A new conjecture on the divisor summatory function offering a much higher prediction accuracy than Dirichlet's divisor problem approach * Wiki-like transdisciplinary article (Open development interval: 2018 - ?) - working paper preprint[1] Version 1.0 (17.06.2018) * Andrei-Lucian Drăgoi[2] (version 1.0 published on: 17.06.2018; last update on: 17.06.2018) Pediatrician specialist with both interdisciplinary and transdisciplinary independent research in physical sciences (Bucharest, Romania), E-mail:
[email protected] * Important note: The latest (free) version of this article can be downloaded from this URL. * Abstract
This paper presents a new conjecture on the divisor summatory function (also in relation with prime numbers), offering a much higher prediction accuracy than Dirichlet's divisor problem approach. Keywords: conjecture; divisor function; divisor summatory function; prime numbers; Dirichlet's divisor problem Important note (1). This atypical URL-rich paper (which maximally exploits the layer of hyperlinks in this document), chooses to use Wikipedia links for all the important terms used. The main motivation for this approach was that each Wikipedia web-article contains all the main reference (included as endnotes) on the most important terms used in this paper: it’s simply the most practical way to cite entire collections of important articles/books without using an overwhelming list of footnote/endnote references. The secondary motivation (for using Wikipedia hyperlinks directly included in keywords) was to assure a “click-away“ distance to short encyclopedic monographs on all the (important) terms used in this paper, so that the flow of reading to be minimally interrupted. Important note (2). This paper also exploits the advantages of the hierarchic tree-like model of presenting informational content which is very easy to be kept updated and well organized.
[1] Online paper preprint DOI: 10.13140/RG.2.2.19770.49603 [2] Andrei-Lucian Dragoi research pages on: ResearchGate, Academia.edu, ViXra, GSJournal;
2 I.
Introduction
1) Introduction to the divisor function. From the number theory literature and given a matrix 1 1 2 2 3 2 Mn = which counts (on its 2nd column) the number of (trivial plus non-trivial) positive 4 3 ⋮ ⋮ n dn divisors ( d n ) for each (non-zero) natural number n ≥ 1 (A000005 OIES sequence), d n (in simplified notation and usually noted d ( n ) or τ ( n ) ; aka “the divisor function” d ( n ) = (ν 1 + 1)(ν 2 + 1) ... (ν k + 1) with νk
ν 1 ,ν 2 ...ν k being the exponents of the prime factorization[URL2] of n = p1ν1 p2ν 2 ... pk
k
νj
= ∏ pj
; also known
j =1
in sigma notation as the special case σ 0 ( n ) = ∑ d 0 = d ( n ) of σ x ( n ) = ∑ d x d |n
[URL]
with x being a real
d |n
or complex number) appears in a number of remarkable identities (including relationships on the Riemann zeta function and the Eisenstein series of modular forms) and has some well-known properties like [URL1, URL2, URL3, URL4] :
a)
d n < 2 n , for any (non-zero) natural number n ≥ 1 ; n
b) In 1838, P. G. L Dirichlet showed that the (arithmetic) average number of divisors d n( av ) = ∑ d k / n k =1
has the property d n( av ) ≅ d n ( av )( D ) = ln ( n ) + ( 2γ − 1) n
(
function (DSF) Sn = ∑ d k = n ⋅ d n ( av ) k =1
n
)
[URL-MathWorld]
and that divisor summatory
(with the simplified notation S n replacing the standard
notation of DSF D(n) = ∑ d (k ) ) has the property that Sn( D ) = n ln ( n ) + ( 2γ − 1) + O nθ k =1
( )
for any
natural number n ≥ 1 (this “ S n ” predicted by Dirichlet was abbreviated as S n( D ) or “DSn” so that to be distinguished from Sn and to be compared with the other predicted S n( pr ) proposed by the conjecture presented in this paper) [URL1, URL2, URL3], with the following explanations, definitions and notations: i) Euler–Mascheroni (gamma) constant γ (the limiting difference between the harmonic series and ∞ 1 n −1 the natural logarithm) is predefined as γ = lim ∑ k − ln ( n ) = ∫ − 1/ x ≅ 0.5772 n→∞ k =1 1 floor ( x) (with floor function floor ( x) predefined as the greatest integer i ≤ x and x being a real number)
3 ii) the big O (Bachmann–Landau or asymptotic) notation is predefined such as: f ( x) ∈ O( g ( x )) or f ( x) = O( g ( x)) if and only if there exists both a real constant c > 0 and a finite real number x0 , such that f ( x) ≤ c ⋅ g ( x ) for any real number x ≥ x0
( )
iii) the theta exponent (θ ) of non-leading term O nθ of DSn is the “target” of Dirichlet's divisor problem (DDP) which is: to find the smallest value of θ (noted θ min ) for which
(θ
Sn − n ln ( n ) + ( 2γ − 1) = O n
min +t
)
for any t > 0 . Until present, it is widely conjectured that
conj.
θ min = 1 / 4 and θ min was already formally demonstrated to be in double closed interval ?
[1 / 4, 131 / 416(≅ 0.3149)] by M.N. Huxley in 2003. DDP is one of the major arithmetical problems still unsolved up to present, but the new conjecture presented in this paper offers a practical alternative method to approximate S n (also in relation with prime numbers), independently to DSn and its predicted S n( D ) . iv) One consequence of S n( D ) definition is that a randomly chosen (non-zero) natural number n ≥ 1 has an expected number of divisors
(
)
d n ≅ d n ( av ) ≅ ln ( n ) + ( 2γ − 1) ≅ ln ( n ) < 2 n , which
n
implies that Sn ( D ) ≅ ∑ ln(k ) ≅ ln(n !) . The graph of the ratio d n / ln( n) ≅ 1 (with red linear trend k =1
line
added)
and
the
graph of
its
absolute
error
(in
base-10
logarithmic
log10 1 − d n / ln( n) (which indicates an interesting oscillating accuracy of the
approximation
d n / ln( n) ≅ 1 ) are presented next. 4 3.5 3 2.5
d n / ln( n)
2
1.5 1 0.5 0 n
scale)
n
Figure Intro-1a. The values of the ratio d n / ln( n) for the natural number n ∈ 1,10 4 .
4
1
0 1
607 1213 1819 2425 3031 3637 4243 4849 5455 6061 6667 7273 7879 8485 9091 9697
-1
log10 1 − d n / ln(n) -2
-3
-4
-5 n
Figure Intro-1b. The values of log10 1 − d n / ln( n) for the natural number n ∈ 1,10 4 .
***
5 II.
Some new conjectures on the divisor function
1) New conjectures on the divisor function. The author of this paper discovered some new conjectures n
on Sn = ∑ d k with relatively high accuracy in predicting S n and which can be used as alternatives to k =1
Dirichlet’s DSn: see next. 2
e Sn / n 2) Conjecture no. 1 (C1). En = ≅ 1 , an approximate equality which becomes progressively more 3 n exact with the growth of the natural number n ≥ 1 to infinity. C1 additionally states that En < 1 for any natural number n ≥ 1082 . See the next graph.
1.6 1.5 1.4 1.3
En
S /n n =e 3 n
2 1.2 1.1 1 0.9 1
915
1829 2743 3657 4571 5485 6399 7313 8227 9141 n
Figure C1-1a. The values of En for the natural numbers n ∈ 1,10 4 a) Interestingly, the values of the function log10 1 − En
(which measures the closeness of En ( ≅ 1) to
value 1: the absolute error measured logarithmically), tends to stabilize its values very close to -1.5 so that En ≅ 1 − 10−3/2 ≅ 0.968 for n ≥ 1082 : see the next graph.
6
n 0 1
657 1313 1969 2625 3281 3937 4593 5249 5905 6561 7217 7873 8529 9185 9841
-1
-2
-3
log10 1− En
-4
-5
-6
-7
Figure C1-1b. The values of log10 1 − En for the natural numbers n ∈ 1,10 4 3) Actually, the value x = 1 − 10−3/2 ≅ 0.968 appears as the real “target” around which En tends to stabilize: see the next graph of the function log10 En − x . 0 1
586 1171 1756 2341 2926 3511 4096 4681 5266 5851 6436 7021 7606 8191 8776 9361 9946
-1
-2
-3
log10 En − x -4
-5
-6
-7 n
Figure C1-1c. The values of log10 En − x for the natural numbers n ∈ 1,10 4
7
a) Redefinition (1). Based on the progressive decrease of log10 En − x , C1 can be refined and rewritten 2
e Sn / n −3/2 as En = ≅ 0.968 . Note. C1 allows to rapidly predict (with relative high ≅ x = 1 − 10 3 n
(
)
(
)
accuracy) the value of S n for any n ≥ 1 , such as Sn ≅ n ln 3 xn . Defining the predicted (pr)
(
)
Sn ( pr ) = n ln 3 xn , the ratio
Sn ( pr ) / Sn ≅ 1 is graphed below. The absolute error measured
logarithmically as log10 1 − Sn ( pr ) / S n is also graphed below (with a red linear trend line added):
from this (second) graph, one can observe that log10 1 − S n( pr ) / S n ≅ − log10 (n) , which is equivalent to 1 − Sn ( pr ) / Sn ≅ 1 / n and S n ( pr ) / S n ≅ 1 − 1 / n (as also seen from the graph of Sn ( pr ) / Sn ≅ 1 ) 1.1 1.08 1.06 1.04 1.02
Sn ( pr ) Sn
1 0.98 0.96 0.94 0.92 0.9 1
656 1311 1966 2621 3276 3931 4586 5241 5896 6551 7206 7861 8516 9171 9826 n
Figure C1-1d. The values of the ratio Sn ( pr ) / Sn ≅ 1 for the natural numbers n ∈ 1,10 4
8
n
0 n -1
-2
-3
log10 1 − S n ( pr ) / S n -4
-5
-6
-7
-8
Figure C1-1e. The values of log10 1 − Sn ( pr ) / S n for the natural numbers n ∈ 1,10 4 b) Redefinition (2). Based on the additional property Sn ( pr ) / Sn ≅ 1 − 1 / n (which is equivalent to
(
)
S n ≅ Sn ( pr ) / (1 − 1 / n ) ), C1 can be refined as Sn ≅ n ln 3 xn / (1 − 1 / n ) , with x = 1 − 10−3/2 ≅ 0.968
(
)
and natural number n ≥ 1 . (Re)defining Sn ( pr 2) = n ln 3 xn / (1 − 1 / n ) , the ratio Sn ( pr 2) / S n ≅ 1 and its associated log10 1 − S n ( pr 2) / Sn is graphed next.
9
1.01
1.005
1
S n ( pr 2) / S n
0.995
0.99 1
672 1343 2014 2685 3356 4027 4698 5369 6040 6711 7382 8053 8724 9395 n
Figure C1-2a. The values of the ratio Sn ( pr 2) / S n ≅ 1 for the natural numbers n ∈ 1,10 4 . n
0 1
716 1431 2146 2861 3576 4291 5006 5721 6436 7151 7866 8581 9296
-1 -2 -3
log10 1 −
S n ( pr 2) -4
Sn
-5 -6 -7 -8
Figure C1-2b. The values of log10 1 − S n ( pr 2) / Sn for the natural numbers n ∈ 1,104 . c) Redefinition
(3).
The
graph
log10 1 − Sn ( pr 2) / S n ≅ − log10 (n) ,
of
log10 1 − S n ( pr 2) / Sn
which
is
(
equivalent
one to
can
also
observe
1 − Sn ( pr 2) / S n ≅ 1 / n
that and
)
S n ( pr 2) / S n ≅ 1 − 1 / n : this implies that n ln 3 xn / (1 − 1 / n ) ≅ S n (1 − 1 / n ) , which is equivalent to
(
)
Sn ≅ n ln 3 xn / (1 − 1 / n )
2
so
that
a
predicted
S n ( pr 3)
can
be
further
refined
as
10
( ) Sn (sup) = n ln ( 3 n ) / (1 − 1 / n )
2
S n ( pr 3) = n ln 3 xn / (1 − 1 / n ) .
(
2
C1
additionally
states
that
the
function
is a superior limit for S n for any natural number n ≥ 1082 , so that >
)
Sn = O Sn (sup) : see the next graph of the ratio Sn (sup) / S n ≅ 1 . ( n ≥1082)
1.009
1.007
1.005
Sn (sup) / Sn ≅ 1 1.003
1.001
0.999 1
661 1321 1981 2641 3301 3961 4621 5281 5941 6601 7261 7921 8581 9241 9901 n
Figure C1-3a. The values of the ratio Sn (sup) / S n ≅ 1 for the natural numbers n ∈ 1,10 4 . d) Redefinition (4). S n( pr 3) supports further refining with even higher accuracies, by using an
(
)
“accessory” function f (n) = n − n /ln( n ) , so that Sn ( pr 4) = n ln 3 xn / (1 − 1 / n )
2− f ( n )
(
.
)
e) The predicted (arithmetic) average number of divisors d n ( av )( pr 3) = S n ( pr 3) / n = ln 3 xn / (1 − 1 / n )
(
)
and d n ( av )( pr 4) = S n ( pr 4) / n = ln 3 xn / (1 − 1 / n )
2− f ( n )
2
(with f (n) = n − n /ln( n ) ) can be compared with
DSn prediction d n( av )( D ) = ln ( n ) + ( 2γ − 1) . For example, d n (av )( pr 3) generates much more accurate n
predictions for
d n( av ) = ∑ d k / n = Sn / n k =1
than
d n( av )( D )
does: for comparison, the ratios
d n ( av )( pr 3) / d n ( av ) and d n ( av )( D ) / d n ( av ) are graphed next in red and blue respectively.
11
1.8 1.7
d n ( av )( D ) / d n ( av )
1.6 1.5 1.4 1.3 1.2 1.1 1
d n( av )( pr 3) / d n ( av )
0.9 0.8 1
574 1147 1720 2293 2866 3439 4012 4585 5158 5731 6304 6877 7450 8023 8596 9169 9742 n
Figure C1-4. A comparison between the ratios d n( av )( pr 3) / d n ( av ) (in red) and d n( av )( D ) / d n( av ) (in blue), to
emphasize the much higher accuracy of C1 when compared to DSn in predicting d n( av ) for the natural numbers n ∈ 1,10 4 . f) Remark. C1 is also indirectly related to the prime-number theorem because an important element which “slows down” the progressive growth of Sn (and the growth of the exponential En implicitly, which is conjectured to remain subunitary for any n ≥ 1082 ) is the frequency of prime numbers (a frequency mainly defined by the prime number theorem as n / Pn ≅ 1 / ln( Pn ) , also based on the prime-
counting function Pn , usually noted π (n) ) which primes (p) all have d p = 2 , a d p value which acts like a “brake” and slowing the growth of S n and En implicitly.
4) Final conclusion. Conjecture 1 (C1) has a major advantage of Dirichlet’s estimation of d n ( av ) (DSn) (and Sn = n ⋅ d n ( av )
implicitly), as C1 predicts d n ( av ) (and Sn implicitly) with much higher accuracy:
(
)
d n ( av )( pr 3) = S n ( pr 3) / n = ln 3 xn / (1 − 1 / n )
with f (n) = n − n /ln( n ) .
2
and
(
)
d n ( av )( pr 4) = S n ( pr 4) / n = ln 3 xn / (1 − 1 / n )
2− f ( n )
,
