Approximation by special values of Dirichlet series

Let $S$ be a non-empty finite set of prime numbers and, for each $p$ in $S$ , let $\\bm{Z}_p$ denote the ring of $p$ -adic integers. Let $F$ be an abelian extension over the rational field such that the Galois group of $F$ over some subfield of $F$ with finite degree is topologically isomorphic to the additive group of the direct product of $\\bm{Z}_p$ for all $p$ in $S$ . We shall prove that each of certain arithmetic progressions contains only finitely many prime numbers $l$ for which the $l$ -class group of $F$ is nontrivial. This result implies our conjecture in [3] that the set of prime numbers $l$ for which the $l$ -class group of $F$ is trivial has natural density $1$ in the set of all prime numbers.

We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

Background: The twin prime conjecture is regarded as a classic puzzle in number theory and one of the most well-known conjectures that has always perplexed us. Mathematician David Hilbert presented 23 key mathematical problems and conjectures to be solved at the International Congress of Mathematicians in 1900. He included the Bernhard Riemann conjecture, the Twin Prime Conjecture, and the Goldbach's conjecture in the eighth of 23 mathematical problems. Methods: Based on the "Differential Incremental Equilibrium Theory" [1], the infinite set of infinite prime numbers is divided, the increment equation of infinite prime numbers is established, and the tree-like set of prime numbers is obtained. Find the twin primes with a minimum unit [1\(\to\)1] of 2. Results: When a set of prime numbers is infinitely divided, there are 2[1\(\to\)1] pairs of prime numbers whose gap is equal to 2 and gap is not equal to 2. We gives a complete proof of the twin prime conjecture. It shows that the importance of "Differential Incremental Equilibrium Theory" [1] and infinite classification in twin prime conjecture. In a higher-level ideology, the set infinite partition classification confirms that the minimum unit is 2. It's a new way to prove Twin Prime Conjecture. Conclusion: This paper gives a complete proof of the establishment of the Twin Prime Conjecture.

An infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp [2] asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh–Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.

Some problems of the number theory are associated with the zeros of special functions, such as the Riemann zeta function ζ(s), Dirichlet L-functions L(s, χ) and others. The Riemann zeta function is the most famous. On the half-plane ℜs > 1, the Riemann zeta function is defined by Dirichlet series ζ(s) = ∑ +∞ n=1 n −s . In 1859, Riemann conjectured that non-trivial zeros of the Riemann zeta function lie on the critical line ℜs = 1 2 . Hardy was the first to prove in 1914 that an infinity of zeros are on the critical line. In 1942, Selberg obtained lower bound of the correct order of magnitude for the number zeros of the Riemann zeta functions on intervals of critical line [T, T + H], H = T 0.5+e . In 1984, A. A. Karatsuba proved Selberg’s result for shorter intervals of critical line [T, T + H], H = T 27 82 +e . For arithmetic Dirichlet series satisfying a functional equation of Riemann type but admitting no Euler product expansions, lower bounds of the correct order of magnitude for the number of their zeros on intervals of the critical line ℜs = 1/2 have not been obtained so far. The first to show that the critical line contains abnormally many zeros of an arithmetic Dirichlet series without Euler products was Voronin, who proved in 1980 that interval (0, T] of critical line contains more than cT e 1 20 √ ln ln ln ln T zeros of the Davenport–Heilbronn function. In 1989 A. A. Karatsuba developed a new method for obtaining lower bounds for the number zeros of certain Dirichlet series in intervals of critical line; by using this method, he substantially strengthened Voronin’s result. In 1991 Karatsuba solved (by his 1989 method) the problem of estimating the number zeros of linear combinations of functions which are analogous the Hardy function. In the present paper we prove a theorem similar to the theorem of A. A. Karatsuba (in 1991), but only for almost all intervals of the form (T, T + H), H = Xe , where e is an arbitrary positive number, and X ≤ T ≤ 2X, X > X0(e).

Abel's theorem on power series is well known: If the series += a, converges and has the sum S, then the power series 2o anX converges for I x < 1 and its sum tends to S as x tends to 1 through real values. The converse theorem is not unconditionally true: it may occur that the power series O anoX converges for I x 1 < 1 and its sum tends to a finite limit S as x tends to 1, but this series does not converge when x = 1. Nevertheless Tauber [12]' proved in 1897 that, if nan , or even if (1 /n) a, tends to zero when n tends to + so, the assumption that f(x) = + an X tends to S when x tends to 1 through real smaller values implies the convergence of the series +O a. to the sum S. This result was the origin of a long series of works which led to various converses of Abel's theorem and its extensions to Dirichlet series and Laplace integrals. We shall quote here only some of them. Let us consider a Dirichlet series EOw ake , where the ak are real or complex numbers and the Xk are real numbers which satisfy

&lt;p&gt;As a generalization of the XVI-th - XVII-th centuries magic squares, we&nbsp;present the Smarandache magic square of order n, which is: 2 square array of rows of&nbsp;elements of A arranged so that the law I applied to each horizontal and vertical row&nbsp;and diagonal give the same result.&lt;/p&gt;

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in \mathbf{Z}$, we study the set $\bigcap_{i=1}^{n}\lbrace p\in\mathcal{P}-\Sigma_{X}, N_{X}(p)\neq a_{i} [\bmod p]\rbrace$ where $\Sigma_{X}$ is the finite set of primes of bad reduction for $X$. In case $\dim X\leq 3$, we show the set is either empty or has positive lower-density. We also address the question of the size of the smallest prime in that set. Using sieve methods, we obtain for example an upper bound for the size of the least prime of $\lbrace p\in\mathcal{P}, p\nmid N_{X}(p)\rbrace$ on average in particular families of hyperelliptic curves.

We generalize the classical definition of zeta-regularization of an infinite product. The extension enjoys the same properties as the classical definition, and yields new infinite products. With this generalization we compute the product over all prime numbers answering a question of Ch. Soule. The result is 4π2. This gives a new analytic proof, companion to Euler’s classical proof, that the set of prime numbers is infinite.

Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp. The famous sum-product problem of Erdős and Szemeredi asks the following. Let $A$ be a set of $n$ real numbers. Define the _sumset_ $A+A$ of $A$ to be the set $\{x+y:x,y\in A\}$ and the _product set_ $A.A$ to be the set $\{xy:x,y\in A\}$. Must at least one of $A+A$ and $A.A$ have size $n^{2-o(1)}$? Since neither set can have size greater than $n^2$, this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was $n^{4/3}$, up to logarithmic factors [3], though the exponent has now been very slightly improved [2]. More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if $A$ is a subset of a finite field ${\mathbb F}_p$ and $p^{\delta} < |A| < p^{1-\delta}$, then one of $A+A$ and $A.A$ must have size at least $n^{1+\epsilon}$ for some $\epsilon$ that depends on $\delta$ only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that $\mathbb F_p$ does not. This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if $A$ has a large difference set $D=A-A$, it is perfectly possible for $D+D$ to be much bigger than $D$. Nevertheless, when $A$ is a set of real numbers, it still implies, as the main result of the paper shows, that the product set $D.D$ has size at least $|D|^{1+c}$, where $c>0$ is an absolute constant. The same is true of the quotient set $D/D$. The paper also proves a result of a similar kind in the finite field $\mathbb F_p$ provided that $D$ is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup $H$ of $\mathbb F_p$ that is not too large is not the difference set of any subset of $\mathbb F_p$. (The precise assumption required is that $H$ has order at most $p^{4/5-\epsilon}$.) [1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343) [2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771) [3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040) [Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)

Let K be a number field, P the set of prime numbers, and {ρ`}`∈P a compatible system (in the sense of Serre [19]) of semisimple, n-dimensional `-adic representations of Gal(K/K). Denote the Zariski closure of ρ`(Gal(K/K)) in GLn,Q` by G` and its Lie algebra by g`. It is known that the identity component G ◦ ` is reductive and the formal character of the tautological representation G` ↪→ GLn,Q` is independent of ` (Serre). We use the theory of abelian `-adic representations to prove that the formal character of the tautological representation of the derived group (G` ) der ↪→ GLn,Q` is likewise independent of `. By investigating the geometry of weights of this faithful representation, we prove that the semisimple parts of g` ⊗ C satisfy an equal-rank subalgebra equivalence for all ` which is equivalent to the number of An := sln+1,C factors for n ∈ {6, 9, 10, 11, ...} and the parity of the number of A4 factors in g` ⊗ C are independent of `.

LetP1 andP2 be two sets of prime numbers and let ω(m,Pi)=#{p: p/m, pePi} (i=1,2) be two related additive functions ofm. For an irreducible positive fractionm/n, defineh(m/n)=ω(m, P1)+ω(n, P2). In this paper the local frequenciesvx{h(m/n)=s}=#{m/n ∈ Fx:h(m/n)=s}/#Fx are considered, whereFx denotes the classical Farey series. Using the mean-value theorem for multiplicative functions of rational argument, a local limit theorem forvx{h(m/n)=s} is proved.

An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and v if and only if ∣u - v∣D where D is a subset of the positive integers. It is known that x(G(Z,D) )=4 where P is a set of Prime numbers. So we can allocate the subsets D of P to four classes, accordingly as is 1 or 2 or 3 or 4. In this paper we have considered the open problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of primes like Additive primes, Deletable primes, Wedderburn-Etherington Number primes, Euclid-Mullin sequence primes, Motzkin primes, Catalan primes, Schroder primes, Non-generous primes, Pell primes, Primeval primes, Primes of Binary Quadratic Form, Smarandache-Wellin primes, and Highly Cototient number primes. We also have indicated the membership of a number of special classes of prime numbers in class 2 category.

In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in a fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest.

Szemerédi's famous theorem on arithmetic progressions asserts that every subset of integers of positive asymptotic density contains arithmetic progressions of arbitrary length. His remarkable theorem has been developed into a major new area of combinatorial number theory. This is the topic of the present survey.