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\begin{center}
\vskip 1cm{\LARGE\bf Sum of Reciprocals of Germain Primes
}
\vskip 1cm
\large
Samuel S. Wagstaff, Jr. \\
Center for Education and Research in Information Assurance and Security\\
and \\
Department of Computer Sciences\\
Purdue University\\
West Lafayette, IN 47907-1398\\
USA \\
\href{mailto:ssw@cerias.purdue.edu}{\tt ssw@cerias.purdue.edu} \\
\end{center}

\vskip 0.2in

\begin{abstract}
A prime $p$ is a Germain prime if $2p+1$ is also prime.
We compute the sum of the reciprocals of the Germain primes
and related sequences.
Since we do not know whether there are infinitely many
Germain primes, all we can do is bound the sum in an interval.
\end{abstract} 


\section{Introduction}

In 1919, Brun \cite{Brun} proved that the sum
\[
B=\frac{1}{3}+ \frac{1}{5}+ \frac{1}{5}+ \frac{1}{7}+ \frac{1}{11}+
\frac{1}{13}+ \frac{1}{17}+ \frac{1}{19}+ \frac{1}{29}+ \frac{1}{31}+ \cdots
\]
of the reciprocals of all twin primes either converges
or is a finite sum.
Various authors \cite{SW}
have estimated Brun's constant $B$.
Sebah \cite{Sebah} computed the sum of the reciprocals
of all twin primes $<10^{16}$ and found that $B>1.830$.
Klyve \cite{Kly} showed that $B<2.347$ as reported in
Crandall and Pomerance \cite{CP}.
Platt and Trudgian \cite{PT} used the number of twin primes
$<4\cdot10^{18}$ computed by Oliveira e Silva \cite{OS} and found that
$1.840503 < B < 2.288513$.
Several authors \cite{Brent,Bow,Nice1,Nice,OS,PT,Sebah,SW} agree that the most
probable value of $B$ is about $1.90216$, but this estimate is not rigorous.
These results show how little we know about the distribution
of twin primes, even whether there are infinitely many of them.
The twin primes are Sequence \seqnum{A001097} in the OEIS \cite{OEIS}.

A prime $p$ is a Germain prime if $2p+1$ is also prime.
They are named after Sophie Germain, who proved nearly 200 years ago
that the first case of Fermat's Last Theorem is true when
the exponent is a Germain prime.
The Germain primes are Sequence \seqnum{A005384} in the OEIS.
Computing the sum of the reciprocals of the Germain primes
is similar to computing Brun's constant $B$.
First, it is easy to prove using a sieve that this sum
either converges or is a finite sum.
Second, we do not know whether there are infinitely many
Germain primes, so the best lower bound for the sum is
the partial sum up to the limit to which we can compute it directly.
One can compute a most probable value for the sum
just as for $B$.

Little extra work is required to study the primes $p$
for which $2^kp+1$ is also prime for any fixed $k\ge1$.
The same methods apply to the sum of the reciprocals of the
primes $p$ for which $p+2^k$ is also prime.
See Lee and Park \cite{Lee} for the case of $p$, $p+8$ both prime, for example.

\section{Notation and easy results}

Throughout this work $p$ denotes a prime.
Let
\[
c_2=\prod_{p>2}\left(1-\frac{1}{(p-1)^2}\right)\approx
0.6601618158468695739278121100
\]
be the twin prime constant as computed by Harley (See Lee and Park \cite{Lee}).

Let $a$ and $b$ be positive integers with $\gcd(a,b)=1$ and $2\mid ab$.
Let $\Sc_{a,b}=\{p : ap+b \text{ is prime}\}$.
For real $x>0$, let $S_{a,b}(x)$ denote the sum of $\frac{1}{p}$
over all primes $p\le x$ with $p\in\Sc_{a,b}$.
Also let $S'_{a,b}(x)$ be the sum
of $\frac{1}{p}+\frac{1}{ap+b}$
over all primes $p\le x$ with $p\in\Sc_{a,b}$.
Clearly one sum is finite if and only if the other is finite.
Let $S_{a,b}=\lim_{x\rightarrow\infty} S_{a,b}(x)$
and $S'_{a,b}=\lim_{x\rightarrow\infty} S'_{a,b}(x)$.

Thus $S'_{1,2}=B$, Brun's constant, and
\[
S_{2,1}=\frac{1}{2}+ \frac{1}{3}+ \frac{1}{5}+
\frac{1}{11}+ \frac{1}{23}+ \frac{1}{29}+ \frac{1}{41}+ \frac{1}{53}+ \cdots
\]
is the sum of the reciprocals of the Germain primes.  Also,
\[
S_{4,1}=\frac{1}{3}+ \frac{1}{7}+ \frac{1}{13}+
\frac{1}{37}+ \frac{1}{43}+ \frac{1}{67}+ \frac{1}{73}+ \frac{1}{79}+ \cdots
\]
and
\[
S_{8,1}=\frac{1}{2}+ \frac{1}{5}+ \frac{1}{11}+ \frac{1}{17}+
\frac{1}{29}+ \frac{1}{71}+ \frac{1}{101}+ \frac{1}{107}+ \cdots\,.
\]
See Sequences \seqnum{A023212} and \seqnum{A023228} in the OEIS.
The case of $S_{1,8}$ was studied by Lee and Park \cite{Lee}.

We show first that the limits $S_{a,b}$ and $S'_{a,b}$ always exist.
Let $\pi_{a,b}(x)$ be the number of primes $p\le x$ for which
$ap+b$ is also prime.
\begin{theorem}
Suppose $\gcd(a,b)=1$ and $2\mid ab$.
Define $S_{a,b}(x)$, $S'_{a,b}(x)$, $S_{a,b}$ and $S'_{a,b}$ as above.
Then the limits in the definitions of $S_{a,b}$ and $S'_{a,b}$ exist.
\end{theorem}
\begin{proof}
Since $S_{a,b}(x)$ is the sum of positive terms we need only
show that it is bounded.
According to Theorem 3.12 of Halberstam and Richert \cite{HR}, we have
\[
\pi_{a,b}(x) \le
\frac{cx}{(\log x)^2}\left(1+O\left(\frac{\log\log x}{\log x}\right)\right).
\]
for some constant $c$.  Therefore,
\begin{align*}
S_{a,b}(x)&=
\sum_{p\le x, p\in\Sc_{a,b}} \frac{1}{p}=
\sum_{t=1}^x \frac{\pi_{a,b}(t)-\pi_{a,b}(t-1)}{t} \\
&\le \sum_{t=2}^x \left(\frac{1}{t}-\frac{1}{t+1}\right)\pi_{a,b}(t)
\quad\text{ (since } \pi_{a,b}(1)=0) \\
&\le \sum_{t=2}^x
\frac{1}{t(t+1)}\frac{ct}{(\log t)^2}
\left(1+O\left(\frac{\log\log t}{\log t}\right)\right)
=O(1)
\end{align*}
as $x\rightarrow\infty$.
The proof for $S'_{a,b}$ is identical.
\end{proof}

Fix $x_0$.  Since $1/p>0$ always, we have
$S_{a,b}\ge S_{a,b}(x_0)$ for every $(a,b)$.
These are the best lower bounds for $S_{a,b}$ that we know
how to compute.
Table \ref{tablow} shows these lower bounds for a few values of $x_0$.
We computed lower bounds for Brun's constant for comparison,
although a better lower bound is found in Platt and Trudgian \cite{PT}.
The sums were computed in double precision, but only
nine decimal places are shown in the table.
Table \ref{tabcnt} shows the corresponding values of $\pi_{a,b}$.
The data for twin primes is exactly the same as Nicely \cite{Nice1} found.

The most probable value for $S_{a,b}$ is based on the
Hardy-Littlewood \cite{HL} (or see
Bateman and Horn \cite{BH}) approximation to $\pi_{a,b}(x)$.
This heuristic estimate is
\[
\pi_{a,b}(x)\approx
2c_2\int_2^x\frac{dt}{(\log t)^2}\prod_{p>2,\,p\mid ab}\frac{p-1}{p-2}
\]
for fixed coprime integers $a$, $b$.
When $ab$ is a power of 2,
which we always assume in this work,
the product over primes $p>2$ is 1. 
Assuming this heuristic and using partial summation, one finds
\[
S_{a,b}-S_{a,b}(x_0)\approx 2c_2\int_{x_0}^{\infty}\frac{dt}{t(\log t)^2}=
\frac{2c_2}{\log x_0}\,.
\]
Thus, after computing $S_{a,b}(x_0)$, the most probable value for
$S_{a,b}$ is $S_{a,b}(x_0)+2c_2/\log x_0$.
The constant $2c_2$ is replaced by $4c_2$ for $S'_{1,2}$
because we add the reciprocals of two (consecutive) primes
in that case.
These values are shown in Table \ref{tabprob}.

\begin{table}[H]
\begin{center}
\begin{tabular}{r|rrrr} 
$x_0$ & $S'_{1,2}(x_0)$ & $S_{2,1}(x_0)$ & $S_{4,1}(x_0)$ & $S_{8,1}(x_0)$ \\ 
\hline
$10^2$ & 1.330990366 & 1.268745760 & 0.654987903 & 0.898299886 \\
$10^3$ & 1.518032463 & 1.350207371 & 0.718127473 & 0.985924806 \\
$10^4$ & 1.616893558 & 1.395040698 & 0.758673876 & 1.021448995 \\
$10^5$ & 1.672799585 & 1.422243022 & 0.783466978 & 1.045080342 \\
$10^6$ & 1.710776931 & 1.440222094 & 0.800867822 & 1.061448591 \\
$10^7$ & 1.738357044 & 1.453223143 & 0.813394490 & 1.073447436 \\
$10^8$ & 1.758815621 & 1.463079361 & 0.822866962 & 1.082581870 \\
$10^9$ & 1.774735958 & 1.470771610 & 0.830302440 & 1.089777846 \\
$10^{10}$ & 1.787478503 & 1.476946485 & 0.836290746 & 1.095594911 \\
$10^{11}$ & 1.797904311 & 1.482013752 & 0.841220054 & 1.100393501 \\
$10^{12}$ & 1.806592419 & 1.486246659 & 0.845347507 & 1.104420686 \\
$2\cdot10^{12}$ & 1.808931050 & 1.487387756 & 0.846461664 & 1.105509109 \\
$5\cdot10^{12}$ & 1.811852563 & 1.488814135 & 0.847855248 & 1.106871419 \\
$10^{13}$ & 1.813943761 & 1.489835793 & 0.848854028 & 1.107848344 
\end{tabular}
\caption{Some values of $S'_{1,2}(x)$ and $S_{a,b}(x)$.} \label{tablow}
\end{center}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{r|rrrr} 
$x_0$ & $\pi_{1,2}(x_0)$ & $\pi_{2,1}(x_0)$ & $\pi_{4,1}(x_0)$ & $\pi_{8,1}(x_0)$ \\ 
\hline
$10^2$ & 8 & 10 & 9 & 6 \\
$10^3$ & 35 & 37 & 31 & 34 \\
$10^4$ & 205 & 190 & 176 & 161 \\
$10^5$ & 1224 & 1171 & 1057 & 1019 \\
$10^6$ & 8169 & 7746 & 7422 & 7090 \\
$10^7$ & 58980 & 56032 & 53709 & 51464 \\
$10^8$ & 440312 & 423140 & 407198 & 392990 \\
$10^9$ & 3424506 & 3308859 & 3198946 & 3095744 \\
$10^{10}$ & 27412679 & 26569515 & 25773602 & 25030013 \\
$10^{11}$ & 224376048 & 218116524 & 212205881 & 206597032 \\
$10^{12}$ & 1870585220 & 1822848478 & 1777532673 & 1734464714 \\
$2\cdot10^{12}$ & 3552770943 & 3464491157 & 3380477421 & 3300430590 \\
$5\cdot10^{12}$ & 8312493003 & 8112446112 & 7921805792 & 7740062386 \\
$10^{13}$ & 15834664872 & 15462601989 & 15107599980 & 14768799353 
\end{tabular}
\caption{Some values of $\pi_{a,b}(x_0)$.} \label{tabcnt}
\end{center}
\end{table}

\begin{table}[H]
\begin{center}
\begin{tabular}{r|rrrr} 
$x_0$ & $S'_{1,2}$ & $S_{2,1}$ & $S_{4,1}$ & $S_{8,1}$ \\ 
\hline
$10^2$ & 1.904399633 & 1.555450394 & 0.941692537 & 1.185004520 \\
$10^3$ & 1.900305309 & 1.541343794 & 0.909263896 & 1.177061229 \\
$10^4$ & 1.903598191 & 1.538393015 & 0.902026193 & 1.164801312 \\
$10^5$ & 1.902163292 & 1.536924875 & 0.898148831 & 1.159762196 \\
$10^6$ & 1.901913353 & 1.535790305 & 0.896436033 & 1.157016803 \\
$10^7$ & 1.902188263 & 1.535138753 & 0.895310100 & 1.155363045 \\
$10^8$ & 1.902167938 & 1.534755520 & 0.894543121 & 1.154258029 \\
$10^9$ & 1.902160239 & 1.534483751 & 0.894014581 & 1.153489986 \\
$10^{10}$ & 1.902160356 & 1.534287412 & 0.893631672 & 1.152935837 \\
$10^{11}$ & 1.902160541 & 1.534141868 & 0.893348170 & 1.152521616 \\
$10^{12}$ & 1.902160630 & 1.534030764 & 0.893131612 & 1.152204792 \\
$2\cdot10^{12}$ & 1.902160522 & 1.534002492 & 0.893076401 & 1.152123846 \\
$5\cdot10^{12}$ & 1.902160560 & 1.533968133 & 0.893009246 & 1.152025417 \\
$10^{13}$ & 1.902160571 & 1.533944198 & 0.892962433 & 1.151956749 
\end{tabular}
\caption{Most probable values of $S'_{1,2}$ and $S_{a,b}$.} \label{tabprob}
\end{center}
\end{table}
The consistency of the estimates in the columns instills confidence,
although there is no proof that they are anywhere near the true values.
One might consider ``higher order'' terms in this approximation,
but Shanks and Wrench \cite{SW} tell why this is unlikely to
provide a closer estimate; see Sections 3 and 5 of their paper.

\section{Upper bound on the limit}

The upper bound is based on Lemma 5 of Riesel and Vaughan \cite{RV},
which we quote here using Inequality (3.20) from their proof,
as was done by Klyve \cite{Kly}.
\begin{theorem}
\label{lem5}
Let $a$ and $b$ be integers with $a>0$, $b\neq0$ and $\gcd(a,b)=1$.
Let
\[
R(x,a,b)=\sup_I \sum_{p\in I,ap+b\text{\ \rm prime}} 1,
\]
where the supremum is taken over all intervals of length $x$.
Suppose that $L$ and $C=C(L)$ are related by Table \ref{tabLC}.
Then, whenever $x\ge e^L$ we have
\[
R(x,a,b) < \left(\frac{16c_2 x}{(\log x)(C+\log x)}+2\sqrt{x}\right)
\prod_{p>2,\,p\mid ab}\frac{p-1}{p-2}\,.
\]
\end{theorem}

\begin{table}[H]
\begin{center}
\begin{tabular}{r|l||r|l} 
$L$ & $C$ & $L$ & $C$ \\ \hline
24 & 0.97 & 48 & 8.2054 \\
25 & 2.31 & 60 & 8.302 \\
26 & 3.40 & 82 & 8.3503 \\
27 & 4.28 & 100 & 8.3708 \\
28 & 5.00 & 127 & 8.3905 \\
29 & 5.58 & 147 & 8.404 \\
31 & 6.45 & 174 & 8.4102 \\
34 & 7.24 & 214 & 8.4201 \\
36 & 7.56 & 278 & 8.4301 \\
42 & 8.04 & 396 & 8.44004 \\
44 & 8.11 & 690 & 8.45001 
\end{tabular}
\caption{$L$ and $C$ of Theorem \ref{lem5}.} \label{tabLC}
\end{center}
\end{table}
The product in Theorem \ref{lem5} is 1 when $ab$ is a power of 2,
as we always assume.  Klyve \cite{Kly} found it more convenient not to have
the $2\sqrt{x}$ term in the theorem.  He gave an alternate
version, which we quote here.
\begin{theorem}
\label{lem5a}
Let $a$ and $b$ be coprime positive integers with $ab$ a power of 2.
Suppose that $L$ and $D=D(L)$ are related by Table \ref{tabLD}.
Then, whenever $x\ge e^L$ we have
\[
\pi_{a,b}(x) < \frac{16c_2 x}{(\log x)(D+\log x)}\,.
\]
The same bound applies when $\pi_{a,b}(x)$ is replaced
by the count over any other interval of length $x\ge e^L$.
\end{theorem}

In Table \ref{tabLD} we correct a tiny error in Klyve \cite{Kly}.
He had $D=2.30$ when $L=25$; it should be $D=2.296$.
\begin{table}[H]
\centering
\resizebox{4cm}{!}{%
\begin{tabular}{r|l||r|l} 
$L$ & $D$ & $L$ & $D$ \\ \hline
24 & 0.95 & 48 & 8.20 \\
25 & 2.296 & 60 & 8.30 \\
26 & 3.39 & 82 & 8.35 \\
27 & 4.27 & 100 & 8.37 \\
28 & 4.99 & 127 & 8.39 \\
29 & 5.57 & 147 & 8.40 \\
31 & 6.44 & 174 & 8.41 \\
34 & 7.23 & 214 & 8.42 \\
36 & 7.55 & 278 & 8.43 \\
42 & 8.03 & 396 & 8.44 \\
44 & 8.10 & 690 & 8.45 
\end{tabular}
}
\caption{$L$ and $D$ of Theorem \ref{lem5a}.} \label{tabLD}
\end{table}

Note that the upper bound in Theorem \ref{lem5a} is independent
of $a$ and $b$, so long as $ab$ is a power of 2.

\begin{theorem}
\label{upper}
We have $1.4898 < S_{2,1} < 1.8027$,
$0.8488 < S_{4,1} < 1.1617$ and $1.1078 < S_{8,1} < 1.4208$.
\end{theorem}
\begin{proof}
The lower bounds come from Table \ref{tablow} with $x_0=10^{13}$.

Let $a$ and $b$ be positive integers with $\gcd(a,b)=1$ and $2\mid ab$.
Let $0<M<N$.  Stieltjes integration by parts yields
\begin{equation}\label{eq1}
\sum_{t=M}^N\frac{\pi_{a,b}(t) - \pi_{a,b}(t-1)}{t}=
\frac{\pi_{a,b}(N)}{N} - \frac{\pi_{a,b}(M)}{M} +
\int_M^N \frac{\pi_{a,b}(t)}{t^2}dt\,.
\end{equation}
Now
\[
S_{a,b}=S_{a,b}(x_0)+ 
\sum_{t=x_0}^{\infty}\frac{\pi_{a,b}(t) - \pi_{a,b}(t-1)}{t}\,.
\]
We will divide the interval $[x_0,\infty)$ into segments with
boundaries at $e^L$ for $L$ in Table \ref{tabLD} to take advantage
of the constants $D(L)$ in that table.
We will use Stieltjes integration to bound $S_{a,b}(N)-S_{a,b}(M)$
on each interval $[M,N)$.

Use Theorem \ref{lem5a} to bound the integral above.
If $L$, $L'$ are consecutive entries in Table \ref{tabLD},
then the integral over $[M,N)=[e^L,e^{L'})$ becomes
\begin{align*}
\int_{e^L}^{e^{L'}}\frac{\pi_{a,b}(t)}{t^2}dt &
\le \int_{e^L}^{e^{L'}}\frac{16c_2 t}{t^2(\log t)(D(L)+\log t)}\,dt \\
& = 16c_2\int_L^{L'}\frac{ds}{s(s+D(L))}\qquad(s=\log t) \\
& = \left.\frac{16c_2}{D(L)}(\log s - \log(s+D(L))\right|_L^{L'} \\
& = \frac{16c_2}{D(L)}\log\left(\frac{L'(L+D(L))}{L(L'+D(L))}\right)\,.
\end{align*}
Note that $\log x_0\approx29.993606$.
If $L\le\log x_0<L'$ in Table \ref{tabLD},
then $L=29$, $L'=31$ and
the first integral is bounded by
\[
\frac{16c_2}{D(L)}\log\left(\frac{L'(\log(x_0)+D(L))}
{\log(x_0)(L'+D(L))}\right)\approx0.010262\,.
\]
The last integral is bounded by
\[
\frac{16c_2}{8.45}\log\left(\frac{8.45+690}{690}\right)\approx0.015216\,.
\]
The values of these upper bounds are shown in Table \ref{tabint}.
\begin{table}[H]
\begin{center}
\begin{tabular}{|cc|c||cc|c|} 
$M$ & $N$ & Upper Bound & $M$ & $N$ & Upper Bound \\ \hline
   $x_0$ & \strut $e^{31}$ & 0.0102619142161059 &
$e^{100}$ & $e^{127}$ & 0.0208935182022260 \\
$e^{31}$ & $e^{34}$ & 0.0250838160282704 &
$e^{127}$ & $e^{147}$ & 0.0106594237333466 \\
$e^{34}$ & $e^{36}$ & 0.0143023911370928 &
$e^{147}$ & $e^{174}$ & 0.0105916018816358 \\
$e^{36}$ & $e^{42}$ & 0.0350847550483840 &
$e^{174}$ & $e^{214}$ & 0.0108704126795364 \\
$e^{42}$ & $e^{44}$ & 0.0096317734983264 &
$e^{214}$ & $e^{278}$ & 0.0109807270912696 \\
$e^{44}$ & $e^{48}$ & 0.0170051336041258 &
$e^{278}$ & $e^{396}$ & 0.0110369750969163 \\
$e^{48}$ & $e^{60}$ & 0.0381486495943858 &
$e^{396}$ & $e^{690}$ & 0.0111777020604581 \\
$e^{60}$ & $e^{82}$ & 0.0421828971523529 &
$e^{690}$ & $\infty$ & 0.0152151240070501 \\
$e^{82}$ & $e^{100}$ & 0.0212203397984129 & & & 
\end{tabular}
\caption{$M$, $N$ and the upper bound for $\int_M^N\pi_{a,b}(t)t^{-2}dt$.}
\label{tabint}
\end{center}
\end{table}
The sum of all upper bounds gives
\[
\int_{x_0}^{\infty}\frac{\pi_{a,b}(t)}{t^2}dt<0.3143472\,,
\]
which is the same for every $(a,b)$ with $ab$ a power of 2.
The sum of the terms $\pi_{a,b}(N)/N -\pi_{a,b}(M)/M$ in Equation (\ref{eq1})
telescopes to give $-\pi_{a,b}(x_0)/x_0$.
Adding these values to the lower bounds from Table \ref{tablow} gives
\begin{align*}
S_{2,1}&<1.4898358-0.0015463+0.3143472=1.8026367,\\
S_{4,1}&<0.8488541-0.0015108+0.3143472=1.1616905, \quad\hbox{ and}\\
S_{8,1}&<1.1078484-0.0014769+0.3143472=1.4207187.
\end{align*}
\end{proof}
Better upper and lower bounds would result from using a larger $x_0$.
One probably can derive slightly smaller upper bounds (with the same $x_0$)
by assuming the ERH as Klyve \cite{Kly} did for $S'_{1,2}$.

\section{Acknowledgments}
I thank Harold Diamond and
Carl Pomerance for valuable correspondence concerning this research.
I am grateful to Nick Egbert for noticing the typo in Klyve's
thesis \cite{Kly}.

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+ 1/41 + 1/43 + 1/59 + 1/61 + \cdots$, ou les d{\'e}nominateurs
sont `nombres premieres jumeaux,' est convergente ou finie,
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{\bf43} (1919), 100--104, 124--128.

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T. R. Nicely,
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\bibitem{Nice}
T. R. Nicely,
A new error analysis for {Brun's} constant,
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T. Oliveira e Silva,
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D. Platt and T. Trudgian,
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\end{thebibliography}

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\noindent
2010 \emph{Mathematics Subject Classification}:~Primary 11N05.
Secondary 11L20, 11N37, 11N55. 

\medskip

\noindent
\emph{Keywords}:~Brun's constant, Germain prime, twin prime, sum of reciprocals.

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\noindent
(Concerned with sequences
\seqnum{A001097},
\seqnum{A005384},
\seqnum{A023212}, and
\seqnum{A023228}.)

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\vspace*{+.1in}
\noindent
Received August 16 2021;
revised version received October 2 2021.  
Published in {\it Journal of Integer Sequences}, October 2 2021.

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\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
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