Mathematics

We investigate lower bounds for the variance in arithmetic progressions of certain multiplicative functions "close" to 1 . Specifically, we consider α N -fold divisor functions, when α N is a sequence of positive real numbers approaching 1 in a suitable way or α N =1 , and the indicator of y -smooth numbers, for suitably large parameters y . As a corollary, we will strengthen a previous author's result on the first subject and obtain matching lower bounds to some Barban-Davenport-Halberstam type theorems for y -smooth numbers. Incidentally, we will also find a lower bound for the variance in arithmetic progressions of the prime factors counting functions ?(n) and Ω(n) .

We present a new random approximation method that yields the existence of a discrete Beurling prime system P={ p 1 , p 2 ,?�} which is very close in a certain precise sense to a given non-decreasing, right-continuous, nonnegative, and unbounded function F . This discretization procedure improves an earlier discrete random approximation method due to H. Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704]. We obtain several applications. Our new method is applied to a question posed by M. Balazard concerning Dirichlet series with a unique zero in their half plane of convergence, to construct examples of very well-behaved generalized number systems that solve a recent open question raised by T. Hilberdink and A. Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling prime system with regular primes but extremely irregular integers was constructed.

Improving on some recent results of Matomäki and of Wright, we show that the number of Carmichael numbers to X in a coprime residue class exceeds X 1/(6logloglogX) for all sufficiently large X depending on the modulus of the residue class.

The Ohno relation is a well-known relation among multiple zeta values. Hirose, Onozuka, Sato, and the author investigated the sum related to the Ohno relation and presented two types of new relations and five conjectural formulas. This paper proves one of these formulas.

In this paper, we enhance the lower bound for the number of balancing non-Wieferich primes in arithmetic progression. In particular, for any given integer r?? there are ?�logx balancing non-Wieferich primes p?�x with the truth of the abc conjecture for the number field Q( 2 ????) . This improves the more recent result of Y. Wang and Y. Ding.

In this note we show a simple formula for the coefficients of the polynomial associated to the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, as a by-product, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind.

In the present paper we explore a way to represent numbers with respect to the base ??3 2 using the set of digits {0,1,2} . Although this number system shares several properties with the classical decimal system, it shows remarkable differences and reveals interesting new features. For instance, it is related to the field of 2 -adic numbers, and to some ``fractal'' set that gives rise to a tiling of a non-Euclidean space.

The pair correlation is a localized statistic for sequences in the unit interval. Pseudo-random behavior with respect to this statistic is called Poissonian behavior. The metric theory of pair correlations of sequences of the form ( a n α ) n≥1 has been pioneered by Rudnick, Sarnak and Zaharescu. Here α is a real parameter, and ( a n ) n≥1 is an integer sequence, often of arithmetic origin. Recently, a general framework was developed which gives criteria for Poissonian pair correlation of such sequences for almost every real number α , in terms of the additive energy of the integer sequence ( a n ) n≥1 . In the present paper we develop a similar framework for the case when ( a n ) n≥1 is a sequence of reals rather than integers, thereby pursuing a line of research which was recently initiated by Rudnick and Technau. As an application of our method, we prove that for every real number θ>1 , the sequence ( n θ α ) n≥1 has Poissonian pair correlation for almost all α∈R .

Let k?? be a fixed integer, and P N be the set of primes no more than N . We prove that if set A??P N contains no patterns p 1 , p 1 +( p 2 ?? ) k , where p 1 , p 2 are prime numbers, then |A| | P N | ?? loglogN logN ) 1 12 k 3 .

In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers { h n } n?? is an elliptic divisibility sequence if it verifies the recurrence relation h m+n h m?�n h 2 r = h m+r h m?�r h 2 n ??h n+r h n?�r h 2 m for every natural number m?�n?�r . The second definition says that a sequence of integers { β n } n?? is an elliptic divisibility sequence if it is the sequence of the square roots (chosen with an appropriate sign) of the denominators of the abscissas of the iterates of a point on a rational elliptic curve. It is well-known that the two sequences are not equivalent. Hence, given a sequence of the denominators { β n } n?? , in general does not hold β m+n β m?�n β 2 r = β m+r β m?�r β 2 n ??β n+r β n?�r β 2 m for m?�n?�r . We will prove that the recurrence relation above holds for { β n } n?? under some conditions on the indexes m , n , and r .