The sums ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn1.gif] {$S_{q}(N)$} are defined by the equality ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn2.gif] {$S_{q}(N)=s_q(1)+\dots+s_q(N-1)$} for all positive integers ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn3.gif] {$N$} and ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn4.gif] {$q\geqslant2$} , where ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn5.gif] {$s_q(n)$} is the sum of digits of the integer ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn6.gif] {$n$} written in the system with base ##IMG## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn7.gif] {$q$} . In 1975 Delange generalised Trollope's formula and proved...

中文翻译：

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Delange函数的全局极值，数字和和凹函数的边界

和## IMG ## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn1.gif] {$ S_ {q}（N）$}由等式定义## IMG ## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn2.gif] {$ S_ {q}（N）= s_q（1）+ \ dots + s_q（N-1）$}用于所有正整数## IMG ## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn3.gif] {$ N $}和## IMG ## [http://ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn4.gif] {$ q \ geqslant2 $}，其中## IMG ## [http：/ /ej.iop.org/images/1064-5616/211/3/336/MSB_211_3_336ieqn5.gif] {$ s_q（n）$}是整数## IMG ##的数字总和[http：// ej .iop.org / images / 1064-5616 / 211/3/336 / MSB_211_3_336ieqn6.gif] {$ n $}是在系统中以## IMG ##为基础编写的[http://ej.iop.org/images/ 1064-5616 / 211/3/336 / MSB_211_3_336ieqn7.gif] {$ q $}。1975年，Delange推广了Trollope的公式，并证明...