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Does this multiplicative function have a name? If so, what is known about it?
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It is well-known that the Euler $phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $phi(mn) = phi(m)phi(n)$. Thus it is defined by its values on prime powers. We know that $phi(p^k) = p^k-1 (p-1)$ for all primes $p$.
What about the multiplicative function $psi$ defined on the primes by $psi(p^k) = p^k-1 (p+1)$? Does it have a name? If so, what's known about it?
For example, can one evaluate $sum_n leq X psi(n)$?
nt.number-theory analytic-number-theory arithmetic-functions
edited 49 mins ago
Martin Sleziak
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
add a comment |Â
up vote
1
down vote
favorite
It is well-known that the Euler $phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $phi(mn) = phi(m)phi(n)$. Thus it is defined by its values on prime powers. We know that $phi(p^k) = p^k-1 (p-1)$ for all primes $p$.
What about the multiplicative function $psi$ defined on the primes by $psi(p^k) = p^k-1 (p+1)$? Does it have a name? If so, what's known about it?
For example, can one evaluate $sum_n leq X psi(n)$?
nt.number-theory analytic-number-theory arithmetic-functions
edited 49 mins ago
Martin Sleziak
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
1
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
It is well-known that the Euler $phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $phi(mn) = phi(m)phi(n)$. Thus it is defined by its values on prime powers. We know that $phi(p^k) = p^k-1 (p-1)$ for all primes $p$.
What about the multiplicative function $psi$ defined on the primes by $psi(p^k) = p^k-1 (p+1)$? Does it have a name? If so, what's known about it?
For example, can one evaluate $sum_n leq X psi(n)$?
nt.number-theory analytic-number-theory arithmetic-functions
edited 49 mins ago
Martin Sleziak
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
It is well-known that the Euler $phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $phi(mn) = phi(m)phi(n)$. Thus it is defined by its values on prime powers. We know that $phi(p^k) = p^k-1 (p-1)$ for all primes $p$.
What about the multiplicative function $psi$ defined on the primes by $psi(p^k) = p^k-1 (p+1)$? Does it have a name? If so, what's known about it?
For example, can one evaluate $sum_n leq X psi(n)$?
nt.number-theory analytic-number-theory arithmetic-functions
nt.number-theory analytic-number-theory arithmetic-functions
edited 49 mins ago
Martin Sleziak
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
edited 49 mins ago
Martin Sleziak
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
edited 49 mins ago
Martin Sleziak
2,78432028
edited 49 mins ago
Martin Sleziak
2,78432028
edited 49 mins ago
Martin Sleziak
2,78432028
2,78432028
asked 51 mins ago
Stanley Yao Xiao
7,93242682
asked 51 mins ago
Stanley Yao Xiao
7,93242682
asked 51 mins ago
Stanley Yao Xiao
7,93242682
7,93242682
1
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago
add a comment |Â
1
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago
1
1
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
$psi$ is the multiplicative convolution of $mu^2$ and the identity function, hence its Dirichlet series is
$$sum_n=1^inftyfracpsi(n)n^s=fraczeta(s)zeta(s-1)zeta(2s),qquadRe(s)>2.$$
This implies by Perron's formula and standard bounds that
$$sum_n leq X psi(n)simfraczeta(2)2zeta(4)X^2=frac152pi^2X^2.$$
answered 26 mins ago
GH from MO
56.7k5139215
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
$psi$ is the multiplicative convolution of $mu^2$ and the identity function, hence its Dirichlet series is
$$sum_n=1^inftyfracpsi(n)n^s=fraczeta(s)zeta(s-1)zeta(2s),qquadRe(s)>2.$$
This implies by Perron's formula and standard bounds that
$$sum_n leq X psi(n)simfraczeta(2)2zeta(4)X^2=frac152pi^2X^2.$$
answered 26 mins ago
GH from MO
56.7k5139215
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
add a comment |Â
up vote
4
down vote
accepted
$psi$ is the multiplicative convolution of $mu^2$ and the identity function, hence its Dirichlet series is
$$sum_n=1^inftyfracpsi(n)n^s=fraczeta(s)zeta(s-1)zeta(2s),qquadRe(s)>2.$$
This implies by Perron's formula and standard bounds that
$$sum_n leq X psi(n)simfraczeta(2)2zeta(4)X^2=frac152pi^2X^2.$$
answered 26 mins ago
GH from MO
56.7k5139215
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
$psi$ is the multiplicative convolution of $mu^2$ and the identity function, hence its Dirichlet series is
$$sum_n=1^inftyfracpsi(n)n^s=fraczeta(s)zeta(s-1)zeta(2s),qquadRe(s)>2.$$
This implies by Perron's formula and standard bounds that
$$sum_n leq X psi(n)simfraczeta(2)2zeta(4)X^2=frac152pi^2X^2.$$
answered 26 mins ago
GH from MO
56.7k5139215
$psi$ is the multiplicative convolution of $mu^2$ and the identity function, hence its Dirichlet series is
$$sum_n=1^inftyfracpsi(n)n^s=fraczeta(s)zeta(s-1)zeta(2s),qquadRe(s)>2.$$
This implies by Perron's formula and standard bounds that
$$sum_n leq X psi(n)simfraczeta(2)2zeta(4)X^2=frac152pi^2X^2.$$
answered 26 mins ago
GH from MO
56.7k5139215
edited 19 mins ago
answered 26 mins ago
GH from MO
56.7k5139215
answered 26 mins ago
GH from MO
56.7k5139215
answered 26 mins ago
GH from MO
56.7k5139215
56.7k5139215
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
add a comment |Â
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $sum_nle x varphi_1(n)=fraczeta(2)2zeta(4)x^2+O(xlog x)$. (Apostol denotes the function by $varphi_1$.)
– Martin Sleziak
18 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
In the field of automorphic forms, this function is often denotes $nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms).
– Peter Humphries
10 mins ago
add a comment |Â
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1
I found using Approach0 that this function is mentioned in this post: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. The topic of that post is exactly deriving some kind of asymptotic formula for $sum_n leq X psi(n)$.
– Martin Sleziak
47 mins ago
$psi(n) = # mathbbP^1(mathbbZ/nmathbbZ)$ is the number of points on the projective line.
– Chris Wuthrich
16 mins ago
I am not sure to which extent this is interesting, but here is another context in which this function appears: How can we show the equality $[SL_2(mathbb Z): Gamma_0(N)]=Nprod_pmid Nleft(1+frac1pright)$?. (As you can probably guess, I found the question using Approach0.)
– Martin Sleziak
13 mins ago