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LSpice
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In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables""Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers that are sums of two squares, there exists a continuous function $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$$$L(x)=\exp\left(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}}\right)$$ and $c$ is a suitable positive constant. This result is obtained by using the Selberg-DelangeSelberg–Delange method. My question is how to obtain an explicit expression for $\lambda$ and how to find a good bound for the sums on the right side, i.e. on $\sum_{n\le x}\beta(n)$.
Many thanks in avance.

In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers sums of two squares, there exists a continuous function $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$ and $c$ is a suitable positive constant. This result is obtained by using the Selberg-Delange method. My question is how to obtain an explicit expression for $\lambda$ and how to find a good bound for the sums on the right side, i.e. on $\sum_{n\le x}\beta(n)$.
Many thanks in avance.

In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers that are sums of two squares, there exists a continuous function $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp\left(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}}\right)$$ and $c$ is a suitable positive constant. This result is obtained by using the Selberg–Delange method. My question is how to obtain an explicit expression for $\lambda$ and how to find a good bound for the sums on the right side, i.e. on $\sum_{n\le x}\beta(n)$.

Minor Math Jaxing and grammar improvement.
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Daniele Tampieri
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Explicit values ofexpression for a function in number theory

In their paper "moyenne"Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers sums of two squares, itthere exists a continuonscontinuous function on [0,1] $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$ and $c$ is a suitable positive constant. This result is due toobtained by using the Selberg-delangeDelange method. My question concernsis how to obtain an explicit values ofexpression for $\lambda$ and Thishow to find a good bound for the sums of $\beta(n)$ less or equal toon the right side, i.e. on $x.$$\sum_{n\le x}\beta(n)$.
Many thanks in avance.

Explicit values of a function

In their paper "moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers sums of two squares, it exists a continuons function on [0,1] $\lambda$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$ and $c$ is a suitable positive constant. This result is due to Selberg-delange method. My question concerns an explicit values of $\lambda$ and This to find a good bound for the sums of $\beta(n)$ less or equal to $x.$ Many thanks in avance.

Explicit expression for a function in number theory

In their paper "Moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers sums of two squares, there exists a continuous function $\lambda$ on $[0,1]$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$ and $c$ is a suitable positive constant. This result is obtained by using the Selberg-Delange method. My question is how to obtain an explicit expression for $\lambda$ and how to find a good bound for the sums on the right side, i.e. on $\sum_{n\le x}\beta(n)$.
Many thanks in avance.

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Explicit values of a function

In their paper "moyenne de certains fonctions arithmétiques sur les entiers friables", Tenenbaum and Wu proved that for the case of the function $\beta$ which is the indicator function of integers sums of two squares, it exists a continuons function on [0,1] $\lambda$ such that $\lambda(0)>0$ and $$\sum_{n \leq x} \beta(n)= \int_0^{1/2} x^{1-t} \frac{\lambda(t)}{\sqrt{t}} dt+O\left(\frac{x}{L(x)^c}\right)\quad (x\geq 3).$$ Where $$L(x)=\exp(\frac{\log(x)^{3/5}}{\log_2(x)^{1/5}})$$ and $c$ is a suitable positive constant. This result is due to Selberg-delange method. My question concerns an explicit values of $\lambda$ and This to find a good bound for the sums of $\beta(n)$ less or equal to $x.$ Many thanks in avance.