Aleksandar Ivic

• Ph. D.
• Academician at Serbian Academy of Sciences and Arts, Knez Mihailova 35, Belgrade, Serbia

It is shown explicitly how the sign of Hardy’s function Z(t) depends on the parity of the zero-counting function N(T). Two existing definitions of this function are analyzed, and some related problems are discussed.

Let S(t):=[Formula presented]arg⁡ζ([Formula presented]+it). We prove that, for T27/82+ε⩽H⩽T, we have mes{t∈[T,T+H]:S(t)>0}=[Formula presented]+O([Formula presented]), where the O-constant is absolute. A similar formula holds for the measure of the set with S(t)<0, where logk⁡T=log⁡(logk−1⁡T). This result is derived from an asymptotic formula for th...

This paper deals primarily with the analytic continuation of $G(s) = \sum_{\gamma>0}\gamma^{-s}$, where $\gamma$ denotes ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$. Several old and new results are presented, including an equivalent form of the Riemann hypothesis in terms of the analytic continuation of $G(s)$. New results ar...

Talk given on September 3, 2018 in Bonn, at the conference ELAZ2018. This was held at the Max Planck Institute for Mathematics in Bonn.

Let $S(t) \;:=\; \frac{\displaystyle 1}{\displaystyle \pi}\arg \zeta(\frac{1}{2} + it)$. We prove that, for $T^{\,27/82+\varepsilon} \le H \le T$, we have $$ {\rm mes}\Bigl\{t\in [T, T+H]\;:\; S(t)>0\Bigr\} = \frac{H}{2} + O\left(\frac{H\log_3T}{\varepsilon\sqrt{\log_2T}}\right), $$ where the $O$-constant is absolute. A similar formula holds for th...

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critic...

Hardy-Ramanujan Journal-(yyyy),-submitted dd/mm/yyyy, accepted dd/mm/yyyy, revised dd/mm/yyyy Hardy-Ramanujan Journal-(yyyy),-submitted dd/mm/yyyy, accepted dd/mm/yyyy, revised dd/mm/yyyy Abstract. Let γ denote the imaginary parts of complex zeros ρ = β + iγ of ζ(s). The problem of analytic continuation of the function G(s) := γ>0 γ −s to the left...

The aim is to give estimates for $$\sum_{T<\gamma\le T+H}|\zeta(1/2 + i\gamma)|^2\qquad(1 \ll H = H(T) \le T),$$ where $\gamma >0$ denotes ordinates of complex zeros of $\zeta(s)$. Some related integrals are also discussed. If the RH holds, the above some vanishes, but the accent is on unconditional estimates.

This is the first part of a review lecture presenting some of my recent results and problems on Hardy's function $Z(t)$

This is the second part of a lecture on Hardy;s function, given at the NTNU, Trondheim, March 15, 2018

It is shown explicitly how the sign of Hardy's function Z(t) depends on the parity of the zero-counting function N(T). Two existing definitions of this function are analyzed, and some related problems are discussed.

It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|\zeta(1/2+i\gamma)|^2$$ and some related integrals is presented, where $\gamma\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. It is shown unconditionally that the above sum is $\,\ll H\log^2T\log\log T\,$ for $\,T^{2/3}\log^4T \ll H \le T...

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is investigated, and its Laurent expansion at the pole $s=1$ is obtained. Estimates for the second moment on the critic...

Let as usual $Z(t) = \zeta(1/2+it)t\chi^{-1/2}(1/2+it)$ denote Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$. Assuming the Riemann hypothesis upper and lower bounds for some integrals involving $Z(t)$ and $Z'(t)$ are proved. It is also proved that $$ H(\log T)^{k^2} \ll_{k,\alpha} \sum_{T<\gamma\le T+H}\max_{\gamma\le \tau_\gamma\le \gamma...

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^X\Delta(x)\Delta_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than...

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^X\Delta(x)\Delta_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than...

We investigate bounds for the multiplicities m(β+iγ), where β+iγ (β≥12,γ>0) denotes complex zeros of ζ(s). It is seen that the problem can be reduced to the estimation of the integrals of the zeta-function over "very short" intervals. A new, explicit bound for m(β+iγ) is also derived, which is relevant when β is close to unity. The related Karatsub...

An asymptotic formula for $$\displaystyle{ \int _{T/2}^{T}Z^{2}(t)Z(t + U)dt\qquad (0 <U = U(T)\leqslant T^{1/2-\varepsilon }) }$$ is derived, where $$\displaystyle{ Z(t):= \zeta \left ({1 \over 2} + it\right )\big(\chi \left ({1 \over 2} + it\right )\big)^{-1/2}\quad (t \in \mathbb{R}),\quad \zeta (s) =\chi (s)\zeta (1 - s) }$$ is Hardy’s function...

We investigate bounds for the multiplicities $m(\beta+i\gamma)$, where $\beta+i\gamma\,$ ($\beta\ge \1/2, \gamma>0)$ denotes complex zeros of $\zeta(s)$. It is seen that the problem can be reduced to the estimation of the integrals of the zeta-function over "very short" intervals. A new, explicit bound for $m(\beta+i\gamma)$ is also derived, which...

Several problems and results involving the multiplicities of the Riemann zeta-function $\zeta(s)$ are discussed, as well as the related Karatsuba conjectures. The connection with mean values over short intervals is established.

This is primarily an overview article on some results and problems involving the classical Hardy function Z(t):= ζ(1/2 + it)(χ(1/2 + it))−1/2, ζ(s) = χ(s)ζ(1 − s). In particular, we discuss the first and third moments of Z(t) (with and without shifts) and the distribution of its positive and negative values. A new result involving the distribution...

In this overview paper, presented at the meeting ELAZ2014, Hildesheim, July 28–August 1, 2014, we present some selected works of the eminent mathematician Wolfgang Schwarz. This choice is personal and reflects the common research interest of the author and Prof. Schwarz.

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le T}{(\gamma_{n+1}-\gamma_n)}^k \qquad(k>0) $$ are proved.

Some results and problems on Hardy's function $$ Z(t) = \zeta(1/2+it)t\bigl(\chi(\hf+it)\bigr)^{-1/2}, \;\zeta(s) = \chi(s)\zeta(1-s) $$ are presented.

We investigate the distribution of positive and negative values of Hardy's function $$ Z(t) := \zeta(1/2+it){\chi(1/2+it)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular we prove that $$ \mu\bigl(I_{+}(T,T)\bigr) \;\gg T\; \qquad \hbox{and}\qquad \mu\bigl(I_{-}(T, T)\bigr) \; \gg \; T, $$ where $\mu(\cdot)$ denotes the Lebesgue measur...

We investigate the distribution of positive and negative values of Hardy's function $$ Z(t) := \zeta(1/2+it){\chi(1/2+it)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular we prove that $$ \mu\bigl(I_{+}(T,T)\bigr) \;\gg T\; \qquad \hbox{and}\qquad \mu\bigl(I_{-}(T, T)\bigr) \; \gg \; T, $$ where $\mu(\cdot)$ denotes the Lebesgue measur...

Some results and problems on Hardy's classical function $$ Z(t) = \zeta(1/2+it)\chi^{-1/2}(1/2+it),\quad \zeta(s) = \chi(s)\zeta(1-s) $$ are presented.

Let d(n) be the number of divisors of n, let Δ(x) :=∑n≤xd(n) − x(log x + 2γ − 1) denote the error term in the classical Dirichlet divisor problem, and let ζ(s) denote the Riemann zeta-function. Several upper bounds for integrals of the type ∫0TΔk(t) = ζ 1 2 + it2mdt(k,m ∈ ℕ) are given. This complements the results of [A. Ivić and W. Zhai, On some m...

Hardy's function $Z(t)$-results and problems Aleksandar Ivi´cIvi´c 1 Abstract. This is primarily an overview article on some results and problems involving the classical Hardy function $Z(t) := ζ(1 2 + it) χ(1 2 + it)^{ −1/2} , ζ(s) = χ(s)ζ(1 − s)$. In particular, we discuss the first and third moment of $Z(t)$ (with and without shifts) and the dis...

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt<< HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt << T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/...

An asymptotic formula for $$ \int_{T/2}^{T}Z^2(t)Z(t+U)\,dt\qquad(0< U = U(T) \le T^{1/2-\varepsilon}) $$ is derived, where $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}\quad(t\in\Bbb R), \quad \zeta(s) = \chi(s)\zeta(1-s) $$ is Hardy's function. The cubic moment of $Z(t)$ is also discussed, and a mean value result is presented which su...

Several upper bounds for $\int_0^T\Delta^k(t)|\zeta(1/2+it)|^{2m}dt$, where $k,m$ are fixed integers, are given. As usual, $\Delta(t)$ is the error term in the classical Dirichlet divisor problem.

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$ \int_0^T\Delta^k(t)|\zeta(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N) $$ are...

Lecture in Beijing 2012 on results concerning the mean values of the Riemann zeta-function on the critical line.

Lecture in Beijing 2012 on the general additive problem, important in the applications to power moments of the Riemann zeta-function on the critical line.

The lecture centering on the scientific work of the great mathematician Wolfgang Schwarz.

Results on $d(n)$ (the divisor function) and $\Delta(x)$ (the error term in the divisor problem) are presented, Budapest 2013, at the conference honouring Paul Erd\H os' s 100th birthday.

Several results on $\Delta(x)$ (the error term in the divisor problem) and$|\zeta(1/2+it)|^2$ in "short" intervals are presented.

Joint work with Wenguang Zhai (Beijing) on integrals of powers of $\D(x)$ (the error term in the divisor problem) and $|\zeta(1/2+it)|^2$.

The "hybrid" moments $$ \int_T^{2T}|\zeta(1/2+it)|^k{(\int_{t-G}^{t+G}|\zeta(1/2+ix)|^\ell dx)}^m dt $$ of the Riemann zeta-function $\zeta(s)$ on the critical line $\Re s = 1/2$ are studied. The expected upper bound for the above expression is $O_\epsilon(T^{1+\epsilon}G^m)$. This is shown to be true for certain specific values of the natural numb...

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. It is shown that $$ \int_0^T\Delta(t)|\zeta(1/2+it)|^2\,dt \ll T(\log T)^{4}. $$ Fu...

This is primarily an overview article (Lecture given during the conference “Number Theory and its Applications Workshop” in Xi’an (China), October 23–28, 2014.) dealing with the large values of \(\vert \zeta (\frac{1} {2} + it)\vert\). This approach allows one to obtain upper bounds for moments (mean values) of \(\vert \zeta (\frac{1} {2} + it)\ver...

In this overview paper, presented at the meeting DANS14, Novi Sad, July3-7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of $|\zeta(1/2+it)|$, and a discussion of two functional equations connected to a work of Prof. Bogoljub Stankovi\'c.

Let denote the error term in the Dirichlet divisor problem, and the error term in the asymptotic formula for the mean square of If with then we discuss bounds for third, fourth and fifth power moment of We also prove that always changes sign in for and obtain (conditionally) the existence of its large positive, or small negative values.

In this overview we give a detailed discussion of power moments of ζ(s), when s lies on the critical line. The survey includes early results, the mean square and mean fourth power, higher moments, conditional results and some open problems. © 2014 Springer Science+Business Media New York. All rights reserved.

The purpose of this text is twofold. First we discuss some divisor problems involving Paul Erd\H os (1913-1996), whose centenary of birth is this year. In the second part some recent results on divisor problems are discussed, and their connection with the powers moments of $|\zeta(\frac{1}{2}+it)|$ is pointed out. This is an extended version of the...

We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as $T\rightarrow \infty$, when $1\leq j \leq 6$, where $\zeta(s)$ is the Riemann zeta-function. The obtained ranges imp...

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) := -\Delta(x) + 2\Delta(2x) - \frac{1}{2}\Delta(4x)$ and $\int_0^T E^*(t)\,dt = \frac{3}{4}\pi T + R(T)$, then...

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - 1/2\Delta(4x)$ and we set $\int_0^T E^*(t)\,dt = 3\pi T/4 + R(T)$, then we obtain $$ \int_T^{T+...

Hardy's Z-function, related to the Riemann zeta-function ζ(s), was originally utilised by G. H. Hardy to show that ζ(s) has infinitely many zeros of the form ½+it. It is now amongst the most important functions of analytic number theory, and the Riemann hypothesis, that all complex zeros lie on the line ½+it, is perhaps one of the best known and mo...

We obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line \$\operatorname {Re} s= \frac {1}{2}\$.

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

We prove that, for a fixed j∈ℕ, there exists σ 0 =σ 0 (j) (<1) such that ∫ 0 T ζ1 2+it 4 |ζ(σ+it)| 2j dt≪ j,ε T 1+ε holds for σ>σ 0 . We also indicate how to obtain an asymptotic formula for the above integral, for the range of σ>σ 1 =σ 1 (j), where σ 0 <σ 1 <1.

We prove that, for a fixed j ∈ N, there exists σ0 = σ0(j) (< 1) such that  T 0     ζ 1 2 + it    4 |ζ(σ + it)| 2jdt j,ε T1+ε holds for σ>σ0. We also indicate how to obtain an asymptotic formula for the above integral, for the range of σ>σ1 = σ1(j), where σ0 < σ1 < 1.

Let . We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for and . These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.

If $$ \Delta(x) \;:=\; \sum_{n\leqslant x}c_n - Cx\qquad(C>0) $$ denotes the error term in the classical Rankin-Selberg problem, then we obtain a non-trivial upper bound for the mean square of $\Delta(x+U) - \Delta(x)$ for a certain range of $U = U(X)$. In particular, under the Lindel\"of hypothesis for $\zeta(s)$, it is shown that $$ \int_X^{2X} \...

We obtain a new upper bound for the sum I pound (ha parts per thousand currency signH) Delta (k) (N, h) when 1 a parts per thousand currency sign H a parts per thousand currency sign N, k a a"center dot, k a parts per thousand yen 3, where Delta (k) (N, h) is the (expected) error term in the asymptotic formula for I pound (N < na parts per thousand...

A detailed account of Prof. G.V. Milovanović’s mathematical work, on the occasion of his 60th birthday, is given. He obtained important results in several fields of Numerical Analysis and Approximation Theory. The topics to which he made his most important contributions include: Orthogonal polynomials and systems; Polynomials (extremal problems, in...

Some relations involving the Mellin and Laplace transforms of powers of the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are obtained. In particular, we discuss some mean square identities and their consequences.

Some problems involving the classical Hardy function $$ Z\left( t \right) = \zeta \left( {\frac{1} {2} + it} \right)\left( {\chi \left( {\frac{1} {2} + it} \right)} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} , \zeta \left( s \right) = \chi \left( s \right) \zeta \left( {1 - s} \right) $$, are discussed....

We obtain the approximate functional equation for the Rankin-Selberg zeta-function on the 1/2-line.

International audience Various properties of the Mellin transform function $$\mathcal{M}_k(s):= \int_1^{\infty} Z^k(x)x^{-s}\,dx$$ are investigated, where $$Z(t):=\zeta(\frac{1}{2}+it)\,\chi(\frac{1}{2}+it)^{-1/2},~~~~\zeta(s)=\chi(s)\zeta(1-s)$$ is Hardy's function. Connections with power moments of $|\zeta(\frac{1}{2}+it)|$ are established, and n...

If $Z(t) = \chi^{-1/2}(1/2+it)\zeta(1/2+it)$ denotes Hardy's function, where $\zeta(s) = \chi(s)\zeta(1-s)$ is the functional equation of the Riemann zeta-function, then it is proved that $$ \int_0^T Z(t)\d t = O_\e(T^{1/4+\e}). $$ Comment: 7 pages

We obtain, for T ε ≤U=U(T)≤T 1/2−ε , asymptotic formulas for òT2T(E(t+U)-E(t))2dt, òT2T(D(t+U)-D(t))2dt,\int_{T}^{2T}\Bigl(E(t+U)-E(t)\Bigr)^{2}{\mathrm{d}}{t},\qquad \int_{T}^{2T}\Bigl(\Delta (t+U)-\Delta (t)\Bigr)^{2}{\mathrm{d}}{t}, where Δ(x) is the error term in the classical divisor problem, and E(T) is the error term in the mean square f...

It is proved that, if k ≥ 2 is a fixed integer and 1 ≪ H ≤ (1/2)X, then $$ \int_{X - H}^{X + H} {\Delta _k^4 \left( x \right) } dx \ll _\varepsilon X^\varepsilon \left( {HX^{{{\left( {2k - 2} \right)} \mathord{\left/ {\vphantom {{\left( {2k - 2} \right)} k}} \right. \kern-\nulldelimiterspace} k}} + H^{{{\left( {2k - 3} \right)} \mathord{\left/ {\vp...

International audience We discuss the mean values of the Riemann zeta-function $\zeta(s)$, and analyze upper and lower bounds for $$\int_T^{T+H} \vert\zeta(\frac{1}{2}+it)\vert^{2k}\,dt~~~~~~(k\in\mathbb{N}~{\rm fixed,}~1<\!\!< H \leq T).$$ In particular, the author's new upper bound for the above integral under the Riemann hypothesis is presented.

Sums of the form \(\Sigma_{n\le x}E^k(n)\ (k\in{\Bbb N}\ \rm{fixed})\) are investigated, where \(E(T) = \int_0^T\vert\zeta (\frac{1}{2} +it) \vert^2{\rm d}t - T(log \frac{T}{2\pi} + 2\gamma -1)\) is the error term in the mean square formula for \(\big\vert\zeta \big(\frac{1}{2} + it\big) \big\vert\). The emphasis is on the case k = 1, which is more...

Assuming the Riemann Hypothesis it is proved that, for fixed $k>0$ and $H = T^\theta$ with fixed $0<\theta \le 1$, $$ \int_T^{T+H}|\zeta(1/2+it)|^{2k} dt \ll H(\log T)^{k^2(1+O(1/\log_3T))}, $$ where $\log_jT = \log(\log_{j-1}T)$. The proof is based on the recent method of K. Soundararajan for counting the occurrence of large values of $\log|\zeta(...

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ≍ T, T ɛ ≦ G ≪ T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma...

Some new results on power moments of the integral J(k)(t, G) = 1/root pi G integral(infinity)(-infinity) vertical bar zeta (1/2 + it + iu)vertical bar(2k) e (-(u/G)2) du (t asymptotic to T, T-E <= G << T, k is an element of N) are obtained when k = 1. These results can be used to derive bounds for moments of vertical bar(zeta (1/2 + it)vertical bar...

It is proved that, for $T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T}$, $$ \int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j \Bigl({\sqrt{T}\over G}\Bigr) + O_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2) $$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed natural number $k$, $$I_k(t,G) = {1\over\sqrt{\p...

Sums of the form $\sum_{n\le x}E^k(n) (k\in{\bf N}$ fixed) are investigated, where $$ E(T) = \int_0^T|\zeta(1/2+it)|^2 dt - T\Bigl(\log {T\over2\pi} + 2\gamma -1\Bigr)$$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. The emphasis is on the case k=1, which is more difficult than the corresponding sum for the divisor problem. The...

We provide upper bounds for the mean square integral $$ \int_X^{2X}(\Delta_k(x+h) - \Delta_k(x))^2 dx \qquad(h = h(X)\gg1, h = o(x) {\roman{as}} X\to\infty) $$ where $h$ lies in a suitable range. For $k\ge2$ a fixed integer, $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of the divisor function $d_k(n)$, genera...

We obtain, for $T^\epsilon \le U=U(T)\le T^{1/2-\epsilon}$, asymptotic formulas for $$ \int_T^{2T}(E(t+U) - E(t))^2 dt,\quad \int_T^{2T}(\Delta(t+U) - \Delta(t))^2 dt, $$ where $\Delta(x)$ is the error term in the classical divisor problem, and $E(T)$ is the error term in the mean square formula for $|\zeta(1/2+it)|$. Upper bounds of the form $O_\e...

An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function (x; ) (0 1), the error term in the Rankin-Selberg problem weighted by -th power of the logarithm. Mean square estimates for (x; ) are proved.

If $$\Delta(x) := \sum\limits_{n{\leq}x}c_n - C_x$$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$\int\limits^X_0 {\Delta}^4(x)dx\, {\ll}_{\varepsilon}\, X^{3+\varepsilon},\quad \int\limits^X_0 {\Delta}^4_1(x)dx\, {\ll}_{\varepsilon}\, X^{11/2+\varepsilon},$$ where Δ1(x) = ∫x 0 Δ(u)du. The latter bound is,...

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then it is proved that $$ \int_0^T|E^*(t)|^3dt \ll_\epsilon T^{3/2+\eps...

Upper bound estimates for the exponential sum are considered, where and is the flrst Fourier coe-cient of the Maass wave form corresponding to the eigenvalue λj to which the Hecke series Hj(s) is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponen...

Dedicated to Professor Matti Jutila on the occasion of his retirement Abstract. A simple proof of the classical subconvexity bound ζ ( 1 2 +it) ≪ε t 1/6+ε for the Riemann zeta-function is given, and estimation by more refined techniques is discussed. The connections between the Dirichlet divisor problem and the mean square of |ζ ( 1 + it) | are ana...

Some new results on power moments of the integral $$ J_k(t,G) = {1\over\sqrt{\pi}G} \int_{-\infty}^\infty |\zeta(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2}du \qquad(t \asymp T, T^\epsilon \le G \ll T, k\in\N) $$ are obtained when $k=1$. These results can be used to derive bounds for moments of $|\zeta(1/2+it)|$.

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$ and we set $\int_0^T E^*(t) dt = 3\pi T/4 + R(T)$, then we obtain $$ R(T)...

This paper gives a survey of known results concerning the Laplace transform $$ L_k(s) := \int_0^\infty |\zeta(1/2+ ix)|^{2k}{\rm e}^{-sx}{\rm d} x \qquad(k \in N, \R s > 0), $$ and the (modified) Mellin transform $$ {\cal Z}_k(s) := \int_1^\infty|\zeta(1/2+ ix)|^{2k}x^{-s}{\rm d} x\qquad(k\in N), $$ where the integral is absolutely convergent for $...

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then we obtain the asymptotic formula $$ \int_0^T (E^*(t))^2 {\rm d} t =...

We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for $f(x)$. Some applications are given, in particular to $|\zeta(1/2+ix)|^{2k}$ and the classical Rankin--Selberg prob...

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta({1\over2}+ix|^{2k}x^{-s}{\rm d} x$ ($k\ge1$ is a fixed integer, $s = \sigma + it$) is used to obtain estimates for $$ \sum_{r=1}^R\int_{t_r-G}^{t_r+G}|\zeta(1/2+it)|^{2k}{\rm d} t\quad(T < t_1 < >... < t_R < 2T), $$ where $t_{r+1} - t_r \ge G (r =1,..., R-1), T^\epsilon \le G \le T...

Several results are obtained concerning the function $\Delta_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding mean square integral. The mean square integral of $\Delta_2(x)$ is investigated in detail.

Some identities for the Riemann zeta-function are proved, using properties of the Mellin transform and M\"untz's identity. Comment: 7 pages

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of\(\left| {\zeta \left( {\frac{1}{2} + it} \right)} \right|\). If E *(t)=E(t)-2πΔ*(t/2π) with\(\Delta *\left( x \right) + 2\Delta \left( {2x} \right) - \frac{1}{2}\Delta \left( {4x} \right)\), then we obtain$$\int_...

Let P(n) denote the largest prime factor of an integer n (≥2), and let N (x) denote the number of natural numbers n such that 2 ≤ n ≤ x, and n does not divide P(n)!. We prove that $$N(x) = x \left(2+O\left(\sqrt{\log_{2}\,x/\!\log x}\,\right) \right)\int_2^x\rho(\log x/\!\log t) {\log t\over t^2} {\rm d} t,$$ where ρ(u) is the Dickman-de Bruijn fun...

Upper bound estimates for the exponential sum $$ \sum_{K<\kappa_j\le K'<2K} \alpha_j H_j^3(1/2) \cos(\k_j\log({4{\rm e}T\over \kappa_j})) \qquad(T^\epsilon \le K \le T^{1/2-\epsilon}) $$ are considered, where $\alpha_j = |\rho_j(1)|^2(\cosh\pi\kappa_j)^{-1}$, and $\rho_j(1)$ is the first Fourier coefficient of the Maass wave form corresponding to t...

The modified Mellin transform ${\cal Z}_k(s) = \int_1^\infty |\zeta(1/2+ix)|^{2k}x^{-s}{\rm d} x (k = 1,2,...)$ is investigated. Analytic continuation and mean square estimates of ${\cal Z}_k(s) $ are discussed, as well as connections with power moments of $|\zeta(1/2+ix)|$, with the special emphasis on the cases $k = 1,2$.

Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma = 1$. A zero-density counting function involving multiplicities is also discussed.