Functions In Short Intervals Research Articles

On the asymptotic formulae for some multiplicative functions in short intervals

We are going to study the mean values of some multiplicative functions connected with the divisor function in short interval of summation. The asymptotics for such mean values will be proved. Considering instead of well-known multiplicative functions, their inverses lead to very weak results of application of standard methods of complex integration. In order to get better estimations, we propose another method which uses as its main tools the density estimates and zero-free region for Riemann ζ-function and Dirichlet L-functions.

Strong Orthogonality between the Möbius Function and Nonlinear Exponential Functions in Short Intervals

Journal Article Strong Orthogonality between the Möbius Function and Nonlinear Exponential Functions in Short Intervals Get access Bingrong Huang Bingrong Huang School of Mathematics, Shandong University, Jinan, Shandong 250100, China Correspondence to be sent to: brhuang@mail.sdu.edu.cn Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 2015, Issue 23, 2015, Pages 12713–12736, https://doi.org/10.1093/imrn/rnv091 Published: 26 March 2015 Article history Received: 02 December 2014 Revision received: 03 March 2015 Accepted: 04 March 2015 Published: 26 March 2015

Random Multiplicative Functions in Short Intervals

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.

Rényi’s Problem in Short Intervals

In this paper, we study Renyi’s problem and other related problems about the additive functions in short intervals. As corollaries, we improve Ivic’s results.

Density theorems and the mean value of arithmetical functions in short intervals

Let Γ=SL2(ℤ) and let ZΓ(s) be the Selberg zeta function. Set πΓ(P)=∑N(P)≤P1, where P is a primitive hyperbolic class of conjugate elements in Γ and N(P) is the norm of P. It is shown that for P1/2+θ=Q, 1≤θ≤1/2, we have $$\pi _\Gamma \left( {P + Q} \right) - \pi _\Gamma \left( P \right) = \int\limits_P^{P + Q} {\frac{{du}}{{\log u}} + O_ \in \left( {QP^{ - \sigma \left( \theta \right) + \in } } \right),} $$ where σ(θ)=θ2/2+O(¸3), θ→0. Thus, a conjecture of Iwaniec (1984) is proved. Similar asymptotic formulas are obtained for the sums ∑P<d≤P+Qh(-d)/ $$\sqrt d $$ and ∑P<n≤P+Qr3(n)/ $$\sqrt n $$ , where h(−d) is the class number of the imaginary quadratic field of discriminant −d<0 and r3(n) is the number of representations of n by the sum of three squares.

Reciprocal sums of a class of additive functions in short intervals

USING the method of probabilistic number-theory, De ~oninck"] , and De Koninck and ~ a l a q b o s ' ~ ] studied the reciprocal sum of the additive function f ( n ) satisfying f ( n )>,to >O ( n 2 2 ) and f ( p -1 and obtained an asymptotic formula, where t o is an absolute positive constant. Let B ( x ) denote the number of n <x not satisfying the inequality loglogn R ( x ) < f ( n ) < loglogn + R ( x ) , ( 1 where R ( x ) is a function tending to infinity. Then in refs. [ I , 21, it is proved that if R ( x ) = o ( loglogx ) and B ( x ) = o ( xlloglogx ) , and

Limiting distributions of additive functions in short intervals

to the characteristic function ~p(t) of F as y tends to infinity. Then cp(t) is continuous in a neighbourhood of t= 0 and ~p (0)= 1. Therefore the problem we have posed is equivalent to the investigation of the y's for which the sum in (5) converges to a nonzero limit for all sufficiently small t. Letfy = f be an additive function. We show that one may choose y = x h(~), where h ( x ) = l e ( x ) with e(x)>0 and e(x)=o(1) as x ~ o , so that thelimitingdistribution of an arbitrary real-valued additive function f(n) in the interval x-y 0 is given, one can construct an additive function f which fulfils the Erd6s--Wintner conditions but possesses no limiting distribution in the interval x-x l -~<n<-x . Assuming that f(p=)-~O as p-~o~ (re=l, 2, ...) we choose h(x)=o(1) where o(1) depends on f (Corollary 2). Cases in which h(x) lies between these extremes are treated in Corollary 3.

A Note on the Distribution Function of Additive Arithmetical Functions in Short Intervals

AbstractLet f be an additive arithmetical function having a distribution F. For any sequence letIn this note, we determine the slowest growing function b so that Qn{b, f) tends weakly to F, for various f.

Multiplicative Functions in Short Intervals

A central problem in probabilistic number theory is to evaluate asymptotically the partial sumsof multiplicative functions f and, in particular, to find conditions for the existence of the “mean value”1.1In the last two decades considerable progress has been made on this problem, and the results obtained are very satisfactory.

On the maximal value of additive functions in short intervals and on some related questions

On the maximal value of additive functions in short intervals and on some related questions