Distribution of large values of the argument of the Riemann zeta function on short intervals

An upper bound for the measure of the set of values t ∈ (T,T + H] for H = T ^27/82+ɛ for which |S(t)| ≥ λ is obtained.     

On the divisor function in short intervals

Let d(n) denote the number of positive divisors of the natural number n. The aim of this paper is to investigate the validity of the asymptotic formula

$\begin{array}{lll}\sum \limits_{x < n \leq x+h(x)}d(n)\sim h(x)\log x\end{array}$

for \({x \to + \infty,}\) assuming a hypothetical estimate on the mean

$\begin{array}{lll} \int \limits_X^{X+Y}(\Delta(x+h(x))-\Delta (x))^2\,{d}x, \end{array}$

which is a weakened form of a conjecture of M. Jutila.

     

Distribution of values of additive functions on short intervals

We consider a family of completely additive functions β_q(n) defined on the set of natural numbers. We find an asymptotic expression for the summation function Σ _n≤x β _q (n study its distribution on short intervals.     

Values of the divisor function on short intervals

In this paper we prove (in a rather more precise form) two conjectures of P. Erdös about the maximum and minimum values of the divisor function on intervals of length k.     

On the divisor function and the Riemann zeta-function in short intervals

We obtain, for T ^ε ≤U=U(T)≤T ^1/2−ε , asymptotic formulas for

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.      

On square-full numbers in short intervals

It is shown that the number of square-full numbers in the interval is asymptotically equal to for every in the range 1/6>0.14254, which extends P.Shiu's range 1/6>0.1526.     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

Mean values for a class of arithmetic functions in short intervals

In this paper, we shall establish a rather general asymptotic formula in short intervals for a class of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable as sums of two squares.     

On the Dirichlet divisor problem in short intervals

We present several new results involving Δ(x+U)?Δ(x), where U=o(x) and $$\varDelta(x):=\sum_{n\leq x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.     

On prime numbers of special kind on short intervals

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .     

Bounding S(t) and S_1(t) on the Riemann hypothesis

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta (s)$ , at the point $s=\frac{1}{2}+it$ . Assuming the Riemann hypothesis, we present two proofs of the bound $$\begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned}$$ for large $t$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $S_1(t) = \int _0^{t} S(u) \> \text{ d}u$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log \log t$ . The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function $f(x) = \arctan \left(\frac{1}{x}\right) - \frac{x}{1 + x^2}$ . This draws upon recent work by Carneiro and Littmann.     

On the normal density of primes in short intervals

Selberg has shown on the basis of the Riemann hypothesis that for every ε > 0 most intervals |x,x+x^?| of length x^? contain approximately $\text{x}{}^{\mathrm{?}}\text{log}\text{x}$ primes. Here by “most” we mean “for a set of values of x of asymptotic density one.” Prachar has extended Selberg's result to primes in arithmetic progressions. Both authors noted that if we assume the quasi Riemann hypothesis, that ζ(s) has no zeros in the domain {$\text{σ>}\text{1}\text{2}\text{+δ}$} for some $\text{δ<}\text{1}\text{2}$, then the same conclusions hold, provided that ε > 2 δ. Here we give a simple proof of these theorems in a general context, where an arbitrary signed measure takes the place of d[ψ(x)?x]. Then we show by a counterexample that this general theorem is the best of its kind: the condition ε > 2δ cannot be replaced by ε = 2δ. In our example, the associated Dirichlet integral is an entire function which remains bounded on the domain {$\text{σ≥}\text{1}\text{2}\text{+δ}$}. Thus its growth and regularity properties are better than those of $\text{ζ′(s)}\text{ζ(s)}$. Nevertheless the corresponding signed measure behaves badly.     

关于函数S(x)及S*(x)的渐近性质

对于任意实数x∈(1,∞),定义S(x)=min{m∈Nx≤m!};x∈[1,∞),S*(x)=max{m∈Nm!≤x}.主要目的是研究这两个函数的渐近性质,并给出了它们的渐近公式.     

On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k}}, \hfill \\ {\left| {{p_j} - \sqrt {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)

     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals for all α ∈ [0,1] whenever . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals S₂(x,y;a)=?_{x < n ￡ x+y}L(n)e(n² a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha}) for all α ∈ [0,1] whenever x^\frac23+e ￡ y ￡ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.