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On digital distribution in some integer sequences

B. D. Craven
1965 Journal of the Australian Mathematical Society
Acknowledgement My thanks are due to the referee for some improvements in the presentation of this paper.  ...  Introduction Although the harmonic series diverges, there is a sense in which it "nearly converges". Let N denote the set of all positive integers, and S a subset of N.  ...  Craven  converges, will be called "harmonically convergent". The sum of the series (2) will then be called the "harmonic sum" of the sequence.  ...

Some open problems concerning the convergence of positive series [article]

Constantin P. Niculescu, Gabriel T. Prajitura
2012 arXiv   pre-print
We discuss some old results due to Abel and Olivier concerning the convergence of positive series and prove a set of necessary conditions involving convergence in density.  ...  Dusart  , shows that the speed of divergence of the series p=prime 1 p is comparable with that of 1 k(ln k+ln ln k) .  ...  A counterexample is provided by the series p=prime 1 p , of inverses of prime numbers, which is divergent (see  or  for a short argument).  ...

Are there Infinitely many Twin Primes?

Edigles Guedes, Raja Rama Gandhi, Srinivas Kishan Anapu
2013 Bulletin of Mathematical Sciences and Applications
We prove that are there infinitely many twin primes.  ...  This demonstrate that the series on the left diverges. Bulletin of Mathematical Sciences and Applications Vol. 5 Step 4.  ...  We prove that the series ∑ ( ) Volume 5 is divergent.  ...

95.06 The harmonic series revisited

J. A. Scott
2011 Mathematical Gazette
The harmonic series revisited In this note we review some algebraic proofs for the divergence of the harmonic series L~. Perhaps the standard proof is the one using groups of 2'terms.  ...  In other words, we have the contradiction Note finally that (although seemingly counter-intuitive) the divergence of the harmonic series may be used to establish divergence for two of its subseries, namely  ...

SOME VALUES OF THE ZETA FUNCTION

S. Zhunisbekov, Taraz State University, Alexandr Shevtsov, Taraz State University
2019 Theoretical & Applied Science
The Zeta-function If we substitute n = 1, we get a harmonic series that diverges.  ...  Introduction Harmonic series are a special case of a more General type of function called the Zeta function ζ(s). The real Zeta function is given for two real numbers r and n [1-2]: ..  ...

Euler and the partial sums of the prime harmonic series

Paul Pollack
2015 Elemente der Mathematik
In this note, we probe Euler's claim there that "the sum of the reciprocals of the prime numbers" is "as the logarithm" of the sum of the harmonic series.  ...  In a 1737 paper, Euler gave the first proof that the sum of the reciprocals of the prime numbers diverges.  ...  Finally, the author expresses his gratitude to the founders of the Euler Archive -Dominic Klyve, Lee Stemkoski, and Erik Toufor making Euler's collected works freely available; see http://www.eulerarchive  ...

The geometric series formula and its applications [article]

Cletus Bijalam Mbalida
2019 arXiv   pre-print
$\zeta(1)$ grows very slowly toward $\tilde\infty$, confirming the divergence of the harmonic series.  ...  By applying the geometric series formula above, it is further proved that the harmonic series $\zeta(1)$ is given by $\zeta(1)=-2\big[\log2+W_n(-\log2)\big]$ and as $n\rightarrow\pm\infty$, the value of  ...  Kofi Adanu, a researcher at the Alabama Transportation Institute, Dr. Larry Gratton and Dr. Jay Baltisberger, Professors at Berea College, for their advice and motivation throughout this work.  ...

On the series of prime reciprocals

James A. Clarkson
1966 Proceedings of the American Mathematical Society
Let pn be the rath prime. We give another proof of the Theorem. The series zZn-i (1/Pn) diverges. Proof. Assume the contrary, and fix k so that (l) E (l/AO < 1/2. Let Q = pip2 ■ ■ ■ pk.  ...  Thus for each r, Sir) = zZSir,j)<zZiU20<l. ) i So zZi-i [1/(1 +iQ)] converges, which in turn implies that the harmonic series does.  ...

Curious convergent series of integers with missing digits [article]

Melvyn B. Nathanson
2021 arXiv   pre-print
This result is extended to much larger families of "missing digits" sets of positive integers with both convergent and divergent harmonic series.  ...  A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges.  ...  This completes the proof. The converse of Lemma 2 is false. The set of prime numbers has density zero, but the sum of the reciprocals of the primes diverges. Theorem 1.  ...

Recreating the Alternating Harmonic Series after deducting a product equal to all the terms multiplied by the second term

Iain Preston
2021 figshare.com
A demonstration of how the Alternating Harmonic Series can be recreated after deducting from this series a product constructed of all the terms of the series, each multiplied by the second term (- 1/2  ...  ), depending on a) the method of deduction, and b) the effect of the Riemann's Series Theorem.  ...  Acknowledgements An expression of gratitude to all those, professionals and amateurs alike, who continue to seek to expand the knowledge of series.  ...

Recreating the Alternating Harmonic Series after deducting a product equal to all the terms multiplied by the second term

Iain Preston
2021 figshare.com
A demonstration of how the Alternating Harmonic Series can be recreated after deducting from this series a product constructed of all the terms of the series, each multiplied by the second term (- 1/2  ...  ), depending on a) the method of deduction, and b) the effect of the Riemann's Series Theorem.  ...  Acknowledgements An expression of gratitude to all those, professionals and amateurs alike, who continue to seek to expand the knowledge of series.  ...

Recreating the Alternating Harmonic Series after deducting a product equal to all the terms multiplied by the second term

Iain Preston
2021 figshare.com
A demonstration of how the Alternating Harmonic Series can be recreated after deducting from this series a product constructed of all the terms of the series, each multiplied by the second term (- 1/2  ...  ), depending on a) the method of deduction, and b) the effect of the Riemann's Series Theorem.  ...  Acknowledgements An expression of gratitude to all those, professionals and amateurs alike, who continue to seek to expand the knowledge of series.  ...

90.15 Square-freedom revisited

J. A. Scott
2006 Mathematical Gazette
, y are the cotangents of the angles of a triangle. The condition merely ensures that, for principal values of the inverse cotangent, A + B + C = nut, where m e {l,2}.  ...  112 THE MATHEMATICAL GAZETTE angles of a triangle, cot A cotfi cotC < cot A + cotfi + cotC, and that the condition V a/3 = 1 is not sufficient to ensure a, /?  ...  Acknowledgement Thanks to the anonymous referee for pointing out that the general formula derived in reference  may also be obtained using Dirichlet convolution.  ...

Valuations, arithmetic progressions, and prime numbers [article]

Shin-ichiro Seki
2018 arXiv   pre-print
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem  ...  The author would like to thank Junnosuke Koizumi for letting the author know Alpoge's work  .  ...  of the harmonic series.  ...

Valuations, arithmetic progressions, and prime numbers

Shin-ichiro Seki, Mathematical Institute, Tohoku University, Japan
2018 Notes on Number Theory and Discrete Mathematics
In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's Theorem  ...  The author is supported in part by the Grant-in-Aid for JSPS Fellows (JP16J01758 and JP18J00151), The Ministry of Education, Culture, Sports, Science and Technology, Japan.  ...  Acknowledgements The author would like to thank Junnosuke Koizumi for letting the author know Alpoge's work  .  ...
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