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Formalized Mathematics, ISSN 1426-2630, 07/2017, Volume 25, Issue 2, pp. 141 - 147
In the article we formalize in the Mizar system [ ] preliminary facts needed to prove the Basel problem [ , ]. Facts that are independent from the notion of... 
Basel problem | 03B35 | 11M06
Journal Article
Formalized Mathematics, ISSN 1426-2630, 07/2017, Volume 25, Issue 2, pp. 149 - 155
A rigorous elementary proof of the Basel problem [6, 1] is formalized in the Mizar system [ ]. This theorem is item #14 from the “Formalizing 100 Theorems”... 
Basel problem | 03B35 | 11M06
Journal Article
Mathematika, ISSN 0025-5793, 07/2020, Volume 66, Issue 3, pp. 612 - 621
In this article, we give, under the Riemann hypothesis, an upper bound for the exponential moments of the imaginary part of the logarithm of the Riemann zeta... 
11M06 (primary)
Journal Article
Mathematische Zeitschrift, ISSN 0025-5874, 2/2019, Volume 291, Issue 1, pp. 661 - 709
We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This... 
Generalized Möbius functions | Modular forms | 11N64 | Zeros on the critical line | Ratios conjecture technique | Mathematics | Mollifier | Secondary 11M06 | Dirichlet polynomial | Holomorphic cusp form | Primary 11M26 | Mathematics, general | Autocorrelation | MATHEMATICS | Generalized Mobius functions | RIEMANN ZETA-FUNCTION | ZEROS
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 10/2016, Volume 368, Issue 10, pp. 6887 - 6914
We investigate the reciprocity law, studied by Conrey and Young, for the second moment of Dirichlet L modulo a prime q with respect to the real topology.... 
MATHEMATICS | MEAN-VALUES | ZETA-FUNCTION | Mathematics - Number Theory
Journal Article
Mathematische Annalen, ISSN 0025-5831, 6/2016, Volume 365, Issue 1, pp. 473 - 496
Let $$\alpha \in (1/2,1)$$ α ∈ ( 1 / 2 , 1 ) be fixed. We prove that $$\begin{aligned} \max _{0 \le t \le T} |\zeta (\alpha +it)| \ge \exp \left(... 
Mathematics, general | Mathematics | 11M06 | 11A05 | MATHEMATICS | GCD SUMS | EXTREME VALUES | SYSTEMS | SERIES | DILATED FUNCTIONS
Journal Article
Journal of the London Mathematical Society, ISSN 0024-6107, 04/2019, Volume 99, Issue 2, pp. 249 - 272
We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of L‐functions multiplied by an oscillating... 
11F72 (primary) | 11L05 | 11M06 (secondary) | MATHEMATICS | SQUARE | VALUES | square | Matematisk analys | values | Mathematical Analysis
Journal Article
Mathematische annalen, ISSN 1432-1807, 2012, Volume 356, Issue 3, pp. 939 - 968
Let $$\pi S(t)$$ denote the argument of the Riemann zeta-function, $$\zeta (s)$$ , at the point $$s=\frac{1}{2}+it$$ . Assuming the Riemann hypothesis, we... 
11M26 | Mathematics, general | 11M36 | Mathematics | 11M06 | 41A30 | MATHEMATICS | EXTREMAL-FUNCTIONS | ZETA-FUNCTION | SELBERG | THEOREMS | Mathematics - Number Theory
Journal Article
Archiv der Mathematik, ISSN 1420-8938, 2018, Volume 112, Issue 1, pp. 53 - 59
We study the mean of the values of the zeta-function on a generalized arithmetic progression on the critical line. 
Value-distribution | Riemann Zeta-Function | 11M26 | Mathematics, general | Mathematics | 11M06 | Nontrivial zeros | MATHEMATICS
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 01/2016, Volume 260, Issue 2, pp. 1456 - 1464
We prove that the Riemann zeta function and the Euler gamma function cannot satisfy a class of algebraic differential equations with functional coefficients... 
Polynomial | Riemann zeta function | Algebraic differential equation | The Euler gamma function | 33B15 | 30D30 | 34M15 | 11M06
Journal Article
Compositio mathematica, ISSN 0010-437X, 3/2006, Volume 142, Issue 2, pp. 307 - 338
Derivation and extended double shuffle (EDS) relations for multiple zeta values (MZVs) are proved. Related algebraic structures of MZVs, as well as a... 
Derivation | Double shuffle relation | Multiple zeta value | Regularization | MATHEMATICS | HARMONIC SERIES | ALGEBRA | regularization | multiple zeta value | derivation | double shuffle relation
Journal Article
International Mathematics Research Notices, ISSN 1073-7928, 2015, Volume 2015, Issue 21, pp. 11419 - 11432
Maier and Rassias computed the moments and proved a distribution result for the cotangent sum c(0)(a/q) := -Sigma(m MATHEMATICS | Mathematics - Number Theory
Journal Article
The Rocky Mountain journal of mathematics, ISSN 0035-7596, 1/2014, Volume 44, Issue 2, pp. 443 - 477
The Stieltjes constants 𝛾𝑘(𝑎) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function ζ(𝑠, 𝑎) about 𝑠 = 1.... 
Dirichlet L functions | Stirling numbers of the first kind | Stieltjes constants | Lerch zeta function | Riemann zeta function | Laurent expansion | Hurwitz zeta function | MATHEMATICS | ZETA-FUNCTION | RIEMANN-XI FUNCTION | 11M35 | 05A10 | 11Y60 | 11M06
Journal Article
Mathematische Zeitschrift, ISSN 0025-5874, 10/2019, Volume 293, Issue 1, pp. 825 - 865
In this work, we obtain an asymptotic formula for the twisted mean square of a Dirichlet L-function with a longer mollifier, whose coefficients are also more... 
Kloosterman sum | Dirichlet L -function | 11M26 | Mathematics, general | Mathematics | 11M06 | Twisted second moment | Simple zeros | Riemann zeta-function | Dirichlet L-function | MATHEMATICS
Journal Article
Mathematische Nachrichten, ISSN 0025-584X, 01/2018, Volume 291, Issue 1, pp. 103 - 108
The paper concerns the uniqueness problem of Riemann zeta‐function. It is showed that the Riemann zeta‐function is uniquely determined in terms of the... 
Riemann zeta‐function | 30D30 | value distribution | Exceptional set | 11M36 | 11M06 | 30D35 | Riemann zeta-function | MATHEMATICS | ZEROS
Journal Article
Mathematica Slovaca, ISSN 0139-9918, 08/2018, Volume 68, Issue 4, pp. 741 - 748
We prove that, under the Riemann hypothesis, a wide class of analytic functions can be approximated by shifts + i ), ∈ ℕ, of the Riemann zeta-function, where... 
universality | Riemann hypothesis | Riemann zeta-function | weak convergence | Primary 11M06 | MATHEMATICS | ZEROS
Journal Article
Experimental Mathematics, ISSN 1058-6458, 09/2012, Volume 21, Issue 3, pp. 235 - 240
We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip.... 
Riemann zeta function | value distribution | arithmetic progression | Primary 11M06 | MATHEMATICS | 11M06
Journal Article
Experimental Mathematics, ISSN 1058-6458, 01/2017, Volume 26, Issue 1, pp. 77 - 92
We denote by the sum of Λ(n)/n for all n ≤ x and congruent to and similarly by ψ(x; q, a) the sum of Λ(n) over the same set. We show that the error term in for... 
explicit formulae | prime numbers in arithmetic progressions | explicit estimates | Primary 11N05, 11M06, 11N13 | Secondary 11N37, 11A25
Journal Article
Inventiones Mathematicae, ISSN 0020-9910, 05/2017, Volume 208, Issue 2, pp. 441 - 499
We prove an asymptotic formula for the number of primes of the shape a2+p4, thereby refining the well known work of Friedlander and Iwaniec (Ann Math (2)... 
Primary: 11M06 | Secondary: 11M26
Journal Article
The Ramanujan journal, ISSN 1572-9303, 2008, Volume 16, Issue 3, pp. 247 - 270
The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and of the second author for... 
33B15 | Functions of a Complex Variable | 11M35 | Field Theory and Polynomials | Double integral | Mathematics | Polylogarithm | Fourier Analysis | Zeta function | 11Y60 | 33B30 | 11M06 | Number Theory | Lerch transcendent | Infinite product | Combinatorics | MATHEMATICS | EULERS CONSTANT | SERIES | polylogarithm | infinite product | zeta function | double integral
Journal Article
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