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On the riemann zeta-function and the divisor problem

Aleksandar Ivić — 2004

Open Mathematics

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 ς 1 2 + i t . If E * t = E t - 2 π Δ * t / 2 π with Δ * x = - Δ x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain 0 T E * t 4 d t e T 16 / 9 + ε . We also show how our method of proof yields the bound r = 1 R t r - G t r + G ς 1 2 + i t 2 d t 4 e T 2 + e G - 2 + R G 4 T ε , where T 1/5+ε≤G≪T, T

On the riemann zeta-function and the divisor problem II

Aleksandar Ivić — 2005

Open Mathematics

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 ζ 1 2 + i t . If E *(t)=E(t)-2πΔ*(t/2π) with Δ * x + 2 Δ 2 x - 1 2 Δ 4 x , then we obtain 0 T E * t 5 d t ε T 2 + ε and 0 T E * t 544 75 d t ε T 601 225 + ε . It is also shown how bounds for moments of | E *(t)| lead to bounds for moments of ζ 1 2 + i t .

On some problems involving Hardy’s function

Aleksandar Ivić — 2010

Open Mathematics

Some problems involving the classical Hardy function Z t = ζ 1 2 + i t χ 1 2 + i t - 1 1 2 2 , ζ s = χ s ζ 1 - s , are discussed. In particular we discuss the odd moments of Z(t) and the distribution of its positive and negative values.

On the number of subgroups of finite abelian groups

Aleksandar Ivić — 1997

Journal de théorie des nombres de Bordeaux

Let T ( x ) = K 1 x log 2 x + K 2 x log x + K 3 x + Δ ( x ) , where T ( x ) denotes the number of subgroups of all abelian groups whose order does not exceed x and whose rank does not exceed 2 , and Δ ( x ) is the error term. It is proved that 1 X Δ 2 ( x ) d x X 2 log 31 / 3 X , 1 X Δ 2 ( x ) d x = Ω ( X 2 log 4 X ) .

On sums of Hecke series in short intervals

Aleksandar Ivić — 2001

Journal de théorie des nombres de Bordeaux

We have K - G k j K + G α j H j 3 ( 1 2 ) ϵ G K 1 + ϵ for K ϵ G K , where α j = ρ j ( 1 ) 2 ( cosh π k j ) - 1 , and ρ j ( 1 ) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ j = k j 2 + 1 4 to which the Hecke series H j ( s ) is attached. This result yields the new bound H j ( 1 2 ϵ k j 1 3 + ϵ .

On mean values of some zeta-functions in the critical strip

Aleksandar Ivić — 2003

Journal de théorie des nombres de Bordeaux

For a fixed integer k 3 , and fixed 1 2 < σ < 1 we consider 1 T ζ ( σ + i t ) 2 k d t = n = 1 d k 2 ( n ) n - 2 σ T + R ( k , σ ; T ) , where R ( k , σ ; T ) = 0 ( T ) ( T ) is the error term in the above asymptotic formula. Hitherto the sharpest bounds for R ( k , σ ; T ) are derived in the range min ( β k , σ k * ) < σ < 1 . We also obtain new mean value results for the zeta-function of holomorphic cusp forms and the Rankin-Selberg series.

On some mean value results for the zeta-function in short intervals

Aleksandar Ivić — 2014

Acta Arithmetica

Let Δ ( 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 |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and 0 T E * ( t ) d t = 3 / 4 π T + R ( T ) , then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dt ( k , 1 H T ) are...

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