Results 151 to 160 of about 31,249 (173)

The sum of divisors function and the Riemann hypothesis

The Ramanujan Journal, 2021
Let \(\sigma(n)\) be the sum of positive divisors of \(n\), and \(\gamma\) denotes the Euler constant. In this paper, the author is motivated by Robin's work asserting that \(\sigma(n)5040\) is equivalent to the Riemann hypothesis (RH), and also Ramanujan's work asserting that under RH the following asymptotic upper bound holds \[ \sigma(n)
Jean-Louis Nicolas
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Remarks on Fibers of the Sum-of-Divisors Function

2015
Let \(\mbox{$\sigma$}\) denote the usual sum-of-divisors function. We show that every positive real number can be approximated arbitrarily closely by a fraction m∕n with \(\sigma (m) =\sigma (n)\). This answers in the affirmative a question of Erdős. We also show that for almost all of the elements v of \(\sigma (\mathbf{N})\), the members of the fiber
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A NEW UPPER BOUND FOR THE SUM OF DIVISORS FUNCTION

Bulletin of the Australian Mathematical Society, 2017
Robin’s criterion states that the Riemann hypothesis is true if and only if $\unicode[STIX]{x1D70E}(n)<e^{\unicode[STIX]{x1D6FE}}n\log \log n$ for every positive integer $n\geq 5041$. In this paper we establish a new unconditional upper bound for the sum of divisors function, which improves the current best unconditional estimate given by Robin. For
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Shifted convolution sums of divisor functions with Fourier coefficients

Journal of Number Theory
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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A survey of the alternating sum-of-divisors function

2011
Dedicated to the memory of my friend and colleague, Professor Antal Bege (1962-2012), 11 ...
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A Note on Sums of Two Squares and Sum-of-divisors Functions

2020
See the abstract in the attached pdf.
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On mean values for the exponential sum of divisor functions

International Journal of Number Theory
In this paper, we study mean values for exponential sums of divisor functions. We improve previous results of [M. Pandey, Moment estimates for the exponential sum with higher divisor functions, C. R. Math. Acad. Sci. Paris  360 (2022) 419–424].
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Erdős’s Work on the Sum of Divisors Function and on Euler’s Function

2013
The following notation will be used throughout Open image in new window log r x is the r times iterated logarithm of x, $$\begin{gathered}\sigma _1 (n) = \sigma (n),\quad \sigma _k (n) = \sigma (\sigma _{k - 1} (n)), \hfill \\\phi _1 (n) = \phi (n),\quad \phi _k (n) = \phi (\phi _{k - 1} (n)), \hfill \\s_1 (n) = s(n) = \quad \sigma (n) -
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Errata to Iterating the Sum-of-Divisors Function

Experimental Mathematics, 1997
Graeme L. Cohen, Herman J. J. te Riele
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On the sum of divisors function

Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio computatorica, 2013
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