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

Abstract

Abstract. We obtain, for T ε ≤ U = U(T ) ≤ T 1/2−ε , asymptotic formulas forwhere ∆(x) is the error term in the classical divisor problem, and E(T ) is the error term in the mean square formula for |ζ( 1 2 + it)|. Upper bounds of the form O ε (T 1+ε U 2 ) for the above integrals with biquadrates instead of square are shown to hold for T 3/8 ≤ U = U(T ) ≪ T 1/2 . The connection between the moments of E(t + U) − E(t) and |ζ( 1 2 + it)| is also given. Generalizations to some other number-theoretic error terms are discussed. IntroductionPower moments represent one of the most important parts of the theory of the Riemann zeta-function ζ(s), defined asand otherwise by analytic continuation. Of particular significance are the moments on the "critical line" σ = 1 2 , and a vast literature exists on this subject (see e.g., the monographs [5], [6], and [23]). In this paper we shall be concerned with moments of the error function1991 Mathematics Subject Classification. 11 M 06, 11 N 37. Key words and phrases. The Riemann zeta-function, the divisor functions, power moments in short intervals, upper bounds.Typeset by A M S-T E X

Resonances and Ω-results for exponential sums related to Maass forms for SL(n,Z)

“…The behaviour of the exponential sums is extremely intriguing: One might expect square-root cancellation, and while this is not an unreasonable assumption in the GL(2) setting when ∆ ≪ M 1/2 , Ernvall-Hytönen [4] has proved that where c ∈ R + is an arbitrary fixed coefficient. This Ω-result extends the earlier work by Ivić [15] (see also [19]), where he has shown that for ∆ = o( √ M ). While upper and lower bounds are relatively widely studied in the GL (2) setting (see e.g.…”

On the variance of sums of divisor functions in short intervals

“…Ivić [7] gives a more precise formula for the left-hand side of (1.3) that includes lower order terms and an error term with a power savings in X. We can also prove more precise formulas than those stated in Theorems 1.1 and 1.2.…”

“…However, this upper bound is most likely not sharp and the true order of magnitude is probably of size (X 1− 1 k /L) · (log L) k 2 −1 . More precise estimates than (1.2) in the case that k = 2 are given by Jutila [8] and Ivić [7] (see also [1] and [9]). In particular, Ivić [7] derives an explicit asymptotic formula for the variance of sums of d 2 (n) = d(n) in short intervals with x < n ≤ x + h and…”

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals

“…The interesting range for us is that of shorter intervals: H < X 1− 1 k . For k = 2, Jutila [20], Coppola and Salerno [8], and Ivić [18,19] show that, for X ǫ < H < X 1/2−ǫ , the mean square of ∆ 2 (x, H) is asymptotically equal to…”