scholarly journals TWO CONDITIONAL RESULTS ABOUT PRIMES IN SHORT INTERVALS

2011 ◽  
Vol 07 (07) ◽  
pp. 1753-1759
Author(s):  
DANILO BAZZANELLA
Keyword(s):  
Asymptotic Formula ◽  
Short Intervals ◽  
Lindelöf Hypothesis

In 1937, Ingham proved that ψ(x + xθ) - ψ(x) ~ xθ for x → ∞, under the assumption of the Lindelöf hypothesis for θ > 1/2. In this paper we examine how the above asymptotic formula holds by assuming in turn two different heuristic hypotheses. It must be stressed that both the hypotheses are implied by the Lindelöf hypothesis.

scholarly journals A symplectic restriction problem

2021 ◽  
Author(s):  
Valentin Blomer ◽  
Andrew Corbett
Keyword(s):  
Asymptotic Formula ◽  
Modular Form ◽  
Half Space ◽  
Imaginary Part ◽  
Dimensional Subspace ◽  
Siegel Modular Form ◽  
3 Dimensional ◽  
Relative Trace Formula ◽  
Lindelöf Hypothesis ◽  
Restriction Problem

AbstractWe investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

Cube-full numbers in short intervals

1992 ◽  
Vol 112 (1) ◽  
pp. 1-5 ◽  
Author(s):  
P. Shiu
Keyword(s):  
Asymptotic Formula ◽  
Short Intervals

AbstractAn innovation by D. R. Heath-Brown on square-full numbers in short intervals is applied to establish an asymptotic formula for the number of cube-full numbers in the interval

scholarly journals On the variance of squarefree integers in short intervals and arithmetic progressions

2021 ◽  
Author(s):  
Ofir Gorodetsky ◽  
Kaisa Matomäki ◽  
Maksym Radziwiłł ◽  
Brad Rodgers
Keyword(s):  
Riemann Hypothesis ◽  
Full Range ◽  
Arithmetic Progressions ◽  
Short Intervals ◽  
Squarefree Integers ◽  
Lindelöf Hypothesis

AbstractWe evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

scholarly journals On two sums related to the Lehmer problem over short intervals

2021 ◽  
Vol 6 (11) ◽  
pp. 11723-11732
Author(s):  
Yanbo Song ◽  
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Lehmer Problem ◽  
Lehmer Numbers

<abstract><p>In this article, we study sums related to the Lehmer problem over short intervals, and give two asymptotic formulae for them. The original Lehmer problem is to count the numbers coprime to a prime such that the number and the its number theoretical inverse are in different parities in some intervals. The numbers which satisfy these conditions are called Lehmer numbers. It prompts a series of investigations, such as the investigation of the error term in the asymptotic formula. Many scholars investigate the generalized Lehmer problems and get a lot of results. We follow the trend of these investigations and generalize the Lehmer problem.</p></abstract>

scholarly journals SHORT INTERVALS ASYMPTOTIC FORMULAE FOR BINARY PROBLEMS WITH PRIME POWERS, II

2019 ◽  
Vol 109 (3) ◽  
pp. 351-370 ◽  
Author(s):  
ALESSANDRO LANGUASCO ◽  
ALESSANDRO ZACCAGNINI
Keyword(s):  
Asymptotic Formula ◽  
Prime Numbers ◽  
Short Intervals ◽  
Asymptotic Formulae ◽  
Representations Of Integers ◽  
Binary Problems

AbstractWe improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, $i=1,\ldots ,s$, are prime numbers, holds in short intervals.

scholarly journals Bykovskii-Type Theorem for the Picard Manifold

2020 ◽  
Author(s):  
Antal Balog ◽  
András Biró ◽  
Giacomo Cherubini ◽  
Niko Laaksonen
Keyword(s):  
Asymptotic Formula ◽  
Explicit Formula ◽  
Short Interval ◽  
Type Theorem ◽  
Exponential Sum ◽  
Interval Estimate ◽  
Gaussian Integers ◽  
Short Intervals ◽  
Prime Geodesic Theorem ◽  
Gauss Circle Problem

Abstract We generalise a result of Bykovskii to the Gaussian integers and prove an asymptotic formula for the prime geodesic theorem in short intervals on the Picard manifold. Previous works show that individually the remainder is bounded by $O(X^{13/8+\epsilon })$ and $O(X^{3/2+\theta +\epsilon })$, where $\theta$ is the subconvexity exponent for quadratic Dirichlet $L$-functions over $\mathbb{Q}(i)$. By combining arithmetic methods with estimates for a spectral exponential sum and a smooth explicit formula, we obtain an improvement for both of these exponents. Moreover, by assuming two standard conjectures on $L$-functions, we show that it is possible to reduce the exponent below the barrier $3/2$ and get $O(X^{34/23+\epsilon })$ conditionally. We also demonstrate a dependence of the remainder in the short interval estimate on the classical Gauss circle problem for shifted centres.

ON THE HARDY–LITTLEWOOD PROBLEM IN SHORT INTERVALS

2008 ◽  
Vol 04 (05) ◽  
pp. 715-723
Author(s):  
ALESSANDRO LANGUASCO ◽  
ALESSANDRO ZACCAGNINI
Keyword(s):  
Asymptotic Formula ◽  
Riemann Hypothesis ◽  
Short Intervals

We study the distribution of Hardy–Littlewood numbers in short intervals both unconditionally and conditionally, i.e. assuming the Riemann Hypothesis (RH). We prove that a suitable average of the asymptotic formula for the number of representations of a Hardy–Littlewood number holds in the interval [n, n + H], where H < X1-1/k+∊ and n ∈ [X, 2X].

ON THE DISTRIBUTION OF r-TUPLES OF SQUAREFREE NUMBERS IN SHORT INTERVALS

2006 ◽  
Vol 02 (02) ◽  
pp. 225-234 ◽  
Author(s):  
D. I. TOLEV
Keyword(s):  
Asymptotic Formula ◽  
Short Interval ◽  
Expected Value ◽  
Short Intervals

We consider the number of r-tuples of squarefree numbers in a short interval. We prove that it cannot be much bigger than the expected value and we also establish an asymptotic formula if the interval is not very short.

scholarly journals GENERALIZED D. H. LEHMER PROBLEM OVER SHORT INTERVALS

2010 ◽  
Vol 53 (2) ◽  
pp. 293-299 ◽  
Author(s):  
PING XI ◽  
YUAN YI
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Asymptotic Properties ◽  
Gauss Sums ◽  
Kloosterman Sums ◽  
Short Intervals ◽  
Lehmer Problem ◽  
Consecutive Integers ◽  
Image Position ◽  
Fixed Positive Integer

AbstractLet n ≥ 2 be a fixed positive integer, q ≥ 3 and c, ℓ be integers with (nc, q)=1 and ℓ|n. Suppose and consist of consecutive integers which are coprime to q. We define the cardinality of a set: The main purpose of this paper is to use the estimates of Gauss sums and Kloosterman sums to study the asymptotic properties of N(, , c, n, ℓ; q), and to give an interesting asymptotic formula for it.

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