.1), we have

$$ \begin{align*}\gamma_q = \gamma + A(q) + B(q) - \gamma_2(q) - \gamma_3(q) - (\gamma_{1,1}(q) + \gamma_{1,2}(q) + \gamma_{1,3}(q)),\end{align*} $$

where

$$ \begin{align*}A(q)&=\sum_{q^*|q} \,{\sum_{\chi^*\mod q^*}} \frac{L'}{L}(1,\chi^*)+\Phi_{\chi^*}(x), \\B(q)&=\sum_{\substack{\chi \bmod q\\ \chi\neq \chi_0}} \Phi_{\chi}(x)-\sum_{q^*|q}\,{\sum_{\chi^*\mod q^*}} \Phi_{\chi^*}(x), \\\gamma_2(q)&=\frac{1}{x-1}\int_1^x \frac{\varphi(q)\psi(t;q,1)-\psi(t)}{t} \,dt, \\\gamma_3(q)&=\frac{1}{x-1}\int_1^x \sum_{\substack{n\leq t\\(n,q) \neq 1}} \frac{\Lambda(n)}{n}\,dt, \\\gamma_{1,1}(q)&=\frac{1}{x-1}\int_1^x\int_1^{\min(q,t)}\bigg(\frac{\varphi(q)\psi(u;q,1)-\psi(u)}{u^2}\,du\bigg)\,dt, \\\gamma_{1,2}(q)&=\frac{1}{x-1}\int_1^x\int_{\min(q,t)}^{\min(x_1,t)}\bigg(\frac{\varphi(q)\psi(u;q,1)-\psi(u)}{u^2}\,du\bigg)\,dt, \\\gamma_{1,3}(q)&=\frac{1}{x-1}\int_1^x\int_{\min(x_1,t)}^t \bigg(\frac{\varphi(q)\psi(u;q,1)-\psi(u)}{u^2}\,du\bigg)\,dt. \end{align*} $$

To complete the proof, for $\varepsilon>0$ , we let $x := q^{100}$ and $x_1 := q^{1 + \varepsilon }$ . Apart from $\gamma _{1,1}(q),$ which gives the $-\log q$ terms in Theorem 1.1, we shall show that these summands are all small.

Estimation of $A(q)$ : By Proposition 1 and Remark (i) of [3],

$$ \begin{align*}\sum_{q=Q}^{2Q} \lvert{A(q)}\rvert =O(Q).\end{align*} $$

Estimation of $B(q)$ : For $B(q)$ , by (26) and Lemma 3 of [3], we simplify

$$ \begin{align*}B(q)&=-\frac{1}{x-1}\int_{1}^x \sum_{q^*|q}\ \sum_{\chi^*\,\mod q^*} \sum_{\substack{n \leq t\\(n,q)>1}} \frac{\Lambda(n)\chi^*(n)}{n} \,dt \\&= -\frac{1}{x-1}\int_{1}^x \sum_{q^*|q}\ \sum_{\chi^*\,\mod q^*}\ \sum_{\substack{p^v\leq t\\ p|q}} \frac{\log p\cdot\chi^*(p^v)}{p^v} \,dt \\&= -\frac{1}{x-1}\int_1^x \sum_{q^*|q}\ \sum_{\substack{p^v\leq t \\ p|q\\ p\nmid q^*}}\ \sum_{\substack{d|(p^v-1,q^*)}}\frac{\log p}{p^v} \cdot \varphi(d) \mu\bigg(\frac{q^*}{d}\bigg) \,dt \\&= -\frac{1}{x-1}\int_1^x \sum_{\substack{p^v\leq t\\ p|q}}\ \sum_{\substack{d|p^v-1}}\frac{\log p}{p^v}\cdot\varphi(d) \sum_{\substack{q^*|q\\d|q^*\\ p\nmid q^*}}\mu\bigg(\frac{q^*}{d}\bigg) \,dt. \end{align*} $$

We note that the innermost sum

$$ \begin{align*}\sum_{\substack{q^*|q\\ d|q^* \\ p\nmid q^*}}\mu\bigg(\frac{q^*}{d}\bigg)\end{align*} $$

is always $0$ or $1$ , so we conclude that $B(q) \leq 0$ for any q. Proposition 2 of [3] gives

$$ \begin{align*}\sum_{q=Q}^{2Q}B(q)=O(Q),\end{align*} $$

and so we have

$$ \begin{align*}\sum_{q=Q}^{2Q}\lvert{B(q)}\rvert =O(Q).\end{align*} $$

Estimation of $\gamma _2(q)$ : By Lemma 8 of [3], uniformly in Q with $u\geq 1,$ we have

$$ \begin{align*}\sum_{q=Q}^{2Q} \psi(u;q,1) \ll u.\end{align*} $$

Therefore,

$$ \begin{align*}\sum_{q=Q}^{2Q} \lvert{\varphi(q)\psi(t;q,1)-\psi(t)}\rvert =O(Qt),\end{align*} $$

and so we conclude that

$$ \begin{align*}\sum_{q=Q}^{2Q} \lvert{\gamma_{2}(q)}\rvert = O(Q).\end{align*} $$

Estimation of $\gamma _3(q)$ : By definition, $\gamma _3$ is positive, so by (36) of [3],

$$ \begin{align*}\sum_{q=Q}^{2Q} \lvert{\gamma_{3}(q)}\rvert =O(Q).\end{align*} $$

Estimation of $\gamma _{1,1}(q)$ : Since $\psi (u;q,1)=0$ for $u<q$ , we have

$$ \begin{align*} \gamma_{1,1}(q)=-\frac{1}{x-1}\int_1^x \bigg(\int_1^{\min(q,t)}\frac{\psi(u)}{u^2} \,du \bigg) \,dt. \end{align*} $$

Dividing both sides of (41) of [3] by Q,

$$ \begin{align*}\gamma_{1,1}(q)=-\log q + O(1).\end{align*} $$

Estimation of $\gamma _{1,2}(q)$ : By the same proof as (42) of [3], we have

$$ \begin{align*} \sum_{q=Q}^{2Q} \lvert{\gamma_{1,2}(q)}\rvert \ll \varepsilon Q\log Q. \end{align*} $$

Summing the above estimates, we conclude unconditionally that

$$ \begin{align*} \frac{1}{Q}\sum_{q=Q}^{2Q} \lvert{\gamma_q-\log q}\rvert =\frac{1}{Q}\sum_{q=Q}^{2Q} \lvert{\gamma_{1,3}(q)}\rvert + O(\varepsilon \log Q). \end{align*} $$

Estimation of $\gamma _{1,3}(q)$ : If we assume Conjecture EH holds, then we have (as in Lemma 7 of [3]) that

$$ \begin{align*} \sum_{\substack{q\le 2Q\\ (q,a)=1}}\varphi(q)\bigg\vert{\psi(x;q,a)-\frac{\psi(x)}{\varphi(q)}}\bigg\vert=O_A\big(Qx(\log x)^{-A+2}\big). \end{align*} $$

Therefore,

$$ \begin{align*} \frac{1}{Q} \sum_{q=Q}^{2Q}\lvert{\gamma_{1,3}(q)}\rvert=O_{\epsilon,A}(\log^{-A} Q). \end{align*} $$

By combining these estimates, we obtain the main result

$$ \begin{align*} \frac{1}{Q}\sum_{q=Q}^{2Q}\lvert{\gamma_q-\log q}\rvert=o(\log Q). \end{align*} $$