Density functions for prime and relatively prime numbers

Letr ^*(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ^*(x) denote the maximum number ofprime integers in any interval (y,y+x] whereyx. Throughout this paper we assume the primek-tuples hypothesis. (This hypothesis could be avoided by using an alternative sievetheoretic definition of ^*(x); cf. the beginning of Section 1.) We investigate the differencer ^*(x)—^*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr ^*(x)—^*(x)<x ^c for somec>0 (whenx). This improves the previous lower bound of logx. As an upper bound we getr ^*(x)—^*(x)=o[x/(logx)²]. It is known that ^*(x)—(x)>const.[x/(logx)²];.; thus the difference betweenr ^*(x) and ^*(x) is negligible compared to ^*(x)—(x). The results mentioned so far involve the upper bound or maximizing sieve. In Section 2, similar comparisons are made between two types of minimum sieves. One of these is the erasing sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdös and Selfridge [1], involves a kind of minimax for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.     

Special prime numbers and discrete logs in finite prime fields

A set of primes involving numbers such as , where and , is defined. An algorithm for computing discrete logs in the finite field of order with is suggested. Its heuristic expected running time is for , where as , , and . At present, the most efficient algorithm for computing discrete logs in the finite field of order for general is Schirokauer's adaptation of the Number Field Sieve. Its heuristic expected running time is for . Using rather than general does not enhance the performance of Schirokauer's algorithm. The definition of the set and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some .

     

Partition into terms with prime numbers

The survey covers works on the additive number theory during the period 1954–1977. Results pertaining to the classical problems of Goldbach, Hardy-Littlewood, and analogous problems are considered.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 5–33, 1978.     

Distribution of generalized Fermat prime numbers

Numbers of the form are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form . The theoretical distributions of GFN primes, for fixed , are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn's quantitative form of ``Hypothesis H" of Schinzel and Sierpinski. A list of the current largest known GFN primes is included.

     

Class numbers with many prime factors

Here, we construct infinitely many number fields of any given degree d>1 whose class numbers have many prime factors.     

Exceptional characters and prime numbers in short intervals

Under the assumption of the Riemann hypothesis the asymptotic value y/log x is known to hold for the number of primes in the short interval [x - y, x] for for every fixed . We show under the assumption of the existence of exceptional Dirichlet characters the same asymptotic formula holds in the shorter intervals, for some \, in wide ranges of x depending on the characters.     

Quadratic residues and the distribution of prime numbers

D. Shanks [11] has given a heuristical argument for the fact that there are “more” primes in the non-quadratic residue classes modq than in the quadratic ones. In this paper we confirmShanks' conjecture in all casesq<25 in the following sense. Ifl ₁ is a quadratic residue,l ₂ a non-residue modq, ε(n, q, l ₁,l ₂) takes the values +1 or ?1 according ton?l ₁ orl ₂ modq, then $$\mathop {\lim }\limits_{x \to \infty } \sum\limits_p {\varepsilon (p,q,l_1 ,l_2 )} \log pp^{ - \alpha } \exp ( - (\log p)^2 /x) = - \infty$$ for 0≤α<1/2. In the general case the same holds, if all zeros ?=β+yγ of allL(s, χ modq),q fix, satisfy the inequality β²?γ²<1/4.     

Elementary and combinatorial methods for counting prime numbers

Although prime numbers are elementary objects in number theory, the first non-trivial results about their distribution in history rely on analytical methods (see [10]). It was a big surprise when Erd?s [5] and Selberg [12] discovered new proofs of the celebrated prime number theorem without the help of advanced tools from (complex) analysis. However, both approaches, which are not completely unrelated (see [8]), still make use of limits, in particular the real logarithm. In this article we shall introduce a rational logarithm without using any limit, and then derive classical results first due to Euler, Chebyshev and Mertens. Moreover, we revisit all necessary elementary results about prime numbers, sometimes proven in a more combinatorial fashion than usual.     

Prime and prime power divisibility of Catalan numbers

For any prime p, the sequence of Catalan numbers

$\text{a}{}_{\mathrm{n}}\text{=}\text{1}\text{n}\text{2n?2}\text{n?1}$

is divided by the a_n prime to p into blocks B_k(k > 0) of a_n divisible by p. The lengths and positions of the B_k are determined. Additional results are obtained on prime power divisibility of Catalan numbers.     

Discrepancies in the distribution of prime numbers

Chebyshev has noticed a certain predominance of primes of the form 4n + 3 over those of the form 4n + 1. He asserted that $\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{<mtext>?p</mtext><mtext>x</mtext>}}\text{= ?∞}$. This was unproven until today. G. H. Hardy, J. E. Littlewood and E. Landau have shown its equivalence with an analogue to the famous Riemann hypothesis, namely, L(s, χ₁mod 4) ≠ 0, $\text{Re}\text{(s) >}\text{1}\text{2}$. S. Knapowski and P. Turán have given some similar (unproven) relations, e.g., , which are also equivalent to the above. Using Explixit Formulas the author shows that

$\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{log}\text{pp}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}\text{e}{}^{\mathrm{<mtext>?</mtext><mtext>(</mtext><mtext>log</mtext><mtext>2p)</mtext><mtext>x</mtext>}}\text{= ?∞ (1)}$

holds without any conjecture. (In addition, the order of magnitude of divergence is calculated.) It turns out that (1) is only a special case (in several respects). At first, it may be enlarged into

$\text{lim}{}_{\mathrm{x\to \infty }}\text{∑}\text{p > 2}\text{(?1)}{}^{\mathrm{<mtext>(p\; ?\; 1)</mtext><mtext>2</mtext>}}\text{log}\text{pp}{}^{\mathrm{?\alpha }}\text{e}{}^{\mathrm{<mtext>?</mtext><mtext>(</mtext><mtext>log</mtext><mtext>2p)</mtext><mtext>x</mtext><mtext>)</mtext>}}\text{= ?∞, 0?α?}\text{1}\text{2}\text{.}$

Then, it can be generalised to a wider class of progressions. For example, the same is true if one sums over the primes in the classes 3n + 2 and 3n + 1, with a “?” and a “+” sign, respectively. All results of this type depend on the location of the first nontrivial zero of the corresponding L-series. D. Shanks has given some arguments for the predominance of primes in residue classes of nonquadratic type. He conjectured “If m₁ mod k is a quadratic residue and m₂ mod k a non-residue, then there are “more” primes congruent m₂ than congruent m₁ mod k.” This indeed turns out to be true in the sense of (1), not only for k = 3, 4, but for some higher moduli as well. Finally, numerical calculations were made to investigate the behaviour of Δ₃(X) ? π(X, 2 mod 3) ? π(X, 1 mod 3) in the interval 2 ≤ X ≤ 18, 633, 261. No zero was found in this range. In the analogue case of Δ₄(X) ? π(X, 3 mod 4) ? π(X, 1 mod 4) the first sign change occurs at X = 26, 861.     

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inL ^p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2. This paper is a part of the author’s Ph.D. thesis supervised by V. Bergelson.     

Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.     

On a Diophantine inequality involving prime numbers

The Ramanujan Journal - Let $$2&lt; c &lt; \frac{52}{25}$$. In this paper, it is proved that for any sufficiently large real number N, the Diophantine inequality $$|p_1^{c} + p_2^{c} +...     

On a binary Diophantine inequality involving prime numbers

The Ramanujan Journal - Let $$1&lt; c &lt; frac{59}{44},, cne frac{4}{3}$$ . In this paper it is proved that for any sufficiently large real N, for almost all real $$Tin (N,2N]$$ (in...     

Density of Carmichael numbers with three prime factors

We get an upper bound of on the number of Carmichael numbers with exactly three prime factors.