Number Theorem For Arithmetic Progressions Research Articles

On the Universal Presence of Mathematics for Incompletely Predictable Problems in Rigorous Proofs for Riemann Hypothesis, Modified Polignac’s and Twin Prime Conjectures

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

The average value of a certain number-theoretic function over the primes

We consider functions $F:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ for which there exists a positive integer $n$ such that two conditions hold: $F(p)$ divides $n$ for every prime $p$, and for each divisor $d$ of $n$ and every prime $p$, we have that $d$ divides $F(p)$ iff $d$ divides $F(p \mod d)$. Following an approach of Khrennikov and Nilsson, we employ the prime number theorem for arithmetic progressions to derive an expression for the average value of such an $F$ over all primes $p$, recovering a theorem of these authors as a special case. As an application, we compute the average number of $r$-periodic points of a multivariate power map defined on a product $Z_{f_1(p)}\times\cdots\times Z_{f_m(p)}$ of cyclic groups, where $f_i(t)$ is a polynomial.

Dirichlet, Sierpiński, and Benford

Sixty years ago, Sierpiński observed that for any positive integers A and B, and any g≥2, there are infinitely many primes whose base g-expansion begins with the digits of A and ends with those of B. Sierpiński's short proof rests on the prime number theorem for arithmetic progressions (PNT for APs). We explain how his result can be viewed as a natural intermediary between Dirichlet's theorem on primes in progressions and the PNT for APs. In addition to being of pedagogical interest, this perspective quickly yields a generalization of Sierpiński's result where the initial and terminal digits of p are prescribed in two coprime bases simultaneously; moreover, the proportion (Dirichlet density) of the corresponding primes is determined explicitly. The same quasielementary method shows that the arithmetic functions φ(n), σ(n), and d(n) obey “Benford's law” in a suitable sense.

A Bombieri–Vinogradov-type theorem with prime power moduli

In 2020, Roger Baker proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal {S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribut

Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

AbstractBuilding on the concept ofpretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Integers free of small prime factors in arithmetic progressions

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

Primes in arithmetic progressions

Strengthening work of Rosser, Schoenfeld, and McCurley, we establish explicit Chebyshev-type estimates in the prime number theorem for arithmetic progressions, for all moduli k ≤ 72 k \le 72 and other small moduli.

On the prime number theorem for arithmetic progressions

Demonstration d'une forme equivalente du theoreme des nombres premiers pour les progressions arithmetiques par une methode de convolution

Sieving the positive integers by small primes

Let Q Q be a set of primes that has relative density δ \delta among the primes, and let ϕ ( x , y , Q ) \phi (x,\,y,\,Q) be the number of positive integers ⩽ x \leqslant x that have no prime factor ⩽ y \leqslant y from the set Q Q . Standard sieve methods do not seem to give an asymptotic formula for ϕ ( x , y , Q ) \phi (x,\,y,\,Q) in the case that 1 2 ⩽ δ > 1 \tfrac {1}{2} \leqslant \delta > 1 . We use a method of Hildebrand to prove that \[ ϕ ( x , y , Q ) ∼ x f ( u ) ∏ p > y p ∈ Q ( 1 − 1 p ) \phi (x,y,Q) \sim x f(u) \prod _{\substack {p > y\\p \in Q}} {\left ( {1 - \frac {1}{p}} \right )} \] as x → ∞ x \to \infty , where u = log ⁡ x log ⁡ y u = \frac {{\log x}}{{\log y}} and f ( u ) f(u) is defined by \[ u δ f ( u ) = { e γ δ Γ ( 1 − δ ) , a m p ; 0 > u ⩽ 1 , e γ δ Γ ( 1 − δ ) + δ ∫ 0 u − 1 f ( t ) ( 1 + t ) δ − 1 d t , a m p ; u > 1. {u^\delta }f(u) = \left \{ {\begin {array}{*{20}{c}} {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}},} \hfill & {0 > u \leqslant 1,} \hfill \\ {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}} + \delta \int _0^{u - 1} {f(t){{(1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end {array} } \right . \] This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where Q Q consisted of all primes.

Explicit Estimates for the Error Term in the Prime Number Theorem for Arithmetic Progressions

We give explicit numerical estimates for the Chebyshev functions $\psi (x;k,l)$ and $\theta (x;k,l)$ for certain nonexceptional moduli k. For values of $\varepsilon$ and b, a constant c is tabulated such that $|\psi (x;k,l) - x/\varphi (k)| < \varepsilon x/\varphi (k)$, provided $(k,l) = 1$, $x \geqslant \exp (c{\log ^2}k)$, and $k \geqslant {10^b}$. The methods are similar to those used by Rosser and Schoenfeld in the case $k = 1$, but are based on explicit estimates of $N(T,\chi )$ and an explicit zero-free region for Dirichlet L-functions.

Explicit estimates for the error term in the prime number theorem for arithmetic progressions

We give explicit numerical estimates for the Chebyshev functions ψ ( x ; k , l ) \psi (x;k,l) and θ ( x ; k , l ) \theta (x;k,l) for certain nonexceptional moduli k. For values of ε \varepsilon and b, a constant c is tabulated such that | ψ ( x ; k , l ) − x / φ ( k ) | &gt; ε x / φ ( k ) |\psi (x;k,l) - x/\varphi (k)| &gt; \varepsilon x/\varphi (k) , provided ( k , l ) = 1 (k,l) = 1 , x ⩾ exp ⁡ ( c log 2 k ) x \geqslant \exp (c{\log ^2}k) , and k ⩾ 10 b k \geqslant {10^b} . The methods are similar to those used by Rosser and Schoenfeld in the case k = 1 k = 1 , but are based on explicit estimates of N ( T , χ ) N(T,\chi ) and an explicit zero-free region for Dirichlet L-functions.

Duality between prime factors and an application to the prime number theorem for arithmetic progressions

We study in this paper a new duality identity between large and small prime factors of integers and its relationship with the prime number theorem for arithmetic progressions. The asymptotic behavior of large prime factors of integers leads to interesting relations involving the Möbius function.

Bombieri's mean value theorem

The purpose of this paper is to give a short proof of an important recent theorem of Bombieri [2] on the mean value of the remainder term in the prime number theorem for arithmetic progressions. Applications of the theorem have been made by Bombieri and Davenport [3], Rodriques [9], and Elliott and Halberstam [5]. For earlier versions of the theorem and a survey of other applications, see Barban [1], and Halberstam and Roth [7, Chapter 4].

Footnote to the Titchmarsh-Linnik divisor problem

The study of Ta(x) was initiated as long ago as 1931 by Titchmarsh [2]. Since the dispersion method is exceedingly complicated, it may be of interest to record that the theorem is a simple consequence of the recent result of Bombieri [3] on the average of the error term in the prime number theorem for arithmetic progressions. For our purpose the following, weaker, form of Bombieri's theorem is sufficient: (i) There exists a positive number B such that

Some assertions equivalent to the prime number theorem for arithmetic progressions

Communications on Pure and Applied MathematicsVolume 2, Issue 2-3 p. 293-308 Article Some assertions equivalent to the prime number theorem for arithmetic progressions Harold N. Shapiro, Harold N. ShapiroSearch for more papers by this author Harold N. Shapiro, Harold N. ShapiroSearch for more papers by this author First published: June ‐ September 1949 https://doi.org/10.1002/cpa.3160020208Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume2, Issue2-3June ‐ September 1949Pages 293-308 RelatedInformation