Research Journal of Mathematics and Statistic 3(1): 39-44, 2011
ISSN: 2040-7505
© Maxwell Scientific Organization, 2011
Received: November 08, 2010
Accepted: December 02, 2010
Published: February 15, 2011
Using Infinite Series Convergence to Prove the Riemann Hypothesis
M.V. Atovigba
Department of Curriculum and Teaching, Benue State University,
P.M.B. 102119 Makurdi Nigeria
Abstract: The study attempts to prove the Riemann hypothesis by showing that the Riemann zeta function
converges to zero and absolutely converges to 0. The work uses the mean and mean deviation approach of
solving quadratic equations, in identifying the roots of an auxiliary energy equation of some second order
homogeneous ordinary differential equation. The energy equation is brought about by simultaneously treating
energy generation by electrons and quarks. A set of complex roots of the auxiliary equation takes the form of
the set of analytic zeros s = ½ + it of the Riemann zeta function H(s). s varies partly as ½ and partly as some
t, t being a function of the mass (m) of particles and k distance (or frequency) achieved by the particle. The
Riemann zeta function is treated as an infinite series which converges to 0 as t tends to infinity and which
converges absolutely to 0 as t tends to infinity.
Key words: Absolute convergence, convergence, critical line, distance, electrons, energy, mass, mean and
mean deviation approach, quarks, Riemann zeta function, speed, zero
whose term is a fraction and attempts to treat the series to
varying values of s, noting particularly that s depends on
t, hence it is actually t that varies as first entries to
produce values of the Riemann zeta function for any n
positive integer. The work studies the behavior of H(s) for
any n = 1, 2, 3, … as t varies, the aim being to locate if
the series tends to zero for the general term n provided t
tends to infinity.
Selberg, in an interview with a web group, Not Even
Wrong (2008), about attempts by him and other notable
mathematicians to solve the Riemann hypothesis,
emphasizes the importance of simplicity in mathematics
and notes that the simple ideas are the ones that will
survive, and that sometimes a good idea does not work,
and what seems like a bad, even idiotic idea may actually
work. This researcher shares Selberg’s view of simplicity
of ideas and goes ahead to adopt sheer fraction analysis to
see if the series that is the Riemann zeta function will
converge to zero.
Proof of the Riemann hypothesis is deemed to be
necessary in order to clear the backlog of problems
considered to be central to mathematics and scientific
research. Hilbert’s address in Paris (1900) emphasizes the
centrality of the Riemann hypothesis to mathematics
when he notes that:
INTRODUCTION
The Riemann Hypothesis states that:
H(s) = 0
(1)
such that the non-trivial zeros of the Riemann zetafunction H(s) must lie on the critical line:
s = ½ +it
(2)
t being any real number, i imaginary number and s …1.
The Wikipedia (2010a) notes that the Riemann
hypothesis is part of Hilbert’s list of 23 unsolved
problems and also one of the Clay Mathematics Institute
Millennium Prize Problems, and avails of suggestions by
Hilbert and Polya that one way to derive the Riemann
hypothesis would be to find a self-adjoint operator from
the existence of which the statement on the real parts of
the zeros of the zeta function would follow when one
applies the criterion on real eigenvalues. This work
similarly bases its origin on some attributes of electrons
and quarks, to provide a quadratic system whose set of
complex roots describe the set of analytic zeros s = ½ + it
of the Riemann zeta function H(s).
Constant (2008 ) in trying to prove the Riemann
Hypothesis remarks that the infinite series can be made to
be zero by finding the limits of the entire series or by
finding that each term is equal to zero. Constant’s
observation is relevant to this work as the work attempts
to simplify the Riemann hypothesis as a series each of
As soon as this proof has been successfully
established, the next problem would consist in testing
more exactly Riemann's infinite series for the number
of primes below a given number and, especially, to
decide whether the difference between the number of
primes below a number x and the integral logarithm
39
Res. J. Math. Stat., 3(1): 39-44, 2011
of x does in fact become infinite of an order not greater
than 1/2 in x. Further, we should determine whether the
occasional condensation of prime numbers which has
been noticed in counting primes is really due to those
terms of Riemann's formula which depend upon the first
complex zeros of the function H(s). After an exhaustive
discussion of Riemann's prime number formula, perhaps
we may sometime be in a position to attempt the rigorous
solution of Goldbach's problem, viz., whether every
integer is expressible as the sum of two positive prime
numbers; and further to attack the well-known question,
whether there are an infinite number of pairs of prime
numbers with the difference 2, or even the more general
problem, whether the linear diophantine equation ax + by
+ c = 0 (with given integral coefficients each prime to the
others) is always solvable in prime numbers x and y.
Hilbert’s address thus demands some urgency for
resolving the Riemann hypothesis which will ultimately
pave the way for further resolutions of other mathematical
problems.
The objective of this study is to provide a simple
proof of the Riemann hypothesis. Specifically, the work
aims at proving that the Riemann zeta function H(s)
converges to zero or absolutely converges to zero for any
n = 1, 2, 3,….
r2 =
1
k 1
+i
−
2
m 4
(3)
of the auxiliary energy equation of a second order
homogenous ordinary differential equation, K standing for
distance travelled by (or frequency of) a particle of m
mass, suggested by Atovigba on the Google Epistemology
Group (2010), which was further investigated by
Atovigba and Chiawa (2010).
The set of complex roots (3) is found using the mean
and mean deviation approach of solving quadratic
equations (Atovigba, 2010) who in his masters in
mathematics education dissertation posits that the roots
of the quadratic equation:
a2x2 +a1x + a0 = 0
(4)
could be expressed as:
r1 = M − D ⎞
⎟
r2 = M + D⎠
(5)
where M is the mean of the roots and D the mean
deviation of each root from M such that:
MATERIALS AND METHODS
D=
This research was carried out between September 5,
2010 and November 5, 2010 at Benue State University
Makurdi, Nigeria while researching and teaching
Mathematics Methods 1.
C
Energy generated by spins of electrons and quarks,
used simultaneously to obtain an auxiliary energy
(quadratic) equation
Mean and mean deviation approach of finding the
zeros of the quadratic equation, which produces
complex (analytic) zeros at some frequencies of the
subatomic particles.
The method used in proving the Riemann hypothesis
is the method of convergence and absolute convergence
of infinite series.
To prove the hypothesis requires, firstly, to identify
some phenomena which generates some complex operator
or the set of analytic zeros
s = ½ +it
a0
a2
(6)
To obtain (3), Atovigba and Chiawa (2010) adopts
submissions by Fitzpatrick (2006) that frequency is
synonymous with energy and that the higher the
frequency then the higher the energy, (and further, that)
the frequency of the quark spin is the square of the spin
frequency of the electron's spin and from this is derived c2
, which makes energy E = mc for electrons and E = mc2
for quarks, where m stands for mass of particles and c
speed of light. Hence, by simultaneous solution, mc =
mc2, which leads to the quadratic equation mc2 - mc = 0
(or mc - mc2 = 0) whose left side is considered to be the
auxiliary energy equation of a particle with mass m
dependent on c as:
The materials used in this research are:
C
M2 −
E(c) = mc2 – mc
(7)
which is the trace of E passing through the origin
P(0,0) and is parabolic (Fig. A, Appendix A) and cuts a
whole host of infinite horizontal planes each of k
perpendicular distance from the plane of origin, say k = 0.
Thus there are infinite planes of variable k distance from
the plane of origin which the electron or quark or atom or
molecule has to travel through. Hence each time E
(2)
as first entries for the Riemann zeta-function. The set of
analytic zeros (2) is adopted from the second complex
solution
40
Res. J. Math. Stat., 3(1): 39-44, 2011
∞
reaches a plane of k distance from the origin, E = k which
makes (7) to become
∑
n− s =
1
E(c) = mc2 – mc = k
1
k 1
= ±i
−
2
m 4
∞
=
k 1
− =t
m 4
∑
∞
(10)
Lim
t→ 0
ς ( s) = 1 + 2 + 3 + .. =
∞
∑n
Lim
t→ 0
−s
=0
∑
1
1
2
1
( )
i ⎞⎟
⎟ =
⎠
⎛ n− t
⎜
⎜ 1
⎝ n2
−
∞
∑
1
⎛ ⎛ 1 ⎞i⎞
⎜⎜ t⎟ ⎟
⎜⎝n ⎠ ⎟
1 ⎟
⎜
⎜ n2 ⎟
⎝
⎠
(13)
n − s = Lim
t→ 0
1
∞
∑
1
⎛ ⎛ 1 ⎞i⎞
⎜⎜ t⎟ ⎟
⎜⎝n ⎠ ⎟
1 ⎟ = 0
⎜
⎜ n2 ⎟
⎝
⎠
(14)
∑n
1
−s
∞
= Lim
t→ 0
∑
1
⎛ ⎛ 1 ⎞i⎞
⎜⎜ t⎟ ⎟
⎜⎝n ⎠ ⎟
1 ⎟ = 0
⎜
⎜ n2 ⎟
⎝
⎠
The Riemann zeta function thus converges absolutely
as can be shown using the ratio test which states that for
the infinite series un to converge absolutely:
(11)
to prove that H(s) = 0 is to prove that H(s) converges to 0,
and this can be done by taking the limit of H(s) as t ÷4.
This is because for any n the system could be fed with any
t a real number, hence the series is of the type un(t).
Specifically, what is required is the proof that:
∑n
∞
∑
n
Absolute convergence of the Riemann zeta function:
Irrespective of the magnitude of n = 1, 2, 3, … , as t ÷4,
the Riemann zeta function absolutely converges as:
1
t→ 0
∑
∞
−s
∞
=
which concludes the proof.
Convergence of the Riemann zeta function:
Given the Riemann zeta function:
−s
1
− − it
2
1
⎛ − it ⎞
⎜n ⎟
⎜ 1 ⎟ =
⎝ n2 ⎠
RESULTS
∞
∑
n
Consequently, as t ÷4, 1/ni ÷ 0 so that (1/ni)i ÷ 0 so that
(1/ni)i / n1/2 ÷ 0.
Hence:
we have (2) as the required complex operator which
is fed into the Riemann zeta function in order to prove the
Riemann Hypothesis, that is, that H(s) = 0.
The Riemann zeta function is an infinite series which
is treated for convergence and absolute convergence as t
gets to infinity having realized that the Riemann zeta
function depends on s, and s further depends on t.
Lim
∞
=
(9)
from where the positive root (3) is obtained. Re-writing
(3) with r1 = s and
−s
1
− ( + it )
2
1
1
−s
∑
n
(8)
which ultimately produces the pair of roots:
r1,2
∞
Lim
n→ ∞
un + 1
=r<1
un
according to Spiegel (2005) who explains that absolute
convergence can be extended to the case where un are
functions of some other variable (say t) denoted by un(t)
when convergence depends on the particular values of t
which could be a set of values of t called the region of
convergence R. Thus absolute convergence of the
Riemann zeta function takes place when:
(12)
1
This can be done by taking the limit as t ÷4; and any of
the two first entries: m mass of a particle, or k distance
travelled, could vary in (10) such that t ÷ 4. Thus:
Lim
t→ ∞
un+ 1 (t )
=r<1
un (t )
The Riemann zeta function H(s) = H (1/2 + it) as an
C
if m is variant with k … 0 and m ÷0, then k/m ÷4,
hence t ÷4.
C if m is constant but moves such that k then t .
Now,
∞
infinite series could be written as un (t ) =
∑n
t =1
41
⎛1 ⎞
− ⎜ + it ⎟
⎝2 ⎠
Res. J. Math. Stat., 3(1): 39-44, 2011
which for the general term converges to 0 when t ÷4 and
converges absolutely to 0 as demonstrated by letting
n = 1 and n + 1 = 2. If n = 1 then:
un (t ) =
⎛ 1⎞
⎜ t⎟
⎝n ⎠
i
1
n2
⎛ 1⎞
⎜ t⎟
⎝1 ⎠
=
involving n steps of complexity. Thus available heat
energy E of such a system at any time t sec is E = H H(s),
and when t tends towards infinity then entropy is
maximized so that there is zero amount of heat available
as an equilibrium state for the machine.
The foregoing remarks relating the Riemann zeta
function to entropy has implications for time as for
isolated machines entropy never decreases but tends to
increase (Wikipedia 2010b), a property that prohibits
perpetual motion machines and suggests an arrow of time,
and this prohibition is seen when E = H .(s) = 0 when
time becomes infinite. Similarly, the findings indicate an
idea of time in that, given the same (concentric) distance
k but different particles or objects of mp masses (p = 1, 2,
3, …), for bodies of larger masses k tends to be negligible
and hence time t tends to be negligible, while the reverse
is the case for bodies of smaller masses. This may be a
way to explain why the average year of planets nearer to
their sun are shorter than that for planets farther away
from the sun.
The Riemann zeta function’s analytic zeros s are such
that within the microscopic radii k = , > 0 a particle of
mass m at any time t attains some speed s = ½ + it of
magnitude:
i
1
12
And for n + 1 = 2,
un+ 1 (t ) =
⎛ 1⎞
⎜ t⎟
⎝n ⎠
i
1
n2
=
⎛ 1⎞
⎜ t⎟
⎝2 ⎠
i
1
22
Consequently,
⎧ ⎛ 1⎞ i ⎫
⎪⎜ ⎟ ⎪
⎪ ⎝ 2⎠ ⎪
⎧ i⎫
⎨ 1 ⎬
⎪ (0) ⎪
⎨ 1 ⎬
⎪ 22 ⎪
⎪ 22 ⎪
⎪⎩
⎪⎭
un + 1 (t )
Lim
= Lim
= ⎩ i ⎭ =
i⎫
t → ∞ un ( t )
t→ ∞ ⎧
⎧1 ⎫
⎛ 1⎞
⎨ ⎬
⎪⎜ t ⎟ ⎪
⎪⎝1 ⎠ ⎪
⎩ 1⎭
⎨ 1 ⎬
⎪ 12 ⎪
⎪⎩
⎪⎭
⎧ ⎫
⎪ 0 ⎪
⎨ 1⎬
⎪⎩ 2 2 ⎪⎭
0
=
= 0< 1
1
{1}
2
s=
DISCUSSION
horizontal) and variable hypotenuse s =
Let t stand for time in t seconds, then
∞
t→ 0
∑n
−s
=0
1 ⎛ k 1⎞
+⎜ − ⎟ =
4 ⎝ m 4⎠
k
m
This implies that particles have their origin at some
point P(0,0) but can reach frequencies that stretch to
infinitely within the open ended right-angled triangular
confinement of fixed base 1/2 unit and opposite variable
side t. t is opposite the angle that s makes with the
Thus the Riemann zeta function is absolutely
convergent and actually converges to 0, which proves the
hypothesis.
Lim
⎛ 1⎞
2
⎜ ⎟ +t =
⎝ 2⎠
k
(Fig. A).
m
This relation could be re-arranged so that k = ± m|s|,
which is some form of momentum since has direction
(red arrows in Fig. A) and is hence some velocity v, and
can be written as k = ± m|s| = mv1 ± mv2 without
contradiction as the particle started from the origin P(0,0)
at which point k = 0 hence v1 = |s|1= 0 hence mv1 = 0.
The findings can be used to investigate particle
annihilation. For instance, when an electron and a
positron collide they both get annihilated into two bosons
each of zero masses such as gamma ray photons
(Wikipedia, 2010c). Thus at annihilation, for any small
k = , > 0 t tends to infinity thus making the zeta function
to collapse to zero as proved in (14) thus indicating
further that the Riemann hypothesis demonstrates that
particle annihilation brings about 100% entropy. This
further implies that bosons do not produce energy but
cause energy entropy since they have zero mass which
makes the Riemann zeta function to tend to zero.
implies that for any n and time t
1
seconds, the Riemann zeta function supports any entropy
theory which describes diminishing values of a measure
of energy, say heat, available to a system which could
diminish until it gets to the zero level after some time t
seconds. Entropy is a measure of the energy not available
for work in a thermodynamic process (Wikipedia, 2010b)
and the system could consequently reach an equilibrium
state (when the system no longer experiences change)
when entropy is said to be maximized, because all
information about the initial conditions of the system is
lost except for the conserved variables. Suppose that H
stands for heat available to a machine, then the product of
H and the Riemann zeta function .(s) will guarantee
amount of heat available to the system at any time t
seconds, depending on the complexity of the machine
42
Res. J. Math. Stat., 3(1): 39-44, 2011
The findings present a distance-mass value for time
as time t seconds is given by:
The findings have educational value for providing
added content and method in estimating energies of
particles and levels of entropy towards an enriched the
mathematics and science education curriculum.
k 1
− =t
m 4
CONCLUSION
Thus time can be expressed as a real-valued function t of
k distance and m mass such that:
t = t(k,m):
The research aimed at proving the Riemann
hypothesis by showing that the Riemann zeta function as
an infinite series sums to zero (0). Thus the most
significant findings of this research is that the Riemann
zeta function H(s) has been found to sum to zero (0) using
infinite series convergence and absolute convergence.
Among the unique findings of the research, the set of
analytic zeros s = ½ + it of the Riemann zeta function is
obtained from the pair of complex zeros of some energy
equation derived from simultaneously combining masses
and frequencies of electrons and quarks, which implies
that the Riemann zeta function could be used to solve
problems involving subatomic particles.
The research findings indicate that the Riemann zeta
function is a suitable factor in any entropy theory.
The findings also indicate that time t depends on
mass m of the particle and distance k from the origin or
point of reference and that time can progress to infinity.
k 1
−
m 4
which, for m given, can assume an infinite value if k
increases to infinity. Hence bodies that are bonded will
share the same idea of time given the same reference
point k (say, the sun) from their point of origin P(0,0).
Time for such bonded bodies of mi masses will then be:
t=
1
k
−
mi 4
∑
1
The findings thus suggest that time hence has nothing to
do with a particle’s speed but everything to do with
distance and the mass in view.
Energy (E)
Appendix: A
ot
Ro
Ki
1
K3
K2
θ
K1
P(0,0)
Speed (C)
ot
Ro
2
C=1/2
E
Fig. A: Energy graph
43
Res. J. Math. Stat., 3(1): 39-44, 2011
Figure A is an illustration of levels of energy (E) achieved by a particle of m mass on making k perpendicular distance from the origin P(0,0), derived
from energy equations with masses/speeds of electrons E = mc and quarks E = mc2, which solved simultaneously gives E = mc2 – mc (or E = mc mc2) which solves when E = 0 and c = 0 or 1, or m = 0. The particle travels cutting k variable perpendicular distance of infinite number of planes
parallel to the horizontal axis through P(0,0). When the horizontal k is not tangential to the E curve, for E = k the roots are complex:
c = s = ½ ±it
where
t=
k 1
−
m 4
is real and i is imaginary.
Let s stand for speed (velocity) of particles since c is average relativity speed hence an approximation of s. s is the critical line whose magnitude (any
of the red lines) is:
2
s=
⎛ 1⎞
2
⎜ ⎟ +t =
⎝ 2⎠
1 ⎛ k 1⎞
+⎜ − ⎟ =
4 ⎝ m 4⎠
k
m
At s = ½, the particle achieves mean energy throughout the system at any given frequency k. All the analytic solutions of E, i.e., s = ½ + it, fall on
this line as the red arrows indicate, even if t gets to infinity.
ACKNOWLEDGEMENT
Fitzpatrick, D.P., 2006. Universities Asleep at the Switch:
A Science Renaissance: A Fresh Look at Quantum
Physics. Retrieved from: http://www.amperefitz.com/
unvasleep2.htm, (Accessed on: January 12, 2010).
Hilbert, D., 1900. Mathematical Problems. Lecture
Delivered Before the International Congress of
Mathematicians at Paris in 1900. Retrieved from:
http://aleph0.clarku.edu/~djoyce/hilbert/problems.h
tml, (Accessed on: October 5, 2010).
Not Even Wrong, 2008. Interview with Atle Selberg.
Retrieved from: http://www.math.columbia.edu/
~woit/wordpress/?p=708, (Accessed on: October 29,
2010).
Spiegel, M.R., 2005. Advanced Mathematics for
Engineers and Scientists. Tata McGraw-Hill, New
Delhi.
Wikipedia, 2010a. Riemann Hypothesis. Retrieved from:
http://en.wikipedia.org/wiki/Riemann_hypothesis,
(Accessed on: October 28, 2010).
Wikipedia, 2010b. Entropy. Retrieved from:
http://en.wikipedia.org/wiki/Entropy, (Accessed on:
November 05, 2010).
Wikipedia, 2010c. Electron-positron Annihilation.
Retrieved from: http://en.wikipedia.org/wiki/
Electron%E2%80%93positron_annihilation,
(Accessed on: November 05, 2010).
Acknowledged is His Excellency, Mr. Gabriel
Suswam, the Executive Governor of Benue State who is
also Visitor to Benue State University who funded this
research.
REFERENCES
Atovigba, M.V., 2010. Effect of use of mean and mean
deviation approach on students’ quadratic equation
solving. Unpublished M.Sc. Thesis, Benue State
University Postgraduate School.
Atovigba, M.V. and M.A. Chiawa, 2010. Model energy
equation: Implication for society education toward
achieving the 7-point agenda of the federal
government. Proceedings of the Mathematical
Association of Nigeria at its annual national
conference that held in Lafia Nigeria August 29 September 3, 2010.
Constant, J., 2008. Proof of Riemann’s Hypothesis.
Retrieved from: http://www.coolissues.com/
mathematics/Riemann/riemann.htm, (Accessed on:
October 29, 2010).
Epistemology Group, 2010. Energy as 2nd ODE.
Retrieved from: http://groups.com/group/
epistemology?hl=en, (Accessed on: October 10,
2010).
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