The TME-EMT project
Home > Projet TME-EMT > Explicit bounds on primes
Explicit bounds on primes
Collecting references:
[
Dusart, 1998 †Dusart, P. 1998
Autour de la fonction qui compte le nombre de nombres premiers
Ph.D. thesis, Limoges, . 173 pp.],
[
Dusart, 2016 †Dusart, Pierre. 2016
Estimates of {$\psi,\theta$} for large values of {$x$} without the Riemann hypothesis
Math. Comp., 85(298), 875--888.],
1. Bounds on primes, in special ranges
The paper
[
Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.],
contains several bounds valid only when the variable is small enough.
In
[
Büthe, 2016 †Büthe, Jan. 2016
Estimating {$\pi(x)$} and related functions under partial RH assumptions
Math. Comp., 85(301), 2483--2498.], the
author proves the next theorem.
Theorem (2016)
Assume the Riemann Hypothesis has been checked up to height
$H_0$. Then when $x$ satisfies $\sqrt{x/\log x}\le H_0/4.92$, we have
- $|\psi(x)-x|\le \frac{\sqrt{x}}{8\pi}\log^2x$ when $x > 59$,
- $|\theta(x)-x|\le \frac{\sqrt{x}}{8\pi}\log^2x$ when $x > 599$,
- $|\pi(x)-\text{li}(x)|\le \frac{\sqrt{x}}{8\pi}\log x$ when
$x > 2657$.
If we use the value $H_0=30\,610\,046\,000$ obtained by D. Platt
in
[
Platt, 2017 †Platt, David J. 2017
Isolating some non-trivial zeros of zeta
Math. Comp., 86(307), 2449--2467.], these
bounds are thus valid for $x\le 1.8\cdot 10^{21}$.
In
[
Büthe, 2018 †Büthe, Jan. 2018
An analytic method for bounding {$\psi(x)$}
Math. Comp., 87(312), 1991--2009.],
the following bounds are also obtained.
Theorem (2018)
We have
- $|\psi(x)-x|\le 0.94\sqrt{x}$ when $11 < x\le 10^{19}$,
- $0<\text{li}(x)-\pi(x)\le\frac{\sqrt{x}}{\log
x}\Bigl(1.95+\frac{3.9}{\log x}+\frac{19.5}{\log^2x}\Bigr)$ when
$2\le x\le 10^{19}$.
2. Bounds on primes, without any congruence condition
The subject really started with the four papers
[
Rosser, 1941 †Rosser, J.B. 1941
Explicit bounds for some functions of prime numbers
Amer. J. Math., 63, 211--232.],
[
Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.],
[
Rosser & Schoenfeld, 1975 †Rosser, J.B., & Schoenfeld, L. 1975
Sharper bounds for the Chebyshev Functions $\vartheta(X)$ and $\psi(X)$
Math. Comp., 29(129), 243--269.]
and
[
Schoenfeld, 1976 †Schoenfeld, L. 1976
Sharper bounds for the Chebyshev Functions $\vartheta(X)$ and $\psi(X)$ II
Math. Comp., 30(134), 337--360.].
We recall the usual notation: $\pi(x)$ is the number of primes up to
$x$ (so that $\pi(3)=2$), the function $\psi(x)$ is the summatory
function of the van Mangold function $\Lambda$,
i.e. $\psi(x)=\sum_{n\le x}\Lambda(n)$, while we also define
$\vartheta(x)=\sum_{p\le x}\log p$.
Here are some elegant bounds that one can find in these papers.
Theorem (1962)
- For $x > 0$, we have
$\psi(x)\le 1.03883 x$ and the maximum of $\psi(x)/x$ is
attained at $x=113$.
- When $x\ge17$, we have $\pi(x) > x/\log x$.
- When $x > 1$, we have $\displaystyle \sum_{p\le x}1/p > \log\log x$.
- When $x > 1$, we have $\displaystyle \sum_{p\le x}(\log p)/p <
\log x$.
There are many other results in these papers.
In
[
Dusart, 1999a †Dusart, P. 1999a
Inégalités explicites pour $\psi(X)$, $\theta(X)$, $\pi(X)$ et les nombres premiers
C. R. Math. Acad. Sci., Soc. R. Can., 21(2), 53--59.],
on can find among other things the inequality
$$
\text{When $x\ge17$, we have } \pi(x) > \frac{x}{\log x -1}.
$$
And also
Theorem (1999)
- When $x\ge e^{22}$, we have
$\displaystyle\psi(x)=x+O^*\Bigl(0.006409\frac{x}{\log
x}\Bigr)$.
- When $x\ge 10\,544\,111$, we have $\displaystyle\vartheta(x)=x+O^*\Bigl(0.006788\frac{x}{\log
x}\Bigr)$.
- When $x\ge 3\,594\,641$, we have $\displaystyle\vartheta(x)=x+O^*\Bigl(0.2\frac{x}{\log^2
x}\Bigr)$.
- When $x > 1$, we have $\displaystyle\vartheta(x)=x+O^*\Bigl(515\frac{x}{\log^3
x}\Bigr)$.
This is improved in
[
Dusart, 2018 †Dusart, P. 2018
Estimates of some functions over primes
Ramanujan J., 45(1), 227--251.],
and in particular, it is shown that the 515 above can be replaced by
20.83 and also that
$$
\text{When $x\ge 89\,967\,803$, we have } \vartheta(x)=x+O^*\Bigl(\frac{x}{\log^3
x}\Bigr).
$$
Bounds of the shape $|\psi(x)-x|\le \epsilon x$ have started appearing
in
[
Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.].
The latest paper is
[
Faber & Kadiri, 2015 †Faber, L., & Kadiri, H. 2015
New bounds for $\psi(x)$
Math. Comp., 84(293), 1339--1357.]
with its corrigendum
[
undefined †undefined
undefined
undefined],
where the explicit density estimate from
[
Kadiri, 2013 †Kadiri, H. 2013
A zero density result for the Riemann zeta function
Acta Arith., 160(2), 185--200.]
is put to contribution, even for moderate
values of the variable. In particular
$$
\text{When $x\ge 485\,165\,196$, we have } \psi(x)=x+O^*(0.00053699\,x).
$$
In
[
Platt & Trudgian, 2021b †Platt, David J., & Trudgian, Timothy S. 2021b
The error term in the prime number theorem
Math. Comp., 90(328), 871--881.],
one find the following estimate
Theorem (2021)
When $x\ge e^{2000}$, we have
$\biggl|\frac{\psi(x)-x}{x}\biggr|\le 235\,(\log
x)^{0.52}\exp-\sqrt{\frac{\log x}{5.573412}}\;.$
Refined bounds for $\pi(x)$ are to be found in
[
Panaitopol, 2000 †Panaitopol, L. 2000
A formula for {$\pi(x)$} applied to a result of Koninck-Ivić
Nieuw Arch. Wiskd. (5), 1(1), 55--56.]
and in
[
Axler, 2016 †Axler, Christian. 2016
New bounds for the prime counting function
Integers, 16, Paper No. A22, 15.].
By comparing in an efficient manner with $\psi(x)-x$,
[
Ramaré, 2013a †Ramaré, O. 2013a
Explicit estimates for the summatory function of ${\Lambda}(n)/n$ from the one of ${\Lambda}(n)$
Acta Arith., 159(2), 113--122.],
obtained the next two results. [There was an error in this paper which
is corrected below].
Theorem (2013)
For $x > 1$, we have
$\sum_{n\le x}\Lambda(n)/n=\log x-\gamma+O^*(1.833/\log^2x)$.
When $x\ge 1\,520\,000$, we can replace the error term by $O^*(0.0067/\log
x)$. When $x\ge 468\,000$, we can replace the error term by $O^*(0.01/\log
x)$.
Furthermore, when $1\le x\le 10^{10}$, this error term can be
replaced by $O^*(1.31/\sqrt{x})$.
Theorem (2013)
For $x\ge 8950$, we have
$$
\sum_{n\le x}\Lambda(n)/n=\log x-\gamma
+\frac{\psi(x)-x}{x}+O^*\Bigl(\frac{1}{2\sqrt{x}}+1.75\cdot 10^{-12}\Bigl)
$$.
[
Mawia, 2017 †Mawia, Ramdin. 2017
Explicit estimates for some summatory functions of primes
Integers, 17, 18pp. A11.]
developed the method to incorporate more functions (and corrected the
initial work), extending results of
[
Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.].
Here are some of his results.
Theorem (2017)
For $x\ge 2$, we have
$$
\sum_{p\le x}\frac1p=\log\log x+B+O^*\Bigl(\frac{4}{\log^3x}\Bigr).
$$
When $x\ge 1000$, one can replace the 4 in the error term by 2.3,
and when $x\ge24284$, by 1. The constant $B$ is the expected one.
Theorem (2017)
For $\log x\ge 4635$, we have
$$
\sum_{p\le x}\frac1p=\log\log
x+B+O^*\Bigl(1.1\frac{\exp(-\sqrt{0.175\log x})}{(\log x)^{3/4}}\Bigr).
$$
When truncating sums over primes, Lemma 3.2 of
[
Ramaré, 2016 †Ramaré, O. 2016
An explicit density estimate for Dirichlet $L$-series
Math. Comp., 85(297), 335--356.]
is handy.
Theorem (2016)
Let $f$ be a C${}^1$ non-negative, non-increasing function over
$[P,\infty[$, where $P\ge 3\,600\,000$ is a real number and such
that $\lim_{t\rightarrow\infty}tf(t)=0$.
We have
\begin{equation*}
\sum_{p\ge P} f(p)\log p
\le (1+\epsilon) \int_P^\infty f(t) dt + \epsilon P f(P) + P
f(P) / (5 \log^2 P)
\end{equation*}
with $\epsilon=1/914$. When we can only ensure $P\ge2$, then a similar
inequality holds, simply replacing the last $1/5$ by a 4.
The above result relies on (5.1*) of
[
Schoenfeld, 1976 †Schoenfeld, L. 1976
Sharper bounds for the Chebyshev Functions $\vartheta(X)$ and $\psi(X)$ II
Math. Comp., 30(134), 337--360.]
because it is easily accessible. However on using
Proposition 5.1 of
[
Dusart, 2016 †Dusart, Pierre. 2016
Estimates of {$\psi,\theta$} for large values of {$x$} without the Riemann hypothesis
Math. Comp., 85(298), 875--888.],
one has access to $\epsilon=1/36260$.
Here is a result due to
[
Treviño, 2012 †Treviño, Enrique. 2012
The least inert prime in a real quadratic field
Math. Comp., 81(279), 1777--1797.].
Theorem (2012)
For $x$ a positive real number. If $x \geq x_0$ then there exist $c_1$
and $c_2$ depending on $x_0$ such that
$$
\frac{x^2}{2\log{x}} +
\frac{c_1 x^2}{\log^2{x}} \leq \sum_{p \leq x} p \leq
\frac{x^2}{2\log{x}} + \frac{c_2 x^2}{\log^2{x}}.
$$
The constants
$c_1$ and $c_2$ are given for various values of $x_0$ in the next
table.
|
|
|---|
| $x_0$ |
$c_1$ |
$c_2$ |
| 315437 |
0.205448 |
0.330479 |
| 468577 |
0.211359 |
0.32593 |
| 486377 |
0.212904 |
0.325537 |
| 644123 |
0.21429 |
0.322609 |
| 678407 |
0.214931 |
0.322326 |
| 758231 |
0.215541 |
0.321504 |
| 758711 |
0.215939 |
0.321489 |
| 10544111 |
0.239818 |
0.29251 |
Further estimates can be found in
[
Axler, 2019 †Axler, Christian. 2019
On the sum of the first {$n$} prime numbers
J. Théor. Nombres Bordeaux, 31(2), 293--311.]
(Proposition 2.7 and Corollary 2.8).
In
[
Deléglise & Nicolas, 2019a †Deléglise, Marc, & Nicolas, Jean-Louis. 2019a
An arithmetic equivalence of the Riemann hypothesis
J. Aust. Math. Soc., 106(2), 235--273.]
(Proposition 2.7 and Corollary 2.8) and
[
Deléglise & Nicolas, 2019b †Deléglise, Marc, & Nicolas, Jean-Louis. 2019b
The Landau function and the Riemann Hypothesis
J. Combinatorics and Numb. Th., 11(2), 45--95.]
(Proposition 2.5, Corollary 2.6, 2.7 and 2.8),
we find among other results the next two.
Theorem (2019)
On setting $\pi_r(x)=\sum_{p\le x}p^r$, we have
$$
\frac{3x^2}{20(\log x)^4}
\le \pi_1(x)-\biggl(
\frac{x^2}{2\log x}
+\frac{x^2}{4(\log x)^2}
\frac{x^2}{4(\log x)^3}
\biggr)
\le
\frac{107x^2}{160(\log x)^4}
$$
where the upper estimate is valid when $x\ge 110117910$
and the lower one when $x\ge905238547$.
We have
$$
\frac{-1069x^3}{648(\log x)^4}
\le \pi_2(x)-\biggl(
\frac{x^3}{3\log x}
+\frac{x^3}{9(\log x)^2}
\frac{x^3}{27(\log x)^3}
\biggr)
\le
\frac{11181x^3}{648(\log x)^4}
$$
where the upper estimate is valid when $x\ge 60173$
and the lower one when $x\ge 1091239$.
$$ \pi_3(x)\le 0.271\frac{x^4}{\log x}\quad\text{for $x\ge 664$},$$
$$ \pi_4(x)\le 0.237\frac{x^5}{\log x}\quad\text{for $x\ge 200$},$$
$$ \pi_5(x)\le 0.226\frac{x^5}{\log x}\quad\text{for $x\ge 44$},$$
For $x\ge 1$ and $r\ge5$, we have
$$ \pi_r(x)\le \frac{\log 3}{3}\bigl(1+(2/3)^r\bigr)
\frac{x^{r+1}}{\log x}$$.
3. Bounds containing $n$-th prime
Denote by $p_n$ the $n$-th prime. Thus $p_1=2,\;p_2=3,\; p_4=5,\cdots$.
The classical form of prime number theorem yields easily
$p_n \sim n \log n.$
[
Rosser, 1938 †Rosser, J.B. 1938
The $n$-th prime is greater than $n\log n$
Proc. Lond. Math. Soc., II. Ser., 45, 21--44.]
shows that this equivalence does not oscillate
by proving that $p_n$ is greater than $n\log n$ for $n\geq 2$.
The asymptotic formula for $p_n$ can be developped as shown in
[
Cipolla, 1902 †Cipolla, M. 1902
La determinatzione assintotica dell`$n^{imo}$ numero primo
Matematiche Napoli, 3, 132--166.]:
$$
p_n\sim n\left(\log n+\log\log n -1+\frac{\log\log n-2}{\log n}
-\frac{(\ln\ln n)^2-6\log\log n +11}{2\log^2 n}+\cdots\right).
$$
This asymptotic expansion is the inverse of the logarithmic integral
$\mbox{Li}(x)$ obtained by series reversion.
But
[
Rosser, 1938 †Rosser, J.B. 1938
The $n$-th prime is greater than $n\log n$
Proc. Lond. Math. Soc., II. Ser., 45, 21--44.]
also proved that for every $n> 1$:
$$
n (\log n + \log \log n - 10) < p_n < n (\log n + \log\log n +8).
$$
He improves these results in
[
Rosser, 1941 †Rosser, J.B. 1941
Explicit bounds for some functions of prime numbers
Amer. J. Math., 63, 211--232.]
: for every $n\geq 55$,
$$
n (\log n + \log \log n - 4) < p_n < n (\log n + \log\log n +2).
$$
This result was subsequently improved by Rosser and Schoenfeld
[
Rosser & Schoenfeld, 1962 †Rosser, J.B., & Schoenfeld, L. 1962
Approximate formulas for some functions of prime numbers
Illinois J. Math., 6, 64--94.]
in 1962 to
$$
n (\log n + \log \log n - 3/2) < p_n < n (\log n + \log\log n -1/2),
$$
for $n > 1$ and $n > 19$ respectively.
The constants were subsequently reduced by
[
Robin, 1983a †Robin, G. 1983a
Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombres premiers et grandes valeurs de la fonction $\omega(n)$ nombre de diviseurs premiers de $n$
Acta Arith., 42, 367--389.].
In particular, the lower bound
$$
n (\log n + \log \log n - 1.0072629) < p_n
$$
is valid for $n>1$ and the constant $1.0072629$ can be replaced by 1 for
$p_k\leq 10^{11}$.
Then
[
Massias & Robin, 1996 †Massias, J.-P., & Robin, G. 1996
Bornes effectives pour certaines fonctions concernant les nombres premiers
J. Théor. Nombres Bordeaux, 8(1), 215--242.]
showed that the lower bound constant equals to 1 was admissible for
$p_k < e^{598}$
and $p_k > e^{1800}$. Finally,
[
Dusart, 1999b †Dusart, P. 1999b
The $k$th prime is greater than $k(\ln k+\ln\ln k-1)$ for $k\geq 2$
Math. Comp., 68(225), 411--415.]
showed
that
$$
n(\log n - \log \log n - 1) < p_n
$$ for all $n > 1$, and also that
$$
p_n < n (\log n + \log\log n - 0.9484)
$$ for $n > 39017$ i.e. $p_n > 467\,473$.
In
[
Carneiro et al., 2019 †Carneiro, Emanuel, Milinovich, Micah, & Soundararajan, Kannan. 2019
Fourier optimization and prime gaps
],
the authors prove the next result.
Theorem (2019)
Under the Riemann Hypothesis we have $p_{n+1}-p_n\le\frac{22}{25}\sqrt{p_n}\log p_n$.
In
[
Axler, 2019 †Axler, Christian. 2019
On the sum of the first {$n$} prime numbers
J. Théor. Nombres Bordeaux, 31(2), 293--311.],
we find (Theorem 1.6 and 1.7) the next result.
Theorem (2019)
We have
$$
\sum_{i\le k}p_i\ge
\frac{k^2}4 +\frac{k^2}{4\log k}
-\frac{k^2(\log\log k-2.9)}{4(\log k)^2}\quad(\text{for $k\ge 6309751$}),
$$
as well as
$$
\sum_{i\le k}p_i\le
\frac{k^2}4 +\frac{k^2}{4\log k}
-\frac{k^2(\log\log k-4.42)}{4(\log k)^2}\quad(\text{for $k\ge 256376$}),
$$
In
[
De Koninck & Letendre, 2020 †De Koninck, Jean-Marie, & Letendre, Patrick. 2020
New upper bounds for the number of divisors function
Colloq. Math., 162(1), 23--52.],
we find in passing (Lemma 4.8) the next result.
Theorem (2020)
We have
$$
\sum_{i\le k}\log p_i\le
k\bigl(\log k+\log\log -3/4\Bigr)\quad(\text{for $k\ge 4$}),
$$
as well as
$$
\sum_{i\le k}\log\log p_i\ge
k\biggl(\log\log k+\frac{\log\log -5/4}{\log k}\biggr)
\quad(\text{for $k\ge319$}).
$$
4. Bounds on primes in arithmetic progressions
Collecting references:
[
McCurley, 1984a †McCurley, K.S. 1984a
Explicit estimates for the error term in the prime number theorem for arithmetic progressions
Math. Comp., 42, 265--285.],
[
McCurley, 1984b †McCurley, K.S. 1984b
Explicit estimates for $\theta(x;3,\ell)$ and $\psi(x;3,\ell)$
Math. Comp., 42, 287--296.],
[
Ramaré & Rumely, 1996 †Ramaré, O., & Rumely, R. 1996
Primes in arithmetic progressions
Math. Comp., 65, 397--425.],
[
Dusart, 2002 †Dusart, P. 2002
Estimates for $\theta(x;k,\ell)$ for large values of $x$
Math. Comp., 71(239), 1137--1168.],
Lemma 10 of [
Moree, 2004 †Moree, P. 2004
Chebyshev's bias for composite numbers with restricted prime divisors
Math. Comp., 73(245), 425--449.],
section 4 of
[
Moree & te Riele, 2004 †Moree, P., & te Riele, H.J.J. 2004
The hexagonal versus the square lattice
Math. Comp., 73(245), 451--473.].
A consequence of Theorem 1.1 and 1.2 of
[
Bennett et al., 2018 †Bennett, Michael A., Martin, Greg, O'Bryant, Kevin, & Rechnitzer, Andrew. 2018
Explicit bounds for primes in arithmetic progressions
Illinois J. Math., 62(1-4), 427--532.]
states that
Theorem (2018)
We have
$\displaystyle
\max_{3\le q\le 10^4}\max_{ x\ge 8\cdot 10^9}\max_{\substack{1\le a\le q,\\
(a,q)=1}}
\frac{\log x}{x}\Bigl|
\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n)
-\frac{x}{\varphi(q)}\Bigr|\le 1/840.
$
When $q\in(10^4, 10^5]$, we may replace $1/840$ by $1/160$ and when
$q\ge 10^5$, we may replace $1/840$ by $\exp(0.03\sqrt{q}\log^3q)$.
Furthermore, we may replace
$\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n)$ by
$\sum_{\substack{p\le x,\\ p\equiv a[q]}}\log p$ with no changes in
the constants.
Similarly, as a consequence of Theorem 1.3 of
[
Bennett et al., 2018 †Bennett, Michael A., Martin, Greg, O'Bryant, Kevin, & Rechnitzer, Andrew. 2018
Explicit bounds for primes in arithmetic progressions
Illinois J. Math., 62(1-4), 427--532.]
states that
Theorem (2018)
We have
$\displaystyle
\max_{3\le q\le 10^4}\max_{ x\ge 8\cdot 10^9}\max_{\substack{1\le a\le q,\\
(a,q)=1}}
\frac{\log^2 x}{x}\Bigl|
\sum_{\substack{p\le x,\\ p\equiv a[q]}}1
-\frac{\textrm{Li}(x)}{\varphi(q)}\Bigr|\le 1/840.
$
When $q\in(10^4, 10^5]$, we may replace $1/840$ by $1/160$ and when
$q\ge 10^5$, we may replace $1/840$ by $\exp(0.03\sqrt{q}\log^3q)$.
Concerning estimates with a logarithmic density, in
[
Ramaré, 2002 †Ramaré, O. 2002
Sur un théorème de Mertens
Manuscripta Math., 108, 483--494.]
and in
[
Platt & Ramaré, 2017 †Platt, D.J., & Ramaré, O. 2017
Explicit estimates: from ${\Lambda}(n)$ in arithmetic progressions to ${\Lambda}(n)/n$
Exp. Math., 26, 77--92.],
estimates for the functions
$\displaystyle\sum_{\substack{n\le x,\\ n\equiv a[q]}}\Lambda(n)/n$
are considered.
Extending computations from the former, the latter paper Theorem 8.1
reads as follows.
Theorem (2016)
We have
$\displaystyle
\max_{q\le 1000}\max_{q\le x\le 10^5}\max_{\substack{1\le a\le q,\\
(a,q)=1}}
\sqrt{x}\Bigl|
\sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}
-\frac{\log x}{\varphi(q)}-C(q,a)\Bigr|\in(0.8533,0.8534)
$
and the maximum is attained with $q=17$, $x=606$ and $a=2$.
The constant $C(q,a)$ is the one expected, i.e. such that
$\sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}
-\frac{\log x}{\varphi(q)}-C(q,a)$ goes to
zero when $x$ goes to infinity.
When $q$ belongs to "Rumely's list", i.e. in one of the
following set:
- $\{k\le 72\}$
- $\{k\le 112, \text{$k$ non premier}\}$
- $\begin{aligned}\{116, 117, &120, 121, 124, 125, 128, 132, 140,
143, 144, 156, 163, \\ &169, 180, 216, 243, 256, 360, 420, 432\}\end{aligned}$
Theorem 2 of
[
Ramaré, 2002 †Ramaré, O. 2002
Sur un théorème de Mertens
Manuscripta Math., 108, 483--494.]
gives the following.
Theorem (2002)
When $q$ belongs to Rumely's list and $a$ is prime to $q$, we have
$\displaystyle
\sum_{\substack{n\le x,\\ n\equiv a[q]}}\frac{\Lambda(n)}{n}
=\frac{\log x}{\varphi(q)}+C(q,a)+O^*(1)
$
as soon as $x\ge1$.
More precise bounds are given.
5. Least prime verifying a condition
[
Bach & Sorenson, 1996 †Bach, E., & Sorenson, J. 1996
Explicit bounds for primes in residue classes
Math. Comp., 65(216), 1717--1735.],
[
Kadiri, 2008 †Kadiri, H. 2008
Short effective intervals containing primes in arithmetic progressions and the seven cube problem
Math. Comp., 77(263), 1733--1748.],
Last updated on June 3rd, 2022, by Olivier Ramaré