Displaying similar documents to “On some mean value results for the zeta-function in short intervals”

Dirichlet series induced by the Riemann zeta-function

Jun-ichi Tanaka (2008)

Studia Mathematica

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The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on ω , the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form ( a p , s ) = p ( 1 - a p p - s ) - 1 for a p in ω . Among other things, using the Haar measure on ω for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

On the Riesz means of n/ϕ(n) - III

Ayyadurai Sankaranarayanan, Saurabh Kumar Singh (2015)

Acta Arithmetica

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Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term E k ( x ) where 1 / k ! n x n / ϕ ( n ) ( 1 - n / x ) k = M k ( x ) + E k ( x ) . For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying k ( l o g x ) 1 + ϵ .

The size of the Lerch zeta-function at places symmetric with respect to the line ( s ) = 1 / 2

Ramūnas Garunkštis, Andrius Grigutis (2019)

Czechoslovak Mathematical Journal

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Let ζ ( s ) be the Riemann zeta-function. If t 6 . 8 and σ > 1 / 2 , then it is known that the inequality | ζ ( 1 - s ) | > | ζ ( s ) | is valid except at the zeros of ζ ( s ) . Here we investigate the Lerch zeta-function L ( λ , α , s ) which usually has many zeros off the critical line and it is expected that these zeros are asymmetrically distributed with respect to the critical line. However, for equal parameters λ = α it is still possible to obtain a certain version of the inequality | L ( λ , λ , 1 - s ¯ ) | > | L ( λ , λ , s ) | .

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

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Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that n x d ² ( n ) = x P ( l o g x ) + E ( x ) , where P(y) is a cubic polynomial in y and E ( x ) = O ( x 3 / 5 + ε ) , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), E ( x ) = O ( x 1 / 2 + ε ) . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce E ( x ) = O ( x 1 / 2 ( l o g x ) l o g l o g x ) . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove E ( x ) = O ( x 1 / 2 ( l o g x ) ) . ...

Mean values related to the Dedekind zeta-function

Hengcai Tang, Youjun Wang (2024)

Czechoslovak Mathematical Journal

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Let K / be a nonnormal cubic extension which is given by an irreducible polynomial g ( x ) = x 3 + a x 2 + b x + c . Denote by ζ K ( s ) the Dedekind zeta-function of the field K and a K ( n ) the number of integral ideals in K with norm n . In this note, by the higher integral mean values and subconvexity bound of automorphic L -functions, the second and third moment of a K ( n ) is considered, i.e., n x a K 2 ( n ) = x P 1 ( log x ) + O ( x 5 / 7 + ϵ ) , n x a K 3 ( n ) = x P 4 ( log x ) + O ( X 321 / 356 + ϵ ) , where P 1 ( t ) , P 4 ( t ) are polynomials of degree 1, 4, respectively, ϵ > 0 is an arbitrarily small number.

On discrete mean values of Dirichlet L -functions

Ertan Elma (2021)

Czechoslovak Mathematical Journal

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Let χ be a nonprincipal Dirichlet character modulo a prime number p 3 and let 𝔞 χ : = 1 2 ( 1 - χ ( - 1 ) ) . Define the mean value p ( - s , χ ) : = 2 p - 1 ψ ( mod p ) ψ ( - 1 ) = - 1 L ( 1 , ψ ) L ( - s , χ ψ ¯ ) ( σ : = s > 0 ) . We give an identity for p ( - s , χ ) which, in particular, shows that p ( - s , χ ) = L ( 1 - s , χ ) + 𝔞 χ 2 p s L ( 1 , χ ) ζ ( - s ) + o ( 1 ) ( p ) for fixed 0 < σ < 1 2 and | t : = s | = o ( p ( 1 - 2 σ ) / ( 3 + 2 σ ) ) .

Lower bounds of discrete moments of the derivatives of the Riemann zeta-function on the critical line

Thomas Christ, Justas Kalpokas (2013)

Journal de Théorie des Nombres de Bordeaux

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We establish unconditional lower bounds for certain discrete moments of the Riemann zeta-function and its derivatives on the critical line. We use these discrete moments to give unconditional lower bounds for the continuous moments I k , l ( T ) = 0 T | ζ ( l ) ( 1 2 + i t ) | 2 k d t , where l is a non-negative integer and k 1 a rational number. In particular, these lower bounds are of the expected order of magnitude for I k , l ( T ) .

The n -th prime asymptotically

Juan Arias de Reyna, Jérémy Toulisse (2013)

Journal de Théorie des Nombres de Bordeaux

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A new derivation of the classic asymptotic expansion of the n -th prime is presented. A fast algorithm for the computation of its terms is also given, which will be an improvement of that by Salvy (1994). Realistic bounds for the error with li - 1 ( n ) , after having retained the first m terms, for 1 m 11 , are given. Finally, assuming the Riemann Hypothesis, we give estimations of the best possible r 3 such that, for n r 3 , we have p n &gt; s 3 ( n ) where s 3 ( n ) is the sum of the first four terms of the asymptotic...