Number Theory

The Tower Function and Applications

Authors: Theophilus Agama
Comments: 4 Pages.

In this paper we study an extension of the Euler totient function to the rationals and explore some applications. In particular, we show that \begin{align} \# \{\frac{m}{n}\leq \frac{a}{b}~|~m\leq a,~n\leq b,~\gcd(m,a)=\gcd(n,b)=1,~\gcd(n,a)>1\nonumber \\~\vee~\gcd(m,b)>1~\vee ~\gcd(m,n)>1\}=\sum \limits_{\substack{\frac{m}{n}\leq \frac{a}{b}\\mn\leq ab\\m>a,n\leq b~\vee~m\leq a,n>b~\vee~\gcd(m,n)>1\\ \gcd(mn,ab)=1}}1\nonumber \end{align} provided $\gcd(a,b)=1$.
Category: Number Theory

An Alternative Angle

Authors: Richard Harris
Comments: 2 Pages.

An alternative, but equivalent, form of Riemann Hypothesis.
Category: Number Theory

Binomial Series and Its Applications in Congruences

Authors: Shazly Abdullah
Comments: 8 Pages.

In this study we used an algebraic method that uses elementary algebra and binomial theorem. To create binomial series ( ) ( ) ( ).This is a type of series that has several properties in variables such as if then ( ) ( ) where ( ) We used these series to prove results in congruences for example,we proved If , where ( ) m,n and ( ) then ( ) ( ) .We also obtained several results in finite series
Category: Number Theory

Proof of Riemann Hypothesis (3)

Authors: Toshihiko Ishiwata
Comments: 34 Pages.

This paper is a trial to prove Riemann hypothesis according to the following process. 1. We make (N+1)/2 infinite series from one equation that gives ζ(s) analytic continuation and 2 formulas (1/2+a+bi, 1/2−a−bi) that show non-trivial zero point of ζ(s). (N = 1, 3, 5, 7, · · · · · · ) 2. We find that a cannot have any value but zero from the above infinite series by performing N → ∞. 3. Therefore non-trivial zero point of ζ(s) must be 1/2 ± bi.
Category: Number Theory

An Exact Formula for the Prime Counting Function

Authors: Jose R. Sousa
Comments: 21 Pages.

This paper discusses a few main topics in Number Theory, such as the M\"{o}bius function and its generalization, leading up to the derivation of a neat power series for the prime counting function, $\pi(x)$. Among its main findings, we can cite the extremely useful inversion formula for Dirichlet series (given $F_a(s)$, we know $a(n)$, which may provide evidence for the Riemann hypothesis, and enabled the creation of a formula for $\pi(x)$ in the first place), and the realization that sums of divisors and the M\"{o}bius function are particular cases of a more general concept. One of its conclusions is that it's unnecessary to resort to the zeros of the analytic continuation of the zeta function to obtain $\pi(x)$.
Category: Number Theory

Lerch's Φ and the Polylogarithm at the Positive Integers

Authors: Jose Risomar Sousa
Comments: 17 Pages.

We review the closed-forms of the partial Fourier sums associated with $HP_k(n)$ and create an asymptotic expression for $HP(n)$ as a way to obtain formulae for the full Fourier series (if $b$ is such that $|b|<1$, we get a surprising pattern, $HP(n) sim H(n)-sum_{kge 2}(-1)^kzeta(k)b^{k-1}$). Finally, we use the found Fourier series formulae to obtain the values of the Lerch transcendent function, $Phi(e^z,k,b)$, and by extension the polylogarithm, $mathrm{Li}_{k}(e^{z})$, at the positive integers $k$.
Category: Number Theory

Lerch's Φ and the Polylogarithm at the Negative Integers

Authors: Jose R. Sousa
Comments: 10 Pages. I improved the writing and fixed typos

At the negative integers, there is a simple relation between the Lerch $Phi$ function and the polylogarithm. Starting from that relation and a formula for the polylogarithm at the negative integers known from the literature, we can deduce a simple closed formula for the Lerch $Phi$ function at the negative integers, where the Stirling numbers of the second kind are not needed. Leveraging that finding, we also produce alternative formulae for the $k$-th derivatives of the cotangent and cosecant (ditto, tangent and secant), as simple functions of the negative polylogarithm and Lerch $Phi$, respectively, which is evidence of the importance of these functions (they are less exotic than they seem). Lastly, we extend formulae for the Hurwitz zeta function only valid at the positive integers to the complex half-plane using this novelty.
Category: Number Theory

A Reformulation of the Riemann Hypothesis

Authors: Jose R. Sousa
Comments: 10 Pages.

We present some novelties on the Riemann zeta function. Using the analytic continuation we created for the polylogarithm, $\mathrm{Li}_{k}(e^{m})$, we extend the zeta function from $\Re(k)>1$ to the complex half-plane, $\Re(k)>0$, by means of the Dirichlet eta function. More strikingly, we offer a reformulation of the Riemann hypothesis through a zeta's cousin, $\varphi(k)$, a pole-free function defined on the entire complex plane whose non-trivial zeros coincide with those of the zeta function.
Category: Number Theory

Inclusive Collatz Problem

Authors: Hajime Mashima
Comments: 8 Pages.

In this paper, I approached the branch size and path increase/decrease in the limit of the Collatz problem(Collatz Conjecture).
Category: Number Theory

Proofs of Twin Prime Number Conjecture and First Hardy-Littlewood Conjecture

Authors: Zhi Li, Hua Li
Comments: 7 Pages.

The twin prime conjecture was proposed by Alfonse de Polignac in 1849 and has not been proven for nearly 300 years. Since there is no mathematical model for prime numbers that can be completely and accurately represented, prime numbers are randomly distributed on the number axis, and twin prime numbers are also randomly distributed. In this paper, the twin prime conjecture is proved by probability and statistics, the twin prime distribution theorem and prime pair distribution theorem are found, and the First Hardy-Littlewood conjecture is further proved.
Category: Number Theory

A Revisit to Lemoine's Conjecture

Authors: Theophilus Agama
Comments: 5 Pages.

In this paper we prove Lemoine's conjecture. By exploiting the language of circles of partition, we show that for all sufficiently large $n\in 2\mathbb{N}+1$ \begin{align} \# \left \{p+2q=n|~p,q\in \mathbb{P}\right \}>0.\nonumber \end{align}This proves that every sufficiently large odd number can be written as the sum of a prime and a double of a prime.
Category: Number Theory

A Proof of Goldbach Conjecture

Authors: Zhi Li, Hua Li
Comments: 6 Pages.

natural numbers can be divided into even numbers and odd numbers, and odd numbers can be divided into composite numbers, odd numbers and prime numbers. Any even number can be decomposed into the sum of a larger prime number and a smaller odd number. The Goldbach conjecture can be proved by calculating whether the cumulative probability that the small odd number is a prime number is much greater than 1 and determining whether the prime number in many small odd numbers is inevitable. In this paper, the Goldbach conjecture is proved by the method of probability and statistics, the Goldbach number theorem is further discovered, and a new method for the study of prime number distribution is created.
Category: Number Theory

An Elementary Approach to the Riemann Hypothesis

Authors: Jose Risomar Sousa
Comments: 7 Pages. A new result was added, evidence for the converge of F(s) when Re(s)>1/2

The Riemann hypothesis is probably true. In this paper I present an approach for it in a very short and condensed way, making use of one of its equivalent problems. But as Carl Sagan once famously said, extraordinary claims require extraordinary evidence. The evidence here is the newly discovered inversion formula for Dirichlet series.
Category: Number Theory

On Consecutive Special Primes

Authors: D. K. K. Janabi
Comments: 2 Pages.

In this short paper, we establish a number-theoretic conjecture about primes with a special property and give a hint for the proof.
Category: Number Theory

Whole Numbers in Specified Arrays and Their Relationships in Multi-Dimensional Locales

Authors: Vimosh Venugopal
Comments: 14 Pages.

The study details specified properties of whole numbers in conjunction with repetitive arrays and sequences. There prevails a common pattern for numbers when they exist in defined structures. The paper extends to the scope of progressions in regard to the specific number relationships and its reach in advanced mathematical studies. The properties of numbers enumerated have its scope in the field of recreational mathematical theories as well.
Category: Number Theory

The Number of Primes

Authors: Pal Doroszlai, Horacio Keller
Comments: 10 Pages.

The prime-number-formula at any distance from the origin has a systematic error, proportional to the square of the number of primes up to the square root of the distance. The proposed completion in the present paper eliminates by a quickly converging recursive formula the systematic error. The remaining error is reduced to a symmetric dispersion, with standard deviation proportional to the number of primes at the square root of the distance.
Category: Number Theory

Fermat's Last Theorem: Basic Case (English, French, Russian)

Authors: Victor Sorokine
Comments: 4 Pages. Each statement is reducible to axioms. Chaque énoncé est réductible à des axiomes. Каждое утверждение сводимо до аксиом..

The third (from the end) digit in the sum of two equivalent Fermat's equalities with the last digits in the numbers A, B, C equal to a, b, c, and n-a, n-b, n-c, is equal to 1, and at the same time it is a single-valued function of the last digits.

Après avoir additionné deux égalités de Fermat avec les derniers chiffres des nombres a, b, c, n-a, n-b, n-c, le troisième (à partir de la fin) chiffre de la somme des puissances est 1, et en même temps il ne peut pas être converti à zéro en changeant les troisièmes chiffres en bases de nombres A, B, C (et les deuxièmes chiffres sont des fonctions à un chiffre des derniers chiffres). *** Третья (от конца) цифра в сумме двух эквивалентных равенств Ферма с последними цифрами в числах A, B, C, равными a, b, c, и n-a, n-b, n-c, равна 1 и при этом она является однозначной функцией лишь последних цифр.
Category: Number Theory

Short Effective Intervals Containing Primes and a Property of the Riemann Zeta Function ζ(1/2+it)

Authors: Kang-Ho Kim
Comments: 10 Pages.

In this paper, we prove the existence of primes in the interval ]x,x+2√x] by inducing an inequality which defines the lower bound of number of primes in the interval ]x,x+2√x] and suggest an opinion for truth of the Lindelöf hypothesis based on the existence of primes in the interval ]x,x+2√x] with the success of Ingham’s preceded work.
Category: Number Theory

Fermat's Last Theorem and Pythagorean Triples

Authors: Michael Griffin
Comments: 14 Pages. contact email mdg46@juno.com

Fermat's Last Theorem is investigated on the set of Pythagorean triples using the ancient Greek formulas of Pythagoras, Euclid, and Plato. These are formulas used to derive natural number solutions of the Pythagorean theorem. Since Euclid’s formula makes all possible triples, proofs that Euclid’s cannot work at higher powers thus prove Fermat’s on the set of triples.
Category: Number Theory