4 pages; similar formulas were proven using Bertrand's Postulate by S. Regimbal, An explicit formula for the k-th prime number, Math. Mag., 48 (1975), 230-232
Using inequalities of Rosser and Schoenfeld, we prove formulas for pi(n) and the n-th prime that involve only the elementary operations +,-,/ on integers, together with the floor function. Pascal Sebah has pointed out that the formula for pi(n) operates in O(n^(3/2)) time. Similar formulas were proven using Bertrand's Postulate by Stephen Regimbal, An explicit formula for the k-th prime number, Mathematics Magazine, 48 (1975), 230-23
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.
Rosser and Schoenfeld remarked that the product $\prod_{p\leq x}(1-1/p)^{-1}$ exceeds $e^γ \log x$ for all $2\leq x\leq 10^8$, and raised the question whether the difference changes sign infinitely often. This was confirmed in a recent paper of Diamond and Pintz. In this paper, we show (under certain hypotheses) that there is a strong bias in the race between the product $\prod_{p\leq x}(1-1/p)^{-1}$ and $e^γ\log x$ which explains the computations of Rosser and Schoenfeld.
We study Kummer's approach towards proving the Fermat's last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.
A new explicit formula is proved for the contribution of the major arcs in the Goldbach and Generalized Twin Prime Problem, in which the level of the major arcs can be chosen very high. This will have many applications in the approximations of the mentioned problems.
In this paper, we propose a new primality test, and then we employ this test to find a formula for π that computes the number of primes within any interval. We finally propose a new formula that computes the nth prime number as well as the next prime for any given number
A new set of formulas for primes is presented. These formulas are more efficient and grow much slower than the two known formulas of Mills and Wright. 3 new formulas are explained.
In this paper we give a new semiprimality test and we construct a new formula for $π^{(2)}(N)$, the function that counts the number of semiprimes not exceeding a given number $N$. We also present new formulas to identify the $n^{th}$ semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots of $N : P_{1}, \; P_{2}....P_{π\left( \sqrt[3]{N}\right) }\leq \sqrt[3]{N}$.
We will derive a function that eliminates any sequence of equidistant numbers from the integer numbers, then we will derive its inverse. Then we will use the Sequence elimination function to eliminate the multiples of the prime numbers from the integer numbers to produce formulas that generate all prime numbers in some interval.
Based on Golomb's arithmetic formulas, Dirichlet series for two classes of twin primes are constructed and related to the roots of the Riemann zeta function in the critical strip.
Under Cramér's conjecture concerning the prime numbers, we prove that for any $x>1$, there exists a real $A=A(x)>1$ for which the formula $[A^{n^x}]$ (where $[]$ denotes the integer part) gives a prime number for any positive integer $n$. Under the same conjecture, we also prove that for any $ε>0$, there exists a positive real number $B$ for which the formula $[B.{n!}^{2+ε}]$ gives a prime number for any sufficiently large positive integer $n$.
In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the unconditional case.
We generalize a formula on the counting of prime geodesics, due to Kuznetsov-Bykovskii, used in the work of Soundararajan-Young on the prime geodesic theorem. The method works over any number field and for any congruence subgroup. We give explicit computation in the cases of principal and Hecke subgroups.
In this paper, a new formula for π^(2)(N) is formulated, it is a function that counts the number of semi-primes not exceeding a given number N. A semi-prime is a natural number that is the product of precisely two prime numbers, the two primes in the product may equal each other. Semi-prime numbers are also a case of almost primes. Since a formula for this is already known, a new identity that uses the prime counting function is created by equating the two functions.
Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to establish the $p$-part of the Birch and Swinnerton-Dyer formula for $E$. In particular, we extend the main result of loc.cit. to primes of multiplicative reduction.
Taking $r>0$, let $π_{2r}(x)$ denote the number of prime pairs $(p, p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $π_{2r}(x)\sim 2C_{2r} {\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders $ω_{2r}(x)=π_{2r}(x)- 2C_{2r} {\rm li}_2(x)$ that corresponds to Riemann's formula for $π(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders $ω_{2r}(x)$ which is supported by numerical results.
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.
We generalize results of Alladi, Dawsey, and Sweeting and Woo for Chebotarev densities to general densities of sets of primes. We show that if $K$ is a number field and $S$ is any set of prime ideals with natural density $δ(S)$ within the primes, then \[ -\lim_{X \to \infty}\sum_{\substack{2 \le \operatorname{N}(\mathfrak{a})\le X\\ \mathfrak{a} \in D(K,S)}}\frac{μ(\mathfrak{a})}{\operatorname{N}(\mathfrak{a})} = δ(S), \] where $μ(\mathfrak{a})$ is the generalized Möbius function and $D(K,S)$ is the set of integral ideals $ \mathfrak{a} \subseteq \mathcal{O}_K$ with unique prime divisor of min…
In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.
We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.
Improving earlier estimates of several authors we show that the number E(X) of Goldbach exceptional even integers (that is, even integers which cannot be written as the sum of two primesw) below X satisfies tho bound E(X) < X^0.72 for sufficiently large X. In the proof crucial role is played by the explicit formula for the contribution of the major arcs proved in Part I of this series (arXiv:1804.05561). In this new version some misprints are corrected and an explanatory sentence is added to the common proof of theorems C and D.
Let $n$ be a positive integer and let $A$ be nonempty finite set of positive integers. We say that $A$ is relatively prime if $\gcd(A) =1$ and that $A$ is relatively prime to $n$ if $\gcd(A,n)=1$. In this work we count the number of nonempty subsets of $A$ which are relatively prime and the number of nonempty subsets of $A$ which are relatively prime to $n$. Related formulas are also obtained for the number of such subsets having some fixed cardinality. This extends previous work for the cases where $A$ is an interval or a set in arithmetic progression. Applications include: a) An exact formul…
We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We then show that the error term in this formula can be sharpened on the assumption of the Riemann hypothesis. Finally, we prove an asymptotic formula for these counting functions in short intervals.
In 1874, Mertens famously proved an asymptotic formula for the product $p/(p-1)$ over all primes $p$ up to $x$. On the other hand, one may expand Mertens' prime product into series over numbers $n$ with only small prime factors. It is natural to restrict such series to numbers $n$ with a fixed number $k$ of prime factors. In this article, we obtain formulae for these series for each $k$, which together dissect Mertens' original estimate. The proof is by elementary methods of a combinatorial flavor.
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors an analogous result by Friedlander-Goldston.
Consider a $q$-Weil polynomial $f$ of degree $2g$. Using an equidistribution assumption that is too strong to be true, we define and compute a product of local relative densities of matrices in $\rm{GSp}_{2g}(\mathbb{F}_\ell)$ with characteristic polynomial $f\mod\ell$ when $g$ is an odd prime. This infinite product is closely related to a ratio of class numbers. When $g=3$ we conjecture that the product gives the size of an isogeny class of principally polarized abelian threefolds.
We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$, with $\ell_1, \ell_2\in\{2,3\}$, $\ell_1+\ell_2\le 5$ are fixed integers, and $n=p^{\ell_1} + m^{\ell_2}$, with $\ell_1=2$ and $2\le \ell_2\le 11$ or $\ell_1=3$ and $ \ell_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.
We shall give an explicit formula for $ψ(x, q, a)$ with an error term of the form $C/\log^αx$ under the condition that $q<\log^{α_1} x$ is nonexceptional, for various values of $α$ and $α_1$. We shall also give an explicit formula for $ψ(x, q, a)$ with error terms $C/\log^A x$ working whether $q$ is exceptional or nonexceptional, but under the condition that $\frac{0.4923A}πq^{1/2}\log^2 q<\log x/\log\log x$. Moreover, we shall give an explicit form of Bombieri-Vinogradov theorem over non-exceptional moduli.
We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $Γ$ has a mass formula for the set of primes not dividing $|Γ|$. We also prove that if $Γ$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $Γ$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting…
In this paper we establish a general asymptotic formula for the sum of the first $n$ prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin. Further we prove a series of results concerning Mandl's inequality on the sum of the first $n$ prime numbers. We use these results to find new explicit estimates for the sum of the first $n$ prime numbers, which improve the currently best known estimates.
We prove upper bounds for the mean square of the remainder in the prime geodesic theorem, for every cofinite Fuchsian group, which improve on average on the best known pointwise bounds. The proof relies on the Selberg trace formula. For the modular group we prove a refined upper bound by using the Kuznetsov trace formula.
We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost al…
We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.
We prove that analogues of the Hardy-Littlewood generalised twin prime conjecture for almost primes hold on average. Our main theorem establishes an asymptotic formula for the number of integers $n=p_1p_2 \leq X$ such that $n+h$ is a product of exactly two primes which holds for almost all $|h|\leq H$ with $\log^{19+\varepsilon}X\leq H\leq X^{1-\varepsilon}$, under a restriction on the size of one of the prime factors of $n$ and $n+h$. Additionally, we consider correlations $n,n+h$ where $n$ is a prime and $n+h$ has exactly two prime factors, establishing an asymptotic formula which holds for …
In this paper we extend the hybrid Euler-Hadamard product model for quadratic Dirichlet $L$-functions associated to irreducible polynomials over function fields. We also establish an asymptotic formula for the first twisted moment in this family of L-functions and then we provide further evidence for the conjectural asymptotic formulas for its moments.
In this paper we study a sequence involving the prime numbers by deriving two asymptotic formulas and finding new upper and lower bounds, which improve the currently known estimates.
Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_{\ell}$-extensions of odd prime degree $\ell$ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics. We also obtain improved upper bounds for the number of such extensions (over Q) of bounded discriminant.
Let $Λ\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}Λ\left(m_{1}\right)Λ\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}Λ\left(n\right)Λ\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\…
We derive heuristically approximate formulas for the negative $k$--moments $M_{-k}(x)$ of the gaps between consecutive primes$<x $ represented directly by $π(x)$ --- the number of primes up to $x$. In particular we propose an analytical formula for the sum of reciprocals of gaps between consecutive primes $<x : ~ M_{-1}(x)\sim \frac{π^2(x)}{x-2π(x)}\log\Big(\frac{x}{2π(x)}\Big) \sim x \log \log(x)/\log^2(x)$. We illustrate obtained results by the enormous computer data up to $x=4\times 10^{18}$.
Let $x$ be a positive integer. We give an asymptotic formula for the number of primes in the set $\{\fl{x/n}, 1 \le n \le x\}$ and give some related results.
In this paper, we exhibit an asymptotic formula for the number of representations of a large integer as a sum of a fixed power of Piatetski-Shapiro primes, thereby establishing a variant of Waring-Goldbach problem with primes from a sparse sequence.
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotarëv's density theorem to the cyclic case.
In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by Büyükboduk--Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-type for non-ordinary primes $p>2$.
We present the geometry lying behind counting twin prime polynomials in $\mathbb{F}_q[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^3 = Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder. The formula we get in degree $3$ is compatible with the Hardy-Littlewood heur…
