scholarly journals Sums of Averages of GCD-Sum Functions II

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Lisa Kaltenböck ◽  
Isao Kiuchi ◽  
Sumaia Saad Eddin ◽  
Masaaki Ueda
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Riemann Zeta Function ◽  
Asymptotic Expansions ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Large Positive Number ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Fixed Positive Integer

AbstractLet $$ \gcd (k,j) $$ gcd ( k , j ) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$ M r ( x ; f ) : = ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ( j , k ) ) for any large real number $$x\ge 5$$ x ≥ 5 , where f is any arithmetical function. Let $$\phi $$ ϕ , and $$\psi $$ ψ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $$M_r(x; \mathrm{id})$$ M r ( x ; id ) , $$M_r(x;{\phi })$$ M r ( x ; ϕ ) and $$M_r(x;{\psi })$$ M r ( x ; ψ ) . Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $$M_r(x;\mathrm{id})$$ M r ( x ; id ) for any large positive number $$x>5$$ x > 5 satisfying $$x=[x]+\frac{1}{2}$$ x = [ x ] + 1 2 .

ON A NEW CIRCLE PROBLEM

2016 ◽  
Vol 103 (2) ◽  
pp. 231-249
Author(s):  
JUN FURUYA ◽  
MAKOTO MINAMIDE ◽  
YOSHIO TANIGAWA
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Dirichlet Character ◽  
Arithmetical Function ◽  
Circle Problem ◽  
The Riemann Zeta Function ◽  
Primitive Dirichlet Character ◽  
Riemann Zeta ◽  
The Mean ◽  
Voronoi Formula

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

scholarly journals THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS

2010 ◽  
Vol 06 (08) ◽  
pp. 1933-1944 ◽  
Author(s):  
SANDRO BETTIN
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Second Moment ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

2018 ◽  
Vol 14 (02) ◽  
pp. 371-382
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
Riemann Zeta ◽  
Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

scholarly journals The Values of the Periodic Zeta-Function at the Nontrivial Zeros of Riemann’s Zeta-Function

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2410
Author(s):  
Janyarak Tongsomporn ◽  
Saeree Wananiyakul ◽  
Jörn Steuding
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Real Line ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Riemann’S Zeta Function

In this paper, we prove an asymptotic formula for the sum of the values of the periodic zeta-function at the nontrivial zeros of the Riemann zeta-function (up to some height) which are symmetrical on the real line and the critical line. This is an extension of the previous results due to Garunkštis, Kalpokas, and, more recently, Sowa. Whereas Sowa’s approach was assuming the yet unproved Riemann hypothesis, our result holds unconditionally.

scholarly journals Weighted value distributions of the Riemann zeta function on the critical line

2021 ◽  
Vol 33 (3) ◽  
pp. 579-592
Author(s):  
Alessandro Fazzari
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Formula ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Matrix Theory ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove a central limit theorem for log ⁡ | ζ ⁢ ( 1 2 + i ⁢ t ) | {\log\lvert\zeta(\frac{1}{2}+it)\rvert} with respect to the measure | ζ ( m ) ⁢ ( 1 2 + i ⁢ t ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta^{(m)}(\frac{1}{2}+it)\rvert^{2k}\,dt} ( k , m ∈ ℕ {k,m\in\mathbb{N}} ), assuming RH and the asymptotic formula for twisted and shifted integral moments of zeta. Under the same hypotheses, we also study a shifted case, looking at the measure | ζ ⁢ ( 1 2 + i ⁢ t + i ⁢ α ) | 2 ⁢ k ⁢ d ⁢ t {\lvert\zeta(\frac{1}{2}+it+i\alpha)\rvert^{2k}\,dt} , with α ∈ ( - 1 , 1 ) {\alpha\in(-1,1)} . Finally, we prove unconditionally the analogue result in the random matrix theory context.

scholarly journals An integral involving the generalized zeta function

1990 ◽  
Vol 13 (3) ◽  
pp. 453-460 ◽  
Author(s):  
E. Elizalde ◽  
A. Romeo
Keyword(s):  
Positive Integer ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Surfaces ◽  
The Riemann Zeta Function ◽  
Special Integral ◽  
Riemann Zeta ◽  
Compact Riemann Surfaces ◽  
Determinants Of Laplacians ◽  
Generalized Zeta Function

A general value for∫abdtlogΓ(t), fora,bpositive reals, is derived in terms of the Hurwitzζfunction. That expression is checked for a previously known special integral, and the case whereais a positive integer andbis half an odd integer is considered. The result finds application in calculating the numerical value of the derivative of the Riemann zeta function at the point−1, a quantity that arises in the evaluation of determinants of Laplacians on compact Riemann surfaces.

On the Riemann–Siegel formula

2007 ◽  
Vol 463 (2086) ◽  
pp. 2557-2568 ◽  
Author(s):  
A Kuznetsov
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Critical Point ◽  
Riemann Zeta Function ◽  
Error Term ◽  
Error Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this article, we derive a generalization of the Riemann–Siegel asymptotic formula for the Riemann zeta function. By subtracting the singularities closest to the critical point, we obtain a significant reduction of the error term at the expense of a few evaluations of the error function. We illustrate the efficiency of this method by comparing it to the classical Riemann–Siegel formula.

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