arXiv subjects
Publications and source records attributed to Janos Pintz.
In a recent work Friedlander studied the problem of how large consecutive prime gaps should be in order that the sum of the reciprocals should be divergent. Supposing a very deep Hypothesis, a generalization of the Hardy--Littlewood prime $k$-tuple conjecture, he gave an almost precise answer for it. In the present work we give an unconditional answer for a much weaker form of the same problem.
The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from a simple but ingenious idea of Halasz the proof uses only classical knowledge about the zeta function, results known since at least hundred years.
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$.
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.
We prove some new log-free density theorems for zeros of Dirichlet L-functions (which accordingly are more sharp than earlier ones near to the boundary line of the critical strip). The results can be applied in several problems of prime number theory.
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.
We show that there exists a bounded pattern of m consecutive primes for any m>0, that means a tuple H_m of m distinct non-negative integers h_i (i=1,2,...m) such that its translations contain arbitrarily long (finite) arithmetic progressions. More precisely, the set of natural numbers n for which all components n+h_i (i=1,2,...m) are consecutive primes contains arbitrarily long (finite) arithmetic progressions. Moreover, the set of m-tuples that satisfy this property represents a positive proportion of all m-tuples. The present result is the generalization of the results of Green-Tao (about the existence of arbitrarily long arithmetic progressions) and of Maynard/Tao (about the existence of infinitely many bounded blocks of m primes, where m is an arbitrary natural number). It also generalizes the author's work which first showed the existence of infinitely many Polignac numbers and they contain arbitrarily long (finite) arithmetic progressions (arXiv: 1305.6289v1, 27 May 2013) which was a common generalization of the above mentioned result of Green-Tao and that of Zhang (about the exisatence of infinitely many bounded gaps between primes).
Erdös conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an interval of type [0,c] with a positive ineffective value c. In the present work we extend this result for a large class of normalizing functions. The only essential requirement is that the function f(n) replacing logn should satisfy f(n)<<lognloglognloglogloglogn/(logloglogn)^2 (with a small implied constant), the well-known Erdös-Rankin bound for the largest known gaps between consecutive primes. The work also proves that apart from a thin set of exceptional functions the original Erdös conjecture holds if logn is replaced by a non-exceptional function f(n). The paper also gives a new proof for a result of Helmut Maier which generalized the Erdös-Rankin bound for an arbitrarily long finite chain of consecutive primegaps. The proof uses a combination of methods of Erdös-Rankin,Maynard-Tao and Banks-Freiberg-Maynard. Since the submission of the present work the very important recent simultaneous and independent works of Ford-Green-Konjagin-Tao (arXiv:1408.4505 [math.NT] and Maynard (aerXiv:1408.5110 [math.NT]) appeared on arXiv and they proved the old conjecture of Erdös which asserts that the lower bound for large gaps exceeds Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. In this new version we prove the same assertions as in the original work for the case when f(n)<<Clognloglognloglogloglogn/(logloglogn)^2 with an arbi8trarily large constant C, in particular we show that there are blocks of m primes for any m such that all gaps between these primes simultaneously satisfy the lower estimate Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. The proof uses the method of Maynard.
In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erdős-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erdös, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively.
We show a smoothed version of Goldston-Pintz-Yildirim's sifting argument to detect small gaps between primes, which has a higly flexible error term. Our argument is applicable to high dimensional Selberg sieve situations as well, although the relevant details are not stated explicitly. In the present v.2, we have added Appendix in which we made a minor correction to our reasoning following formula (5.14). We are indebted to Professor Terrence Tao for pointing out the necessity of this amendment. The text of the original version has not been changed, except for a correction of the title in the item [2] of the references.
As a refinement of the celebrated recent work of Yitang Zhang we show that any admissible k-tuple of integers contains at least two primes and almost primes in each component infinitely often if k is at least 181000. This implies that there are infinitely many gaps between consecutive primes of size at most 2530338, with an improved admissible k-tuple of Andrew W. Sutherland to 2326476.
In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, and Zhang, respectively.
In the present work we investigate the largest possible gaps between consecutive numbers which can be written as the difference of two primes. The best known upper bounds are the same as those concerning the largest possible difference of Goldbach numbers (that is, numbers which can be written as the sum of two primes). Thus, we know that any interval of the form [X, X+X^c] contains numbers which are the difference (or sum, respectively) of two primes, where c=21/800. It is announced in our work that there is a constant C such that for sufficiently large X all intervals of the form [X, X+(logX)^C] contain an even integer which can be written as the difference of two primes. The work contains, as an illustration of the method, the proof of the weaker result that given an arbitrarily small c>0, the interval [X, X+X^c] contains the difference of two primes if X is large enough. Some conditional results are announced too, which are valid under the deep unproved hypothesis that primes have an admissible level of distribution larger than 1/2. The above hypothesis implies, for example, the existence of a large constant C (depending on the admissible distribution level of the primes) such that for sufficiently large values of X the interval [X, X+C] contains at least one even number which can be written as the difference of two consecutive primes in infinitely many ways.
Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the present work it is proved that $p+d$ has an odd number of prime factors for at least one value of d=2,4,...16.
We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic progressions of primes $p$ such that $p+h$ is also prime for each element of the progression. In case of $h=2$ this means that under some plausible unproved hypotheses about regular distribution of the primes and other sequences in arithmetic progressions we really have arbitrarily long arithmetic progressions in the sequence of twin primes.
In the present work the existence of some patterns of primes is shown which generalize the celebrated result of Green and Tao according to which there are arbitrarily long arithmetic progressions in the sequence of primes
In a recent joint work with D.A. Goldston and C.Y. Yildirim we just missed by a hairbreadth a proof that bounded gaps between primes occur infinitely often. In the present work it is shown that adding to the primes a much thinner set, called almost prime powers, the union of the set of primes and almost prime powers contains already infinitely many bounded gaps. More precisely it is shown that if we add to the set of primes either almost prime squares having exactly two, nearly equal prime factors or if we add to the set of primes almost prime cubes having exactly three, nearly equal prime factors, then the resulting set contains already infinitely many bounded gaps.
In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the length of the interval $H$ is sufficiently large as a function of $k$). This result of Gallagher played a key role in our original joint work with D. A. Goldston and C. Y. Yildirim, where we showed the existence of infinitely many small gaps between consecutive primes. In the present work some weaker variants of Gallagher`s result are also proved (in an even easier way) which are still sufficient for the above mentioned applications.