There are many formulations of problems that have been proven to be equivalent to the Riemann Hypothesis in modern mathematics. In this paper we look at the formulation of an inequality derived by Robin in 1984 that proves the Riemann Hypothesis true if and only if Robin's Inequality is true for all $n\geq 5040$. In this paper we look at explicit products of prime numbers and show that if a given $n=p_1^{k_1}p_2^{k_2}\ldots p_m^{k_m}$ satisfies Robin's Inequality, then $n=P_1^{k_1}P_2^{k_2}\ldots P_m^{k_m}$ with $p_j \leq P_j$ also satisfies the inequality. We also then offer two conjectures that if proven individually could imply some interesting properties of the Riemann Hypothesis and Robin's Inequality.
In 1984, Robin showed that the Riemann Hypothesis for $ζ$ is equivalent to demonstrating $σ(n) < e^γn \log \log n$ for all $n > 5040$. Robin's inequality has since been proven for various infinite families of power-free integers: $5$-free integers, $7$-free integers, and $11$-free integers. We extend these results to cover $20$-free integers.
In this paper we give a variant of the Robin inequality which states that $\frac{σ(n)}{n} \leq \frac{e^γ}{2} \log\log n + \frac{0.7398\cdots}{\log\log n}$ for any odd integer $n \geq 3$.
Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality $σ(n):=\sum_{p|n}p<e^γ n \log \log n$ is satisfied for $n > 5040$, where $γ$ denotes the Euler-Mascheroni constant. We show that if the 2-adic order of n is big enough in comparison to the odd part of n then Robin's inequality is satisfied. We also show that if an positive integer $n$ satisfies either $ν_2(n) \leq 19$, $ν_3(n) \leq 12$,$ν_5(n) \leq 7$, $ν_7(n) \leq 6$, $ν_{11}(n) \leq 5$ then Robin's inequality is satisfied, where $ν_p(n)$ is the p-adic order of $n$. In the end we show that $σ(n…
Let $σ(n)$ denotes the sum of divisors function of a positive integer $n$. Robin proved that the Riemann hypothesis is true if and only if the inequality $σ(n) < e^γn \log \log n$ holds for every positive integer $n \geq 5041$, where $γ$ is the Euler-Mascheroni constant. In this paper we establish a new family of integers for which Robin's inequality $σ(n) < e^γn \log \log n$ hold. Further, we establish a new unconditional upper bound for the sum of divisors function. For this purpose, we use an approximation for Chebyshev's $\vartheta$-function and for some product defined over prime numbers.
The conjectured Robin inequality for an integer $n>7!$ is $σ(n)<e^γn \log \log n,$ where $γ$ denotes Euler constant, and $σ(n)=\sum_{d | n} d $. Robin proved that this conjecture is equivalent to Riemann hypothesis (RH). Writing $D(n)=e^γn \log \log n-σ(n),$ and $d(n)=\frac{D(n)}{n},$ we prove unconditionally that $\liminf_{n \rightarrow \infty} d(n)=0.$ The main ingredients of the proof are an estimate for Chebyshev summatory function, and an effective version of Mertens third theorem due to Rosser and Schoenfeld. A new criterion for RH depending solely on $\liminf_{n \rightarrow \infty}D(n)$…
In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $σ(n)<e^γn\log\log n$ holds for every integer $n>5040$, where $σ(n)$ is the sum of divisors function, and $γ$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\cS$ of the natural numbers such that the Robin inequality holds for all but finitely many $n\in\cS$. As a special case, we determine the finitely many numbers of the form $n=a^2+b^2$ that do not satisfy the Robin inequality. In fact, we prove our assertions with the Nicolas inequality $n/ϕ(n)<e^γ\log \log n$; since $σ(n…
For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin's InEquality $X(n):=\frac{σ(n)}{n\ln\ln n}<e^γ$ (RIE). The largest known violation of RIE is $n_8=5040$. In this paper Robin's multipliers are split into logarithmic terms $\mathcal{L}$ and relative divisor sums $\mathcal{G}$. A violation of RIE above $n_{8}$ is proposed to imply oscillations that cause $\mathcal{G}$ to exceed $\mathcal{L}$. To this aim Alaoglu and Erdős's conjecture for the CA numbers algorithm is used and the paper's key points are in section 4.2
Define $s (n) := n^{- 1} σ(n)$ ($σ(n):=\sum_{d|n}d )$ and $ω(n)$ is the number of prime divisors of $n$. One of the properties of $s$ plays a central role: $s (p^a) > s (q^b)$ if $p < q$ are prime numbers, with no special condition on $a, b$ other than $a, b \geqslant 1$. This result, combined with the Multiplicity Permutation theorem, will help us establish properties of the next counterexample (say $c$) to Robin's inequality $s (n) < e^γ \log \log n$. The number $c$ is superabundant, and $ω(c)$ must be greater than a number close to one billion. In addition, the ratio $p_{ω(c)} / \log c$ has…
For a positive integer t>1, an integer N is called t-free if the exponent of any prime factor of N is less than t. Some works shown if N is t-free, then N satisfies Robin's inequality, for t=5, 7, 11, 16. This article shows that the condition of t-free can be resuced to "N cannot be divided by t-th power of 2". I proved that if N cannot be divided by 17-th power of 2, then N satisfies Robin's inequality.
Recall that an integer is $t-$free iff it is not divisible by $p^t$ for some prime $p.$ We give a method to check Robin inequality $σ(n) < e^γn\log\log n,$ for $t-$free integers $n$ and apply it for $t=6,7.$ We introduce $Ψ_t,$ a generalization of Dedekind $Ψ$ function defined for any integer $t\ge 2$ by $$Ψ_t(n):=n\prod_{p | n}(1+1/p+...+1/p^{t-1}).$$ If $n$ is $t-$free then the sum of divisor function $σ(n)$ is $ \le Ψ_t(n).$ We characterize the champions for $x \mapsto Ψ_t(x)/x,$ as primorial numbers. Define the ratio $R_t(n):=\frac{Ψ_t(n)}{n\log\log n}.$ We prove that, for all $t$, there e…
Robin's criterion states that the Riemann hypothesis is equivalent to $σ(n) < e^γn \log\log n$ for all integers $n \geq 5041$, where $σ(n)$ is the sum of divisors of $n$ and $γ$ is the Euler-Mascheroni constant. We prove that the Riemann hypothesis is equivalent to the statement that $σ(n) < \frac{e^γ}{2} n \log\log n$ for all odd numbers $n \geq 3^4 \cdot 5^3 \cdot 7^2 \cdot 11 \cdots 67$. Lagarias's criterion for the Riemann hypothesis states that the Riemann hypothesis is equivalent to $σ(n) < H_n + \exp{H_n}\log{H_n}$ for all integers $n \geq 1$, where $H_n$ is the $n$th harmonic number. W…
Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality sum_{d|n}d<e^{gamma}n loglog n is satisfied for n>=5041, where gamma denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n>=37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreover, that n must be divisible by a fifth power >1. As a consequence we infer that RH holds true if and only if every natural number divisible by a fifth power >1 satisfies Robin's inequality.
For n>1, let G(n)=σ(n)/(n log log n), where σ(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erdős's results.
Let $G(n)=σ(n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^γ$ for all integer $n>5040$. This article divides all colossally abundant numbers in to three disjoint subsets CA1, CA2 and CA3, and shows that Robin hypothesis is true if and only if all CA2 numbers $n>5040$ satisfy Robin inequality.
We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
In the article we establish the Hardy-Littlewood inequality $ π(x + y) \leq π(x) + π(y) $. We also prove that the naturally ordered primes $p_1=2,p_2=3,p_3=5,p_4=7,\dots$ satisfy the inequality $ p_ {a + b}> p_a + p_b $ for all $a, \ b \geq 2$.
In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $π(x)$ holds for every $x \geq N_{\mathcal{R}}$.
In this paper we introduce a new diophantine inequality with prime numbers. Let $1<c<\frac{10}{9}$. We show that for any fixed $θ>1$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the tangent inequality \begin{equation*} \big|p^c_1\tan^θ(\log p_1)+ p^c_2\tan^θ(\log p_2)+ p^c_3\tan^θ(\log p_3) -N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.
Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\varepsilon>0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3$.
Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*} is solvable in primes $p_1,\,p_2,\,p_3$ near to squares.
Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that, for each $i\in\{1,2,3,4\}$, $p_i+2$ has at most $32$ prime factors.
We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| < (\log N)^{-E} , \] where $1 < c < \frac{15}{14}$, $N$ is a sufficiently large real number and $E>0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\left[ \frac{369}{180 - 168 c} \right]$ prime factors, counted with the multiplicity.
In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $π(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.
Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$, which improves the result of Mu.
We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$
We note that the inequalities $0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $π(x)$ came up recently in papers on algebraic number theory.
We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.
We consider the function $G(n)=\frac{σ(n)}{n\log\log n}$ (where $σ(n)=\sum_{d|n}d$) and set an imposed condition on its argument $n$, the fulfillment of which is sufficient for the existence of a prime $p$, at which $G(np)>G(n)$. This inequality is of interest in connection with the Robin's inequality. The paper also presents the results of numerical experiment conducted with superabundant numbers.
We complement the argument of M. Z. Garaev (2009) with several other ideas to obtain a stronger version of the large sieve inequality with sparse exponential sequences of the form $λ^{s_n}$. In particular, we obtain a result which is non-trivial for monotonically increasing sequences $\cal{S}=\{s_n \}_{n=1}^{\infty}$ provided $s_n\le n^{2+o(1)}$, whereas the original argument of M. Z. Garaev requires $s_n \le n^{15/14 +o(1)}$ in the same setting. We also give an application of our result to arithmetic properties of integers with almost all digits prescribed.
A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.
Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$ Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3,\,p_4,\,p_5,\,p_6$ for sufficiently large real number $N$.
We prove a version of Garaev's sum product theorem in the set of finite fields with non-prime order. Because of the presence of subfields, this seems to require some hypotheses on the set. We work under a condition analogous to having Hausdorff dimension less than 1/2. Under these conditions, we obtain a sum-product theorem with exponent 49/48.
It is well known that $li(x)>π(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $γ$ of the Riemann zeta function $ζ(s)$, encoded by the equation $li(x)-π(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_γ\frac{\sin (γ\log x)}γ]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[ψ(x)]>π(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by…
Let $Ω(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $ω(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $ω(N)$ and $Ω(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $Ω\geq \frac{8}{3}ω(N)-\frac{7}{3}$ and if $3 |N$ then $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$
We prove a bound on the number of primes with a given splitting behaviour in a given field extension. This bound generalises the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to $x$.
In this paper, we develop a large sieve type inequality for some special characters whose moduli are squares of primes. Our result gives non-trivial estimate in certain ranges.
We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg sieve.
Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N \end{equation*} is solvable in primes $p_1,p_2,p_3$. Moreover, we also prove that the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N\big|<\log^{-1}N \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4,p_5,p_6$, which improves the previous result $1<c<\frac{37}{18},c\neq2$.
We analyze the inequality $\sqrt{P_{k+1}}-\sqrt{P_{k}}<1,\ k\in\mathbb{N}$, discuss the existence of primes on arbitrary intervals $(r,s),\ r<s,\ r,s\in\mathbb{R}$, and finally address the issue of primes between squares of naturals.
In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered t…
In this paper, using the well known fact that the series of reciprocals of primes diverges, we obtain a general inequality for gaps of consecutive primes that holds for infinitely many primes. As it is shown the key ingredient for this direct approach is a consequence of the the Kummer's characterization of summable sequences of positive terms. Some interesting consequences are then presented. In particular, we show how the twin-prime conjecture is related to our main result.
In this paper we use a theorem first proved by S.W.Golomb and a famous inequality by J.B. Rosser and L.Schoenfeld in order to prove that there exists an exact formula for $π(n)$ which holds infinitely often.
