The Riemann Hypothesis Is Possibly True

Abstract

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is possibly true.

Full text

0:0 November 27, 2021 12:17 a.m. note.tex page 1
The Riemann Hypothesis Is Possibly True
Frank Vega
Abstract. In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta func-
tion has its zeros only at the negative even integers and complex numbers with real part 1
2. The
Riemann hypothesis belongs to the David Hilbert’s list of 23 unsolved problems and it is one
of the Clay Mathematics Institute’s Millennium Prize Problems. The Robin criterion states
that the Riemann hypothesis is true if and only if the inequality σ(n)< eγ×n×log log n
holds for all natural numbers n > 5040, where σ(x)is the sum-of-divisors function and
γ≈0.57721 is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann
hypothesis is true if and only if the inequality Qq≤qn
q
q−1> eγ×log θ(qn)is satisfied for
all primes qn>2, where θ(x)is the Chebyshev function. Using both inequalities, we show
that the Riemann hypothesis is possibly true.
1. INTRODUCTION In mathematics, the Chebyshev function θ(x)is given by
θ(x) = X
q≤x
log q
where q≤xmeans all the prime numbers qthat are less than or equal to x. Let
Nn= 2 ×3×5×7×11 × · · · × qndenotes a primorial number of order nsuch
that qnis the nth prime number. Thus, θ(qn) = log Nn. We define a sequence based
on this function:
Definition. For every prime number qn, we define the sequence of real numbers:
Xn=Qq≤qn
q+1
q
log θ(qn).
We use this limit superior,
Theorem 1. [1].
lim sup
n→∞
Xn=eγ×6
π2.
Say Nicolas(qn)holds provided
Y
q≤qn
q
q−1> eγ×log θ(qn).
The constant γ≈0.57721 is the Euler-Mascheroni constant and log is the natural
logarithm. The importance of this inequality is:
Theorem 2. Nicolas(qn)holds for all prime numbers qn>2if and only if the Rie-
mann hypothesis is true [4].
Besides, we define the following properties of the Riemann zeta function,
November 5, 2021] THE RIEMANN HYPOTHESIS 1

0:0 November 27, 2021 12:17 a.m. note.tex page 2
Theorem 3. [2].
∞
Y
k=1
q2
k
q2
k−1=ζ(2) = π2
6.
Theorem 4. [2]. For a≥1:
Y
q1−1
qa+1 =1
ζ(a+ 1).
As usual σ(n)is the sum-of-divisors function of n[1]:
X
d|n
d
where d|nmeans the integer ddivides n. Define f(n)to be σ(n)
n. We know these
properties for this function:
Theorem 5. [3]. Let Qm
i=1 qai
ibe the representation of nas a product of primes
q1<· · · < qmwith natural numbers as exponents a1, . . . , am. Then,
f(n) = m
Y
i=1
qi
qi−1!×
m
Y
i=1 1−1
qai+1
i.
Theorem 6. [1]. For n > 1:
f(n)<Y
q|n
q
q−1.
Say Robins(n)holds provided
f(n)< eγ×log log n.
The importance of this inequality is:
Theorem 7. Robins(n)holds for all natural numbers n > 5040 if and only if the Rie-
mann hypothesis is true [5]. If the Riemann hypothesis is false, then there are infinitely
many natural numbers n > 5040 such that Robins(n)does not hold [5].
It is known that Robins(n)holds for many classes of numbers n.
Theorem 8. Robins(n)holds for all natural numbers n > 5040 such that n=Nm,
where Nmis a primorial number of order m[1].
Let q1= 2, q2= 3, . . . , qmbe the first mconsecutive primes, then an integer of
the form Qm
i=1 qai
iwith a1≥a2≥ · · · ≥ am≥0is called an Hardy-Ramanujan in-
teger [1]. Based on the theorem 7, we know this result:
Theorem 9. If the Riemann hypothesis is false, then there exist infinitely many natural
numbers n > 5040 which are an Hardy-Ramanujan integer and Robins(n)does not
hold [1].
2 Frank Vega [November 5, 2021

0:0 November 27, 2021 12:17 a.m. note.tex page 3
2. ANCILLARY LEMMAS The following is a key lemma.
Lemma 1. There exists a natural number Nsuch that Xn<eγ×6
π2+εfor all natural
numbers n > N and ε < 6
π2. Only a finite number of elements of the sequence are
greater than eγ×6
π2+ε(this could be an empty set).
Proof. The limit superior of a sequence of real numbers ynis the smallest real number
bsuch that, for any positive real number ε, there exists a natural number Nsuch
that yn< b +εfor all natural numbers n>N. Only a finite number of elements of
the sequence are greater than b+ε(this could be an empty set). Therefore, this is a
consequence of the theorem 1.
Lemma 2. Let Qm
i=1 qai
ibe the representation of an Hardy-Ramanujan integer n >
5040 as a product of the first mprimes q1<· · · < qmwith natural numbers as expo-
nents a1≥a2≥ · · · ≥ am≥0. If Robins(n)does not hold, then Nicolas(qm)holds
indeed.
Proof. When Robins(n)does not hold, then
f(n)≥eγ×log log n.
Let’s assume that Nicolas(qm)does not hold as well. Consequently,
Y
q≤qm
q
q−1≤eγ×log log Nm.
According to the theorem 6,
eγ×log log Nm≥Y
q≤qm
q
q−1
> f(n)
≥eγ×log log n.
However, this implies that Nm> n which is a contradiction since n > 5040 is an
Hardy-Ramanujan integer.
Lemma 3. If some prime number qn>2complies with
Xn≤eγ×6
π2
then Nicolas(qn)does not hold.
Proof. If we have the inequality
Xn≤eγ×6
π2
then this is equivalent to
Y
q≤qn
q+ 1
q≤eγ×6
π2×log θ(qn).
November 5, 2021] THE RIEMANN HYPOTHESIS 3

0:0 November 27, 2021 12:17 a.m. note.tex page 4
If we multiply the both sides by π2
6, so
π2
6×Y
q≤qn
q+ 1
q≤eγ×log θ(qn).
We use that theorem 3 to show that
π2
6×Y
q≤qn
q+ 1
q> Y
q≤qn
q2
q2−1!×Y
q≤qn
q+ 1
q.
Besides,
Y
q≤qn
q2
q2−1!×Y
q≤qn
q+ 1
q=Y
q≤qn
q
q−1
because of
q
q−1=q2
q2−1×q+ 1
q.
Consequently, we obtain that
Y
q≤qn
q
q−1≤eγ×log θ(qn)
and therefore, Nicolas(qn)does not hold.
3. POSSIBLE PROOF OF THE RIEMANN HYPOTHESIS
Theorem 10. The Riemann hypothesis is possibly true.
Proof. Let Qm
i=1 qai
ibe the representation of a sufficiently large Hardy-Ramanujan
integer n > 5040 as a product of the first mprimes q1<· · · < qmwith natural num-
bers as exponents a1≥a2≥ · · · ≥ am≥0. We claim that for every sufficiently large
Hardy-Ramanujan integer n > 5040, then Robins(n)could always hold. Suppose that
Robins(n)does not hold and so, the Riemann hypothesis would be false. Hence,
f(n)≥eγ×log log n.
We use that theorem 5,
m
Y
i=1
qi
qi−1!×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n
which is equivalent to
m
Y
i=1
q2
i
q2
i−1!× m
Y
i=1
qi+ 1
qi!×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n.
4 Frank Vega [November 5, 2021

0:0 November 27, 2021 12:17 a.m. note.tex page 5
If we divide the both sides by log log Nm, then we obtain
m
Y
i=1
q2
i
q2
i−1!×Xm×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n
log log Nm
because of log log Nm= log θ(qm), where Nmis the primorial number of order m.
We know that Xm≤eγ×6
π2is false according to the lemmas 2 and 3. From the lemma
1, we know that there exists a natural number Nsuch that Xm<eγ×6
π2+εfor all
natural numbers m > N and ε < 6
π2. Moreover, only a finite number of elements
of the sequence are greater than eγ×6
π2+ε(this could be an empty set). Under our
assumption, there exist infinitely many Hardy-Ramanujan integers n > 5040 such
that Robins(n)does not hold and Xm<eγ×6
π2+ε. In this way, we obtain that
m
Y
i=1
q2
i
q2
i−1!×(eγ×6
π2+ε)×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n
log log Nm
which is the same as
m
Y
i=1
q2
i
q2
i−1!×6
π2×(eγ+c)×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n
log log Nm
for a sufficiently small constant c=ε×π2
6. That would be equivalent to
Y
q>qm
q2−1
q2!×(eγ+c)×
m
Y
i=1 1−1
qai+1
i≥eγ×log log n
log log Nm
.
Since nis an Hardy-Ramanujan integer, then
Y
q>qm
q2−1
q2!×
m
Y
i=1 1−1
qai+1
i<Y
q1−1
qa1+1 =1
ζ(a1+ 1)
because of the theorem 4, where a1is the highest exponent such that 2a1|n. There-
fore,
(eγ+c)
ζ(a1+ 1) > eγ×log log n
log log Nm
for a sufficiently small constant 0< c < 1. However, this could be false for a suffi-
ciently small value of ε < 6
π2that we could choose, where c=ε×π2
6would be a very
small constant as well. In addition, we know that log log n
log log Nm>1due to the theorem 8.
In conclusion, for every sufficiently large Hardy-Ramanujan integer n > 5040, then
Robins(n)could always hold. By contraposition, the Riemann hypothesis is possibly
true, because of the theorems 7 and 9.
ACKNOWLEDGMENT. The author wishes to thank Richard J. Lipton and Craig Helfgott for helpful com-
ments and his mother, maternal brother and his friend Sonia for their support.
November 5, 2021] THE RIEMANN HYPOTHESIS 5

0:0 November 27, 2021 12:17 a.m. note.tex page 6
RE FE RE NC ES
1. Choie, Y., Lichiardopol, N., Moree, P., Sol´
e, P. (2007). On Robin’s criterion for the Riemann hypothesis.
Journal de Th´
eorie des Nombres de Bordeaux, 19(2): 357–372. http://dx.doi.org/10.5802/jtnb.
591.
2. Edwards, H. M. (2001). Riemann’s Zeta Function. Dover Publications.
3. Hertlein, A. (2018). Robin’s Inequality for New Families of Integers. Integers, 18.
4. Nicolas, J.-L. (1983). Petites valeurs de la fonction d’Euler. Journal of number theory, 17(3): 375–388.
http://dx.doi.org/10.1016/0022-314X(83)90055- 0.
5. Robin, G. (1984). Grandes valeurs de la fonction somme des diviseurs et hypoth`
ese de Riemann. J. Math.
pures appl, 63(2): 187–213.
FRANK VEGA is essentially a back-end programmer who graduated in Computer Science in 2007. In August
2017, he was invited as a guest reviewer for a blind peer-review of a manuscript about Theory of Computa-
tion in the flagship journal of IEEE Computer Society. In October 2017, he contributed as co-author with a
presentation about economy in the 7th International Scientific Conference on economic development and stan-
dard of living (”EDASOL 2017 - Economic development and Standard of living”). In June 2018, this paper
was published by the Journal of Economics and Market Communications (EMC). In February 2017, his book
”Protesta” (a book of poetry and short stories in Spanish) was published by the Alexandria Library Publish-
ing House. In September 2019, a scientific paper entitled ”Triangle Finding” was accepted by the conference
IMCS-55. He was Director of two IT Companies (Joysonic and Chavanasoft) created in Serbia.
CopSonic, 1471 Route de Saint-Nauphary 82000 Montauban, France
vega.frank@gmail.com
6 Frank Vega [November 5, 2021