Displaying similar documents to “Primitive lattice points inside an ellipse”

Some infinite sums identities

Meher Jaban, Sinha Sneh Bala (2015)

Czechoslovak Mathematical Journal

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We find the sum of series of the form ∑ i = 1 ∞ f ( i ) i r for some special functions f . The above series is a generalization of the Riemann zeta function. In particular, we take f as some values of Hurwitz zeta functions, harmonic numbers, and combination of both. These generalize some of the results given in Mező’s paper (2013). We use multiple zeta theory to prove all results. The series sums we have obtained are in terms of Bernoulli numbers and powers of π .

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 ) .

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.

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Hafedh Herichi, Michel L. Lapidus (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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We survey some of the universality properties of the Riemann zeta function ζ ( s ) and then explain how to obtain a natural quantization of Voronin’s universality theorem (and of its various extensions). Our work builds on the theory of complex fractal dimensions for fractal strings developed by the second author and M. van Frankenhuijsen in [60]. It also makes an essential use of the functional analytic framework developed by the authors in [25] for rigorously studying the spectral operator...

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

Aleksandar Ivić (2014)

Acta Arithmetica

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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...

Ergodic Universality Theorems for the Riemann Zeta-Function and other L -Functions

Jörn Steuding (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove a new type of universality theorem for the Riemann zeta-function and other L -functions (which are universal in the sense of Voronin’s theorem). In contrast to previous universality theorems for the zeta-function or its various generalizations, here the approximating shifts are taken from the orbit of an ergodic transformation on the real line.

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 ) ⁵ 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 ) ⁵ ) . ...