Dirichlet Divisor Research Articles

Toward optimal exponent pairs

We quantify the set of known exponent pairs ( k , ℓ ) (k, \ell ) and develop a framework to compute the optimal exponent pair for an arbitrary objective function. Applying this methodology, we make progress on several open problems, including bounds of the Riemann zeta-function ζ ( s ) \zeta (s) in the critical strip, estimates of the moments of ζ ( 1 / 2 + i t ) \zeta (1/2 + it) and the generalised Dirichlet divisor problem.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

On the generalised Dirichlet divisor problem

AbstractWe improve unconditional estimates on , the remainder term of the generalised divisor function, for large . In particular, we show that for all sufficiently large fixed .

The Dirichlet divisor problem over square-free integers and unitary convolutions

We obtain an asymptotic formula for the sum $\tilde{D}_2$ of the divisors of all square-free integers less than or equal to $x$, with error term $O(x^{1/2 + \epsilon})$. This improves the error term $O(x^{3/4 + \epsilon})$ presented in [7] obtained via analytical methods. Our approach is elementary and it is based on the connections between the function $\tilde{D}_2$ and unitary convolutions.

Complex Valued Multiplicative Functions with Bounded Partial Sums

We present a class of multiplicative functions \(f:\mathbb {N}\rightarrow \mathbb {C}\) with bounded partial sums. The novelty here is that our functions do not need to have modulus bounded by 1. The key feature is that they pretend to be the constant function 1 and that for some prime q, \(\sum _{k=0}^\infty \frac{f(q^k)}{q^k}=0\). These combined with other conditions guarantee that these functions are periodic and have sum equal to zero inside each period. Further, we study the class of multiplicative functions \(f=f_1*f_2\), where each \(f_j\) is multiplicative and periodic with bounded partial sums. We show an omega bound for the partial sums \(\sum _{n\le x}f(n)\) and an upper bound that is related with the error term in the classical Dirichlet divisor problem.

Balanced derivatives, identities, and bounds for trigonometric and Bessel series

Motivated by two identities published with Ramanujan's lost notebook and connected, respectively, with the Gauss circle problem and the Dirichlet divisor problem, in an earlier paper, three of the present authors derived representations for certain sums of products of trigonometric functions as double series of Bessel functions [8]. These series are generalized in the present paper by introducing the novel notion of balanced derivatives, leading to further theorems. As we will see below, the regions of convergence in the unbalanced case are entirely different than those in the balanced case. From this viewpoint, it is remarkable that Ramanujan had the intuition to formulate entries that are, in our new terminology, “balanced”. If x denotes the number of products of the trigonometric functions appearing in our sums, in addition to proving the identities mentioned above, theorems and conjectures for upper and lower bounds for the sums as x→∞ are established.

Two-parameter identities for divisor sums in algebraic number fields

In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an “extra” parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with fK(n), the number of integral ideals of norm n in an algebraic number field K. In this paper we establish Riesz sum identities containing an “extra” parameter and involving fK(n), or divisor functions associated with K. Upper bounds for the sums as the upper index tends to infinity are also established.

Piltz divisor problem over number fields à la Voronoï

In this article, we study the Piltz divisor problem, which is sometimes called the generalized Dirichlet divisor problem, over number fields. We establish an identity akin to Voronoï’s formula concerning the error term in the Dirichlet divisor problem.

The mean square discrepancy in the divisor problem

We study the mean square of the error term, or the mean square discrepancy, in the Dirichlet divisor problem. An analysis of the off-diagonal terms in the mean square of the Voronoi summation formula leads to a precise conjecture for this mean square discrepancy. Surprisingly, the discrepancy contains a slowly oscillating, unbounded term.

Uniform bounds of the Piltz divisor problem over number fields

We consider the upper bound of Piltz divisor problem over number fields. Piltz divisor problem is known as a generalization of the Dirichlet divisor problem. We deal with this problem over number fields and improve the error term of this function for many cases. Our proof uses the estimate of exponential sums. We also show uniform results for ideal counting function and relatively $r$-prime lattice points as one of applications.

A logarithmic estimate for harmonic sums and the digamma function, with an application to the Dirichlet divisor problem

Let H_{n} = sum_{r=1}^{n} 1/r and H_{n}(x) = sum_{r=1}^{n} 1/(r+x). Let psi(x) denote the digamma function. It is shown that H_{n}(x) + psi(x+1) is approximated by frac{1}{2}log f(n+x), where f(x) = x^{2} + x + frac{1}{3}, with error term of order (n+x)^{-5}. The cases x = 0 and n = 0 equate to estimates for H_{n} - gamma and psi(x+1) itself. The result is applied to determine exact bounds for a remainder term occurring in the Dirichlet divisor problem.

Shifted Euler constants and a generalization of Euler-Stieltjes constants

The purpose of this article is twofold. First, we introduce the constants ζk(α,r,q) where α∈(0,1) and study them along the lines of work done on Euler constant in arithmetic progression γ(r,q) by Briggs, Dilcher, Knopfmacher, Lehmer and some other authors. These constants are used for evaluation of certain integrals involving error term for Dirichlet divisor problem with congruence conditions and also to provide a closed form expression for the value of a class of Dirichlet L-series at any real critical point. In the second half of this paper, we consider the behaviour of the Laurent Stieltjes constants γk(χ) for a principal character χ. In particular we study a generalization of the “Generalized Euler constants” introduced by Diamond and Ford in 2008. We conclude with a short proof for a closed form expression for the first generalized Stieltjes constant γ1(r/q) which was given by Blagouchine in 2015.

An improved sieve of Eratosthenes

We show how to carry out a sieve of Eratosthenes up to N in space O(N^{1/3} (log N)^{2/3}) and essentially linear time. These bounds constitute an improvement over the usual versions of the sieve, which take space about O(sqrt{N}) and essentially linear time. We can also apply our sieve to any subinterval of [1,N] of length O(N^{1/3} (log N)^{2/3}) in time essentially linear on the length of the interval. Before, such a thing was possible only for subintervals of [1,N] of length >> sqrt{N}. Just as in (Galway, 2000), our approach is related to Diophantine approximation, and also has close ties to Voronoi's work on the Dirichlet divisor problem. The advantage of the method here resides in the fact that, because the method we will give is based on the sieve of Eratosthenes, we will also be able to use it to factor integers, and not just to produce lists of consecutive primes.

Power moments and value distribution of functions

In this paper we study various which one can associate to a given function $f$, or rather to the power moments of $f$. These are motivated by long standing open problems in analytic number theory. We show how these abscissae connect to the distribution of values of $f$ in a very elegant way using convex conjugates. This connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions. Further it may give a new approach to, for example, Dirichlet's divisor problem.

The probability that the number of points on a complete intersection is squarefree

We consider the asymptotic probability that integers chosen according to a binomial distribution will have certain properties: (i) that such an integer is not divisible by the $k$th power of a prime, (ii) that any $k$ of $s$ chosen integers are relatively prime and (iii) that a chosen integer is prime. We also prove an analog of the Dirichlet divisor problem for the binomial distribution. We show how these results yield corresponding facts concerning the number of points on a smooth complete intersection over a finite field.

On the seventh power moment of Δ(x)

Let [Formula: see text] be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text] which improves the previous result.

On representations of error terms related to the derivatives for some Dirichlet series

In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.

Short-interval averages of sums of Fourier coefficients of cusp forms

Let f be a weight k holomorphic cusp form of level one, and let Sf(n) denote the sum of the first n Fourier coefficients of f. In analogy with Dirichlet's divisor problem, it is conjectured that Sf(X)≪Xk−12+14+ϵ. Understanding and bounding Sf(X) has been a very active area of research. The current best bound for individual Sf(X) is Sf(X)≪Xk−12+13(log⁡X)−0.1185 from Wu [17].Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for Sf(X) holds on average over intervals of length X. Jutila [10] improved this result to show that the Classical Conjecture for Sf(X) holds on average over short intervals of length X34+ϵ. Building on the results and analytic information about ∑|Sf(n)|2n−(s+k−1) from our recent work [9], we further improve these results to show that the Classical Conjecture for Sf(X) holds on average over short intervals of length X23(log⁡X)16.

On some upper bounds for the zeta-function and the Dirichlet divisor problem

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

New pathways and connections in number theory and analysis motivated by two incorrect claims of Ramanujan

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoï, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings.The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ(12), and in 1925, S. Wigert obtained a generalization giving a formula for ζ(1k) for any even integer k≥2. We extend the work of Ramanujan and Wigert in this paper.The Voronoï summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σs(n).The modified Bessel functions Ks(x) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σs(n) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.