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On digital distribution in some integer sequences

1965
*
Journal of the Australian Mathematical Society
*

Acknowledgement My thanks are due to

doi:10.1017/s1446788700027750
fatcat:t5qbehkxhfblbi2iymrmid6n7y
*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 [2] converges, will be called "*harmonically*convergent".*The*sum*of**the**series*(2) will then be called*the*"*harmonic*sum"*of**the*sequence. ...##
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Some open problems concerning the convergence of positive series
[article]

2012
*
arXiv
*
pre-print

We discuss some old results due to Abel and Olivier concerning

arXiv:1201.5156v1
fatcat:fromuhwiwrdpbdj2qfiaewfa3m
*the*convergence*of*positive*series*and prove a set*of*necessary conditions involving convergence in density. ... Dusart [9] , 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 [3] or [10] for a short argument). ...##
###
Are there Infinitely many Twin Primes?

2013
*
Bulletin of Mathematical Sciences and Applications
*

We prove that are there infinitely many twin

doi:10.18052/www.scipress.com/bmsa.5.22
fatcat:jseyikpbffhglcxla2345g5nqu
*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

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

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

2015
*
Elemente der Mathematik
*

In this note, we probe Euler's claim there that "

doi:10.4171/em/268
fatcat:mdvnnhnpovdovbmxipnuhhwtoa
*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]

2019
*
arXiv
*
pre-print

$\zeta(1)$ grows very slowly toward $\tilde\infty$, confirming

arXiv:1909.10317v1
fatcat:77noxsk37jc3bamje2g266lbaq
*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

1966
*
Proceedings of the American Mathematical Society
*

Let pn be

doi:10.1090/s0002-9939-1966-0188132-7
fatcat:jf6kcbk2yrc7tn4der5wyjwh7a
*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]

2021
*
arXiv
*
pre-print

This result is extended to much larger families

arXiv:2006.13345v5
fatcat:mqaznmetabc4vclornkwapg4ty
*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

2021
*
figshare.com
*

A demonstration

doi:10.6084/m9.figshare.14959317.v3
fatcat:mrxcznjxzngxbehlhko4kj453m
*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

2021
*
figshare.com
*

A demonstration

doi:10.6084/m9.figshare.14959317.v2
fatcat:5eajo3l67fbs3beyx6ufkvixs4
*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

2021
*
figshare.com
*

A demonstration

doi:10.6084/m9.figshare.14959317.v1
fatcat:zgjxf6kbs5d3rdne5ijm232vhe
*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

2006
*
Mathematical Gazette
*

, y are

doi:10.1017/s0025557200179185
fatcat:wkukszw2gze4nmcs5ab2bskdqi
*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 [1] may also be obtained using Dirichlet convolution. ...##
###
Valuations, arithmetic progressions, and prime numbers
[article]

2018
*
arXiv
*
pre-print

In this short note, we give two proofs

arXiv:1708.08085v2
fatcat:s4fei522abef5gmg6mnd35izm4
*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 [1] . ...*of**the**harmonic**series*. ...##
###
Valuations, arithmetic progressions, and prime numbers

2018
*
Notes on Number Theory and Discrete Mathematics
*

In this short note, we give two proofs

doi:10.7546/nntdm.2018.24.4.128-132
fatcat:v252nv3oybdazgqc556b57fkdu
*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 [1] . ...
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