2.1 The work of Ramanujan

In the process of obtaining the maximal order of the divisor function, Ramanujan [Reference Ramanujan10] extended the result of Mertens beyond the boundary to $s> 0$ . His formula asserts that assuming the Riemann hypothesis (RH) for $\zeta (s)$ ,

$$ \begin{align*} \prod_{p \leqslant x} (1-p^{-s})^{-1} = -\zeta(s) \exp \bigg({\mathrm{Li}}(\vartheta(x)^{1-s})+\frac{2s x^{{1/2}-s}}{(2s-1) \log x} + \frac{S_{s}(x)}{\log x}+O \bigg(\frac{x^{{1/2}-s}}{(\log x)^{2}} \bigg) \bigg), \end{align*} $$

where s is a real variable in $1/2 < s < 1$ , $\vartheta (x) = \sum _{p \leqslant x} \log p$ is the Chebyshev function and

$$ \begin{align*} S_{s}(x) = -s \sum_{\zeta(\rho) = 0} \frac{x^{\kern2pt\rho-s}}{\rho(\rho-s)}. \end{align*} $$

His method first appeared in [Reference Ramanujan8] and was republished with annotations in [Reference Ramanujan10]. There are two manuscripts by him (handwritten by Watson) on sums involving primes (see [Reference Ramanujan9, pages 228–232]).

2.2 Prime number theorem in arithmetic progressions

Let $\chi $ be a primitive Dirichlet character modulo q. Then a Dirichlet L-function $L(s, \chi )$ satisfies the functional equation

$$ \begin{align*} \Lambda(s, \chi) := \bigg(\frac{q}{\pi} \bigg)^{s/2} \Gamma \bigg(\frac{s+\nu}{2} \bigg) L(s, \chi) = \epsilon(\kern2pt\chi) \Lambda(1-s, \overline{\chi}), \end{align*} $$

where $\nu = (1-\chi (-1))/2$ and $\epsilon (\chi ) = i^{-\nu } q^{-1/2} \tau (\kern2pt\chi )$ with the Gauss sum $\tau (\kern2pt\chi )$ . The completed L-function $\Lambda (s, \chi )$ has a meromorphic continuation to ${\mathbb {C}}$ and is entire if $\chi \ne \chi _0$ . Let $\pi (x; q, a)$ be the number of primes p up to x belonging to the arithmetic progression $a \pmod{q}$ . In particular, $\pi (x) := \pi (x; 1, 1)$ and this abbreviation also applies to other functions below. The prime number theorem in arithmetic progressions shows that in any residue class $a \pmod{q}$ , the primes are equidistributed amongst the plausible arithmetic progressions modulo q:

$$ \begin{align*} \pi(x; q, a) \sim \frac{\pi(x)}{\varphi(q)} \end{align*} $$

as $x \to \infty $ whenever $(a, q) = 1$ and $q \geqslant 1$ . An important question here is the uniformity in q that is relevant to the distribution of zeros of $L(s, \chi )$ . The Siegel–Walfisz theorem [Reference Siegel11, Reference Walfisz12] asserts that

(2.1) $$ \begin{align} \pi(x; q, a) = \frac{1}{\varphi(q)} {\mathrm{Li}}(x)+O(x \exp(-c \sqrt{\log x})) \end{align} $$

for any $q \leqslant (\log x)^{A}$ , where c and the implicit constant depend on A alone (not effectively computable if $A \geqslant 2$ ). It is beneficial to weaken the restriction on q for applications. The assumption of GRH yields (2.1) in a wider region $q \leqslant \sqrt {x}/(\log x)^{2+\epsilon }$ . Since we assume GRH throughout this article, we use this restriction on q in our discussion. It is conjectured that

$$ \begin{align*} \pi(x; q, a) = \frac{1}{\varphi(q)} {\mathrm{Li}}(x)+O(x^{1/2+\epsilon}) \end{align*} $$

uniformly for $q \leqslant x^{1/2-\epsilon }$ . For notational convenience, we introduce

$$ \begin{align*} E(x; q, a) = \frac{x}{\varphi(q)}-\vartheta(x; q, a). \end{align*} $$

It is well known that the bound $E(x; q, a) \ll \sqrt {x}(\log x)^{2}$ is equivalent to GRH.

2.3 Summation formulæ

Following Ramanujan, we start with considering the partial summation that if $\Phi ^{\prime }(x)$ is a continuous function, then

$$ \begin{align*} \Phi(p_{1}) \log p_{1}&+\Phi(p_{2}) \log p_{2}+\cdots+\Phi(p_{n}) \log p_{n} \\ &\quad = \Phi(x) \vartheta(x; q, a)-\int_{p_{1}}^{x} \Phi^{\prime}(t) \vartheta(t; q, a) \,dt, \end{align*} $$

where $p_{1} < p_{2} < \cdots < p_{n}$ is an ascending sequence of consecutive primes of the form $a+mq$ and $p_{1}$ (respectively $p_{n}$ ) stands for the smallest (respectively largest) prime below x of such a form. Integrating by parts gives

$$ \begin{align*} \Phi(x) \vartheta(x; q, a)-\int_{p_{1}}^{x} \Phi^{\prime}(t) \vartheta(t; q, a) \,dt &= \mathrm{const} +\frac{1}{\varphi(q)} \int_{p_{1}}^{x} \Phi(t) \,dt-\Phi(x) E(x; q, a) \\ &\quad +\int_{p_{1}}^{x} \Phi^{\prime}(t) E(t; q, a) \,dt, \end{align*} $$

where ‘const’ depends on a, q and $\Phi $ . In what follows, we assume GRH for Dirichlet L-functions, which allows us to work in the region $q \leqslant \sqrt {x}/(\log x)^{2+\epsilon }$ . Taylor’s theorem then leads to

$$ \begin{align*} \int_{p_{1}}^{\varphi(q) \vartheta(x; q, a)} \Phi(t) \,dt &= \int_{p_{1}}^{x} \Phi(t) \,dt - \Phi(x) \varphi(q) E(x; q, a) \\ &\quad + \tfrac{1}{2}(\varphi(q) E(x; q, a))^{2} \Phi^{\prime}(x+O(\!\sqrt{x}(\log x)^{2})). \end{align*} $$

Gathering these identities together,

(2.2) $$ \begin{align} \Phi(p_{1}) \log p_{1} &+\Phi(p_{2}) \log p_{2}+\cdots+\Phi(p_{n}) \log p_{n} = \mathrm{const}+\frac{1}{\varphi(q)} \int_{p_{1}}^{\varphi(q) \vartheta(x; q, a)} \Phi(t) \,dt \notag\\ &+ \int_{p_{1}}^{x} \Phi^{\prime}(t) E(t; q, a) \,dt - \frac{1}{2} \varphi(q) E(x; q, a)^{2} \Phi^{\prime}(x+O(\!\sqrt{x}(\log x)^{2})). \end{align} $$

3.1 Partial Euler products

Let $\chi $ be a primitive character modulo q. We exploit the summation formula (2.2) $\varphi (q)$ times. For our purpose, we assume that $\Phi (x) = \chi (a)/(x^{s}-\chi (a))$ for each a, where $s = \sigma +it$ and $(a, q) = 1$ . It therefore follows that

$$ \begin{align*} &\frac{\chi(2) \log 2}{2^{s}-\chi(2)} +\frac{\chi(3) \log 3}{3^{s}-\chi(3)} + \frac{\chi(5) \log 5}{5^{s}-\chi(5)}+\cdots+\frac{\chi(p) \log p}{p^{s}-\chi(p)} \notag \\ & \quad = \mathrm{const}+\frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \int_{p_{1}}^{\varphi(q) \vartheta(x; q, a)} \frac{\chi(a) \,dt}{t^{s}-\chi(a)} \notag\\ &\qquad - s \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) \int_{p_{1}}^{x} \frac{E(t; q, a) \,dt}{t^{1-s}(t^{s}-\chi(a))^{2}}+O(x^{-\sigma}(\log x)^{4}), \end{align*} $$

where p signifies the largest prime below x. From the explicit formula (see [Reference Iwaniec and Kowalski3]),

(3.1) $$ \begin{align} \varphi(q) E(t; q, a) & = \delta_{2}(q, a) \sqrt{t}+\delta_{3}(q, a) \sqrt[3]{t} + \sum_{\psi \pmod{\!q}} \overline{\psi}(a) \sum_{L(\rho, \psi) = 0} \frac{t^{\rho}}{\rho} \notag\\ &\quad - \sum_{\psi \pmod{\!q}} \overline{\psi}(a) \sum_{L(\rho, \psi) = 0} \frac{t^{\rho/2}}{\rho}+O(t^{1/5}), \end{align} $$

where

$$ \begin{align*} \delta_{m}(q, a) = \# \{x \pmod{q}: x^{m} \equiv a \pmod{q} \}, \end{align*} $$

the outer sums over $\psi $ range over all Dirichlet characters modulo q and the inner sums are over nontrivial zeros of $L(s, \psi )$ . By the Chinese remainder theorem, the functions $\delta _{m}(q, a)$ are multiplicative. Then one sees that the contribution from the fourth term on the right-hand side of (3.1) is bounded by

$$ \begin{align*} \sum_{\psi \pmod{\!q}} \overline{\psi}(a) \sum_{L(\rho, \psi) = 0} \frac{1}{\rho} \int_{p_{1}}^{x} \frac{x^{s+\rho/2-1} \,dx}{(x^{s}-\chi(a))^{2}} \ll \bigg|\sum_{\psi \pmod{\!q}} \sum_{L(\rho, \psi) = 0} \frac{x^{\rho/2-s}}{\rho(\rho/2-s)} \bigg| \ll x^{1/4-\sigma}. \end{align*} $$

Since $(x^{s}-\chi (a))^{-2} = x^{-2s}+O(x^{-3\sigma })$ , the contribution from the third term on the right-hand side of (3.1) becomes

$$ \begin{align*} &\sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) \sum_{\psi \pmod{\!q}} \overline{\psi}(a) \sum_{L(\rho, \psi) = 0} \frac{1}{\rho} \int_{p_{1}}^{x} \frac{x^{s+\rho-1} \,dx}{(x^{s}-\chi(a))^{2}} \\ &\quad = \mathrm{const}+\sum_{L(\rho, \chi) = 0} \frac{x^{\rho-s}}{\rho(\rho-s)}+O(x^{1/2-2\sigma}). \end{align*} $$

Hence, we arrive at the expression

(3.2) $$ \begin{align} &\frac{\chi(2) \log 2}{2^{s}-\chi(2)} +\frac{\chi(3) \log 3}{3^{s}-\chi(3)}+\cdots + \frac{\chi(p) \log p}{p^{s}-\chi(p)} \notag\\ & \quad = \mathrm{const} +\frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \int_{p_{1}}^{\varphi(q) \vartheta(x; q, a)} \frac{\chi(a) \,dt}{t^{s}-\chi(a)} \notag \\ &\qquad - \frac{s}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) \int_{p_{1}}^{x} \frac{\delta_{2}(q, a) \sqrt{t} + \delta_{3}(q, a) \sqrt[3]{t}}{t^{1-s}(t^{s}-\chi(a))^{2}} \,dt \notag\\ &\qquad + S_{s}(x, \chi)+ O(x^{1/2-2\sigma}+x^{1/4-\sigma}). \end{align} $$

3.2 Completion of the proof

From the identity

$$ \begin{align*} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) \delta_{m}(q, a) = \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a)^{m} \end{align*} $$

and the orthogonality relation, we obtain the following lemma.

Lemma 3.1. If $\chi $ is a Dirichlet character modulo q and $m \geqslant 1$ is an integer, then

$$ \begin{align*} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) \delta_{m}(q, a) = \begin{cases} \varphi(q) & \text{if } \chi^{m} = \chi_{0},\\ 0 & \text{otherwise}. \end{cases} \end{align*} $$

We utilise Lemma 3.1 after expanding the integrands of the two integrals on the right-hand side of (3.2). Upon truncating unnecessary terms, the sum of the contributions from these integrals equals

$$ \begin{align*} \mathrm{const} & +\frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \bigg(\frac{\chi(a)}{1-s} (\varphi(q) \vartheta(x; q, a))^{1-s} + \frac{\chi(a)^{2}}{1-2s} (\varphi(q) \vartheta(x; q, a))^{1-2s} \bigg)\\ & + \frac{x^{1-\varphi(q) s}}{1-\varphi(q) s}+\cdots+\frac{x^{1-ns}}{1-ns} - \delta_{\chi^{2} = \chi_{0}} \dfrac{2s x^{1/2-s}}{1-2s}-\delta_{\chi^{3} = \chi_{0}} \dfrac{3s x^{1/3-s}}{1-3s} \\ & - \delta_{\chi^{2} = \chi_{0}} \dfrac{4s x^{1/2-2s}}{1-4s}+O(x^{1/2-2\sigma}), \end{align*} $$

where n is the largest multiple of $\varphi (q)$ not exceeding $[2+1/2\sigma ]$ . It thus follows that

(3.3) $$ \begin{align} & \frac{\chi(2) \log 2}{2^{s}-\chi(2)}+\frac{\chi(3) \log 3}{3^{s}-\chi(3)} + \cdots+\frac{\chi(p) \log p}{p^{s}-\chi(p)} \notag\\ & \quad= -\frac{L^{\prime}(s, \chi)}{L(s, \chi)} + \frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \bigg(\frac{\chi(a)}{1-s} (\varphi(q) \vartheta(x; q, a))^{1-s} + \frac{\chi(a)^{2}}{1-2s} (\varphi(q) \vartheta(x; q, a))^{1-2s} \bigg) \notag\\ &\qquad + \frac{x^{1-\varphi(q) s}}{1-\varphi(q) s} + \frac{x^{1-ns}}{1-ns}-\delta_{\chi^{2} = \chi_{0}} \dfrac{2s x^{1/2-s}}{1-2s}-\delta_{\chi^{3} = \chi_{0}} \dfrac{3s x^{1/3-s}}{1-3s} \notag\\ &\qquad - \delta_{\chi^{2} = \chi_{0}} \dfrac{4s x^{1/2-2s}}{1-4s}+S_{s}(x, \chi)+O(x^{1/2-2\sigma}+x^{1/4-\sigma}). \end{align} $$

Since the left-hand side of (3.3) is the logarithmic derivative of the partial Euler product, the constant is $L'(s, \chi )/L(s, \chi )$ from the work of Conrad [Reference Conrad2]. For the Euler products at $s = 1, 1/2, 1/3$ , $1/4$ , we take the limit of the right-hand side of (3.3). To finish the proof, replace $s = \sigma +it$ with $u+it$ , integrate both sides of the formula (3.3) in u from $\infty $ to $\sigma $ and then relabel $\sigma +it$ as s, giving

(3.4) $$ \begin{align} \log \prod_{p \leqslant x} (1-\chi(p) p^{-s}) & = -\log L(s, \chi)+\frac{1}{\varphi(q)} \ \sideset{}{^{\ast}} \sum_{a \pmod{\!q}} \chi(a) {\mathrm{Li}}((\varphi(q) \vartheta(x; q, a))^{1-s}) \notag\\ & \quad - \delta_{\chi^{2} = \chi_{0}} \dfrac{1}{2} {\mathrm{Li}}(x^{1-2s})+\delta_{\chi^{2} = \chi_{0}} \dfrac{1}{2} {\mathrm{Li}}(x^{1/2-s}) - \dfrac{1}{\varphi(q)} {\mathrm{Li}}(x^{1-\varphi(q) s}) \notag\\ & \quad - \dfrac{1}{n} {\mathrm{Li}}(x^{1-ns})-\frac{x^{1/2-s}+S_{s}(x, \chi)}{\log x} + O \bigg(\frac{x^{1/2-\sigma}}{(\log x)^{2}} \bigg). \end{align} $$

Here we have used

$$ \begin{align*} \int_{\infty}^{\sigma} \frac{x^{a+b(u+it)}}{a+b(u+it)} \,du = \frac{1}{b} {\mathrm{Li}}(x^{a+bs}), \quad \int_{\infty}^{\sigma} S_{u+it}(x, \chi) \,du = -\frac{S_{s}(x, \chi)}{\log x}+O \bigg(\frac{x^{1/2-\sigma}}{(\log x)^{2}} \bigg). \end{align*} $$

Exponentiating (3.4) and classifying the resulting formula into the three cases $0 < \Re (s) < 1/2$ , $\Re (s) = 1/2$ and $\Re (s)> 1/2$ yields the desired identities. This concludes the proof of Theorem 1.1.