Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
The search for mechanical properties of materials reached a highly acclaimed level, when indentations could be analysed on the basis of elastic theory for hardness and elastic modulus. The mathematical formulas proved...The search for mechanical properties of materials reached a highly acclaimed level, when indentations could be analysed on the basis of elastic theory for hardness and elastic modulus. The mathematical formulas proved to be very complicated, and various trials were published between the 1900s and 2000s. The development of indentation instruments and the wish to make the application in numerous steps easier, led in 1992 to trials with iterations by using relative values instead of absolute ones. Excessive iterations of computers with 3 + 8 free parameters of the loading and unloading curves became possible and were implemented into the instruments and worldwide standards. The physical formula for hardness was defined as force over area. For the conical, pyramidal, and spherical indenters, one simply took the projected area for the calculation of the indentation depth from the projected area, adjusted it later by the iterations with respect to fused quartz or aluminium as standard materials, and called it “contact height”. Continuously measured indentation loading curves were formulated as loading force over depth square. The unloading curves after release of the indenter used the initial steepness of the pressure relief for the calculation of what was (and is) incorrectly called “Young’s modulus”. But it is not unidirectional. And for the spherical indentations’ loading curve, they defined the indentation force over depth raised to 3/2 (but without R/h correction). They till now (2025) violate the energy law, because they use all applied force for the indenter depth and ignore the obvious sidewise force upon indentation (cf. e.g. the wood cleaving). The various refinements led to more and more complicated formulas that could not be reasonably calculated with them. One decided to use 3 + 8 free-parameter iterations for fitting to the (poor) standards of fused quartz or aluminium. The mechanical values of these were considered to be “true”. This is till now the worldwide standard of DIN-ISO-ASTM-14577, avoiding overcomplicated formulas with their complexity. Some of these are shown in the Introduction Section. By doing so, one avoided the understanding of indentation results on a physical basis. However, we open a simple way to obtain absolute values (though still on the blackbox instrument’s unsuitable force calibration). We do not iterate but calculate algebraically on the basis of the correct, physically deduced exponent of the loading force parabolas with h3/2 instead of false “h2” (for the spherical indentation, there is a calotte-radius over depth correction), and we reveal the physical errors taken up in the official worldwide “14577-Standard”. Importantly, we reveal the hitherto fully overlooked phase transitions under load that are not detectable with the false exponent. Phase-transition twinning is even present and falsifies the iteration standards. Instead of elasticity theory, we use the well-defined geometry of these indentations. By doing so, we reach simple algebraically calculable formulas and find the physical indentation hardness of materials with their onset depth, onset force and energy, as well as their phase-transition energy (temperature dependent also its activation energy). The most important phase transitions are our absolute algebraically calculated results. The now most easily obtained phase transitions under load are very dangerous because they produce polymorph interfaces between the changed and the unchanged material. It was found and published by high-enlargement microscopy (5000-fold) that these trouble spots are the sites for the development of stable, 1 to 2 µm long, micro-cracks (stable for months). If however, a force higher than the one of their formation occurs to them, these grow to catastrophic crash. That works equally with turbulences at the pickle fork of airliners. After the publication of these facts and after three fatal crashing had occurred in a short sequence, FAA (Federal Aviation Agency) reacted by rechecking all airplanes for such micro cracks. These were now found in a new fleet of airliners from where the three crashed ones came. These were previously overlooked. FAA became aware of that risk and grounded 290 (certainly all) of them, because the material of these did not have higher phase-transition onset and energy than other airplanes with better material. They did so despite the 14577-Standard that does not find (and thus formally forbids) phase transitions under indenter load with the false exponent on the indentation parabola. However, this “Standard” will, despite the present author’s well-founded petition, not be corrected for the next 5 years.展开更多
Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In thi...Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In this paper,we shall study the mean square of E_(f)(x)and establish some new results of E_(f)(x)for some special functions.展开更多
基金Supported by the National Natural Science Foundation of China（11271249） Supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission（1601213） Supported by the Scientific Research Program of Yangtze Normal University（2012XJYBO31）
文摘Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
文摘The search for mechanical properties of materials reached a highly acclaimed level, when indentations could be analysed on the basis of elastic theory for hardness and elastic modulus. The mathematical formulas proved to be very complicated, and various trials were published between the 1900s and 2000s. The development of indentation instruments and the wish to make the application in numerous steps easier, led in 1992 to trials with iterations by using relative values instead of absolute ones. Excessive iterations of computers with 3 + 8 free parameters of the loading and unloading curves became possible and were implemented into the instruments and worldwide standards. The physical formula for hardness was defined as force over area. For the conical, pyramidal, and spherical indenters, one simply took the projected area for the calculation of the indentation depth from the projected area, adjusted it later by the iterations with respect to fused quartz or aluminium as standard materials, and called it “contact height”. Continuously measured indentation loading curves were formulated as loading force over depth square. The unloading curves after release of the indenter used the initial steepness of the pressure relief for the calculation of what was (and is) incorrectly called “Young’s modulus”. But it is not unidirectional. And for the spherical indentations’ loading curve, they defined the indentation force over depth raised to 3/2 (but without R/h correction). They till now (2025) violate the energy law, because they use all applied force for the indenter depth and ignore the obvious sidewise force upon indentation (cf. e.g. the wood cleaving). The various refinements led to more and more complicated formulas that could not be reasonably calculated with them. One decided to use 3 + 8 free-parameter iterations for fitting to the (poor) standards of fused quartz or aluminium. The mechanical values of these were considered to be “true”. This is till now the worldwide standard of DIN-ISO-ASTM-14577, avoiding overcomplicated formulas with their complexity. Some of these are shown in the Introduction Section. By doing so, one avoided the understanding of indentation results on a physical basis. However, we open a simple way to obtain absolute values (though still on the blackbox instrument’s unsuitable force calibration). We do not iterate but calculate algebraically on the basis of the correct, physically deduced exponent of the loading force parabolas with h3/2 instead of false “h2” (for the spherical indentation, there is a calotte-radius over depth correction), and we reveal the physical errors taken up in the official worldwide “14577-Standard”. Importantly, we reveal the hitherto fully overlooked phase transitions under load that are not detectable with the false exponent. Phase-transition twinning is even present and falsifies the iteration standards. Instead of elasticity theory, we use the well-defined geometry of these indentations. By doing so, we reach simple algebraically calculable formulas and find the physical indentation hardness of materials with their onset depth, onset force and energy, as well as their phase-transition energy (temperature dependent also its activation energy). The most important phase transitions are our absolute algebraically calculated results. The now most easily obtained phase transitions under load are very dangerous because they produce polymorph interfaces between the changed and the unchanged material. It was found and published by high-enlargement microscopy (5000-fold) that these trouble spots are the sites for the development of stable, 1 to 2 µm long, micro-cracks (stable for months). If however, a force higher than the one of their formation occurs to them, these grow to catastrophic crash. That works equally with turbulences at the pickle fork of airliners. After the publication of these facts and after three fatal crashing had occurred in a short sequence, FAA (Federal Aviation Agency) reacted by rechecking all airplanes for such micro cracks. These were now found in a new fleet of airliners from where the three crashed ones came. These were previously overlooked. FAA became aware of that risk and grounded 290 (certainly all) of them, because the material of these did not have higher phase-transition onset and energy than other airplanes with better material. They did so despite the 14577-Standard that does not find (and thus formally forbids) phase transitions under indenter load with the false exponent on the indentation parabola. However, this “Standard” will, despite the present author’s well-founded petition, not be corrected for the next 5 years.
基金Supported by the National Natural Science Foundation of China(Grant No.11971476)。
文摘Let f be any arithmetic function and define S_(f)(x):=Σ_(n≤x)f([x/n]).If the function f is small,namely,f(n)﹤﹤n^(ε),then the error term E_(f)(x)in the asymptotic formula of S_f(x)has the form O(x^(1/2+ε)).In this paper,we shall study the mean square of E_(f)(x)and establish some new results of E_(f)(x)for some special functions.
