Results 221 to 230 of about 2,683,293 (266)

Mean Value Inequalities for the Digamma Function

Analysis Mathematica, 2023
Let \(\psi\) be the digamma function. The authors give an upper estimate for \((b-a)\psi(\sqrt{ab})\) and a lower estimate for \((b-a)\psi((a+b)/2)\) in terms of the logarithmic mean \(L(a,b):=(b-a)/\log(b/a)\). \par For all real numbers \(a\) and \(b\) with \(b>a\ge \alpha_0\) we have \[ (b-a)\psi(\sqrt{ab})a>0\) we have \[\left(L(a,b)-a\right)\psi(a)+
Alzer, H., Kwong, M. K.
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On mean values of the zeta‐function

Mathematika, 1984
LetIt is conjectured that for any k,for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, C1 = 1, and c2 = (2π2)-1, but there is not even a conjectural value of ck for any other k. However, it is known that the Riemann hypothesis impliesfor all k ≥ 0 (see Ramachandra [2] and Heath-Brown [1]).
Conrey, J. B., Ghosh, A.
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THE MEAN-VALUES OF ARITHMETICAL FUNCTIONS

The Quarterly Journal of Mathematics, 1949
Es wird zunächst folgender Satz bewiesen: Es seien \(\{a_n\}\), \(\{A_n\}\) zwei Folgen, verknüpft durch \[ A_n = \sum_{m\mid n} a_m,\quad a_n = \sum_{m\mid n} \mu(n) A_m, \] dann gilt \[ \lim_{N\to\infty} N^{-1} \sum_{n=1}^N A_n \rightarrow \lim_{N\to\infty} N^{-1} \sum_{n=1}^N a_nn^{-1}, \qquad (N\to\infty), \] wenn es eine positive nichtabnehmende ...
Atkinson, F. V., Lord Cherwell
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The mean values of multiplicative functions. II

Lithuanian Mathematical Journal, 1997
Let \(g_i: {\mathbb N} \longrightarrow {\mathbb C} \;(i=1,2)\) be arithmetic multiplicative functions. Using the notations \[ M_x(g_1,g_2)={1\over x}\sum\limits _{n\leq x}g_1(n+1) g_2(n), \] \[ S(r,x)=\sum\limits_ ...
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