Research PaperOpen Access

New rate of convergence of series and its application

Wave and Number Research, Newark, CA 94560, USA

https://doi.org/10.1142/S2811007223500098Cited by:0 (Source: Crossref)

Abstract

The existing rate of convergence of series requires a known limit of that series. Direct computation of Brun’s constant as the sum of twin prime reciprocals is challenging as the convergence is agonizingly slow. Here, I introduce the relative increase of the partial sum of a series as a new rate of convergence regardless of the limit of the series. With this concept, I extract a slowly convergent part from the original sum for Brun’s constant so that the remaining one converges faster. Computed data demonstrate the usefulness of the concept. As a result, I obtain an improved lower bound for Brun’s constant using less computing resource and establish an approximate mapping that gives a new estimate for the constant at 1.89558413.

Keywords:

AMSC: 40A25, 11Y60, 11A41

1. Introduction

If a series has a known limit, the rate of convergence of the series is defined also as a limit. However, the definition does not apply to a series converging to an unknown limit. Furthermore, comparing the rates of convergence of two series requires that both series converge to the same known limit. Recently, I did study qualitatively the rates of convergence of a few slowly convergent series with known or unknown limit and realized that normalizing the partial sums of the series by their limits makes sense for convergence comparison.

One famous series with slow convergence is the sum of the reciprocals of twin primes that converges to a finite value as Brun proved [2]. The value plus the reciprocals of the first twin-prime pair is now called Brun’s constant, $B_{2}$ . Brun’s proof is important to sieve methods [3]. For convenience, we use Brun series to refer to this convergent one with the first twin-prime pair included. It is well known that Brun series converges slowly. Steady progress has been made in computing its partial sum and in estimating $B_{2}$ [4,5,6,7,8,9,10,11]. While the upper bound of Brun’s constant is unknown, its lower bound can be computed but is limited by available computing resource. For example, Nicely obtained his lower bound at 1.831808 using the first 19, 831, 847, 025, 792 twin prime pairs [7]. Directly computed lower bound has not been updated for more than two decades.

Another series with slow convergence is the modified prime harmonic series [9,12]. Unlike Brun series, the sum of this series is known.

It is the purpose of this paper to define a new rate of convergence for series with known or unknown limit. The definition will help the study of the series with slow convergence.

2. New Rate of Convergence of Series

Consider the partial sum of a slowly convergent series below :

x N = N \sum n = 1 a n, <math display="block" altimg="eq-00003.gif"><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>=</mo><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo></math> (2.1)

where

$x_{N}$ is the partial sum, n is the index going from 1 to N and

$a_{n}$ is the nth term of the series. To quantify the rate of convergence of the series, we could use the ratio of its partial sum increase to its index increase in analogy to the derivative of a function of a single variable, i.e.

r=ΔxNΔN,<math display="block" altimg="eq-00006.gif"><mi>r</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>N</mi></mrow></mfrac><mo>,</mo></math>(2.2)

where r is the ratio. Unfortunately, for a large N, r is small as

$Δ N = 1$ means that

$r = a_{N}$ or

$Δ N$ as a large denominator means that r may be even smaller. So, r is not useful for comparing the rates of convergence of series. Here, I introduce the partial sum increase relative to the partial sum itself as a new rate of convergence. Mathematically,

r=ΔxNxN.<math display="block" altimg="eq-00010.gif"><mi>r</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></mfrac><mo>.</mo></math>(2.3)

To see how this definition applies to series with known or unknown limit, calculated data for Brun series and modified prime harmonic series are plotted in Fig. 1.

One can immediately see in Fig. 1 that the rates of convergence of both slowly convergent series decrease as the index increases and Brun series converges faster than the modified prime harmonic series in terms of the rates. If we treat $x_{N}$ as a continuous variable and consider $Δ x_{N}$ to be infinitely small, then we can rewrite (2.3) in a differential form

r=dxx=d(lnx).<math display="block" altimg="eq-00013.gif"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mo>ln</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></math>(2.4)

It is interesting to see that (2.4) relates the relative increase of x to the differential of its logarithm value.

3. Improved Lower Bound for Brun’s Constant

According to the properties of convergent series, we have

\infty \sum n = 1 a n - \infty \sum n = 1 b n = \infty \sum n = 1 c n <math display="block" altimg="eq-00014.gif"><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>-</mo><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub></math> (3.1)

and

\infty \sum n = 1 a n = \infty \sum n = 1 b n + \infty \sum n = 1 c n, <math display="block" altimg="eq-00015.gif"><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>\sum</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>\infty</mi></mrow></munderover><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo></math> (3.2)

where

$\sum a_{n}$ ,

$\sum b_{n}$ and

$\sum c_{n}$ are convergent series with n as their common index. Now, let

$\sum a_{n}$ represent Brun series and

$\sum b_{n}$ represent the modified prime harmonic series [9,12], i.e.

∞∑n=1an=13+15+∞∑n=2(1tn+1tn+2),<math display="block" altimg="eq-00021.gif"><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>,</mo></math>(3.3)

where that there are infinitely many twin prime pairs is assumed and

$t_{n}$ and

$t_{n} + 2$ are the nth pair and

∞∑n=1bn=12+∞∑n=21pn(1−n−1∏i=11pi),<math display="block" altimg="eq-00024.gif"><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><munderover accentunder="true" accent="true"><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced><mo>,</mo></math>(3.4)

where

$p_{n}$ is the nth prime. Then, inserting (3.3) and (3.4) into (3.1) produces

∞∑n=1cn=130+∞∑n=2((1tn+1tn+2)−1pn(1−n−1∏i=11pi)).<math display="block" altimg="eq-00026.gif"><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>0</mn></mrow></mfrac><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfenced separators="" open="(" close=")"><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><munderover accentunder="true" accent="true"><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>.</mo></math>(3.5)

Finally, inserting (3.4) and (3.5) into (3.2) leads to

∞∑n=1an=(12+∞∑n=21pn(1−n−1∏i=11pi))+(130+∞∑n=2((1tn+1tn+2)−1pn(1−n−1∏i=11pi))).<math display="block" altimg="eq-00027.gif"><mtable displaystyle="true"><mtr><mtd columnalign="right"><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></munderover><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></mtd><mtd columnalign="center"><mo>=</mo></mtd><mtd columnalign="left"><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><munderover accentunder="true" accent="true"><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd></mtd><mtd></mtd><mtd columnalign="left"><mo>+</mo><mspace width=".17em" class="thinspace"></mspace><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn><mn>0</mn></mrow></mfrac><mo>+</mo><munderover accentunder="true" accent="true"><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><mrow><mi>∞</mi></mrow></munderover><mfenced separators="" open="(" close=")"><mrow><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></mfrac><mfenced separators="" open="(" close=")"><mrow><mn>1</mn><mo>−</mo><munderover accentunder="true" accent="true"><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math>(3.6)

As we know, the series in the first bracket on the right side of (3.6) converges to 1 exactly [9,12]. Effectively, Brun series is now divided into two with one converging to 1. So, the computation of Brun’s constant centers on the other series. If the other series converges faster than Brun series, then an accelerated computation of Brun’s constant is achieved. Computed partial sums for (3.3), (3.4) and (3.5) and their relative increases in percentage (i.e. calculated using (2.3) and multiplied by 100) were tabulated in Table 1. The relative increases clearly indicate that the series in (3.5) converges roughly 60% faster than Brun series and Brun series in turn does 109% faster than that in (3.4). Since the partial sums for all the three series increase monotonously, the last row of the table gives the lower bound of 1.83220763 for Brun’s constant. This bound is based on the partial sum for (3.5) for

$N = 5, 173, 760, 785$ but still larger than Nicely’s 1.831808 for

$N = 19, 831, 847, 025, 792$ . Indeed, the larger lower bound is obtained with less computing resource. Apparently, better computing resource will push the bound higher.

**Table 1. The partial sums for three series $\sum b_{n}$ , $\sum c_{n}$ and $\sum a_{n}$ with increasing limit N of the index.**
N	$\sum_{n = 1}^{N} b_{n}$ (% increase)	$\sum_{n = 1}^{N} c_{n}$ (% increase)	$\sum_{n = 1}^{N} a_{n}$ (% increase)
1,224	0.93908471	0.73371487	1.67279958
8,169	0.95048409 (1.214)	0.76029284 (3.622)	1.710776937 (2.27)
58,980	0.95842751 (0.836)	0.77992953 (2.583)	1.73835704 (1.612)
440,312	0.96418520 (0.601)	0.79463042 (1.885)	1.75881562 (1.177)
3,424,506	0.96857750 (0.456)	0.80615846 (1.451)	1.77473596 (0.905)
27,412,679	0.97202728 (0.356)	0.815451225 (1.153)	1.78747850 (0.718)
224,376,048	0.97480478 (0.286)	0.82309953 (0.938)	1.79790431 (0.583)
986,222,314	0.97644450 (0.275)	0.82768879 (0.935)	1.80413329 (0.577)
1,870,585,220	0.97708771 (0.066)	0.82950471 (0.219)	1.80659242 (0.136)
3,552,770,943	0.97769721 (0.062)	0.83123384 (0.208)	1.80893105 (0.129)
5,173,760,785	0.97803919 (0.035)	0.83220763 (0.117)	1.81024682 (0.073)
$\infty$	1.0	>0.83220763	>1.83220763

4. Estimate for Brun’s Constant

To explore the possibility of establishing a mapping between the partial sums of Brun series and those of the modified prime harmonic series, the convergence of the latter relative to the former is plotted in Fig. 2.

It is seen in Fig. 2 that the relative convergence is close to a constant. So, we can establish an approximate mapping between the two partial sums. Mathematically, let $x = \sum_{n = 1}^{N} b_{n}$ and $y = \sum_{n = 1}^{N} a_{n}$ and consider that both x and y are real and continuous variables. Then, we have the following differential form like (2.4) :

d (ln y) \approx A d (ln x), <math display="block" altimg="eq-00039.gif"><mi>d</mi><mo stretchy="false">(</mo><mo>ln</mo><mi>y</mi><mo stretchy="false">)</mo><mo>\approx</mo><mi>A</mi><mi>d</mi><mo stretchy="false">(</mo><mo>ln</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo></math> (4.1)

where A is a real constant. Integrating (4.1) gives

ln y \approx A ln x + B, <math display="block" altimg="eq-00040.gif"><mo>ln</mo><mi>y</mi><mo>\approx</mo><mi>A</mi><mo>ln</mo><mi>x</mi><mo>+</mo><mi>B</mi><mo>,</mo></math> (4.2)

where B is another real constant. Solving (4.2) with the data of x and y for

$N = 1, 870, 585, 220$ and

$N = 3, 552, 770, 943$ gives

ln y \approx 2.07450287 ln x + 0.63952704 <math display="block" altimg="eq-00043.gif"><mo>ln</mo><mi>y</mi><mo>\approx</mo><mn>2</mn><mo>.</mo><mn>0</mn><mn>7</mn><mn>4</mn><mn>5</mn><mn>0</mn><mn>2</mn><mn>8</mn><mn>7</mn><mo>ln</mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>3</mn><mn>9</mn><mn>5</mn><mn>2</mn><mn>7</mn><mn>0</mn><mn>4</mn></math> (4.3)

y \approx exp (2.07450287 ln x + 0.63952704) . <math display="block" altimg="eq-00044.gif"><mi>y</mi><mo>\approx</mo><mo>exp</mo><mo stretchy="false">(</mo><mn>2</mn><mo>.</mo><mn>0</mn><mn>7</mn><mn>4</mn><mn>5</mn><mn>0</mn><mn>2</mn><mn>8</mn><mn>7</mn><mo>ln</mo><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>3</mn><mn>9</mn><mn>5</mn><mn>2</mn><mn>7</mn><mn>0</mn><mn>4</mn><mo stretchy="false">)</mo><mo>.</mo></math> (4.4)

Approximated values of y by (4.4) along with computed ones for N as small as 1,224 and as large as 5,173,760,785 are tabulated in Table 2. It is seen that the approximated values agree closely with computed ones, with at most 0.53% error for N as small as 1,224. The table also contains predicted Brun’s constant for N going to the infinity. This predicted value agrees with the most accurate estimate for Brun’s constant, known today.

**Table 2. Computed y and approximated y using computed x with increasing N.**
N	Computed x	Computed y	Approximated y (% error)
1,224	0.93908471	1.67279958	1.66386861 (−0.53)
3,424,506	0.96857750	1.77473596	1.77410282 (−0.036)
986,222,314	0.97644450	1.80413329	1.80412616 (−0.0004)
1,870,585,220	0.97708771	1.80659242	1.80659243
3,552,770,943	0.97769721	1.80893105	1.80893105
5,173,760,785	0.97803919	1.81024682	1.81024390 (−0.00016)
$\infty$	1.0(theory)	Impossible	1.89558413 (Unknown)

5. Conclusions and Discussions

A new rate of convergence of series is defined as the relative increase of the partial sum of the series. Computed data indicate that Brun series converges roughly 109% faster than the modified prime harmonic series. Also, the difference between the former and latter does 60% faster than the former itself and gives an improved lower bound for Brun’s constant with less computing resource. Using the rate in a differential form yields an approximate yet excellent mapping between the partial sum for Brun series and that for the modified prime harmonic series. The mapping involves logarithmic function and produces a new estimate for Brun’s constant.

Future work includes computing $\sum_{n = 1}^{N} c_{n}$ for larger N and finding an even better series like (3.4) for improving the lower bound and estimate for Brun’s constant.

Acknowledgments

The author thanks the reviewers for their expert reviews and helpful suggestions for improving the paper.

ORCID

Changhua Wan https://orcid.org/0000-0001-9208-3349

References

1. J. Bowman, Some computational results regarding the prime numbers below 2,000,000,000, Nordisk Tidskr. Informat. (BIT) 13 (1973) 242–244. Google Scholar
2. R. P. Brent, Tables concerning irregularities in the distribution of primes and twin primes up to $1 0^{11}$ , Math. Comput. 30 (1976) 379. Crossref, Google Scholar
3. V. Brun, La serie 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + 1/19 + 1/29 + 1/31 + 1/41 + 1/43 + 1/59 + 1/61 + …, les dénominateurs sont nombres premiers jumeaux est convergente oú finie, Bull. Sci. Math. 43 (1919) 124–128. Google Scholar
4. C.-E. Fröberg, On the sum of inverses of primes and twin primes, Nordisk Tidskr. Informat. (BIT) 1 (1961) 15–20. Google Scholar
5. H. Halberstam and H.-E. Richert, Sieve Methods (Academic Press, New York, 1983). Crossref, Google Scholar
6. T. R. Nicely, Enumeration to $1 0^{14}$ of the twin primes and Brun’s constant, Virginia J. Sci. 46 (1996) 195–204. Google Scholar
7. T. R. Nicely, A new error analysis for Brun’s constant, Virginia J. Sci. 52 (2001) 45–55. Google Scholar
8. P. Sebah and X. Gourdon, Introduction to twin primes and Brun’s constant computation, http://numbers.computation.free.fr/Constants/constants.html. Google Scholar
9. B. Selcoe, $Σ_{n > = 0}$ $a (n) / A 002110 (n + 1) = 1$ (2015), https://oeis.org/A005867. Google Scholar
10. E. S. Selmer, A special summation method in the theory of prime numbers and its application to ‘Brun’s Sum’, Nordisk Mat. Tidskr. 24 (1942) 74–81. Google Scholar
11. D. Shanks and J. W. Wrench, Brun’s constant, Math. Comput. 28 (1974) 293–299. Google Scholar
12. C. Wan, A class of convergent series, Math. Open 2 (2023) 2350002. Link, Google Scholar

Vol. 03

Metrics

Downloaded 993 times

History

Received 26 October 2023

Accepted 6 December 2023

Published: 24 January 2024

Information

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

Keywords

PDF download