Sums of two squares in arithmetic progressions Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Sums of two squares in (certain) integral domainsIs there another simple formula for the sum-of-squares function?Exact formula for the number of integers in an interval which are the sum of two squares.Arithmetic Progressions of SquaresSums of two squares: What is known about the distribution of r(n)?Primes in arithmetic progressions in number fieldsJacobi's theorem on sums of two squares (reference request)Average class number formula for imaginary quadratic fields with prime discriminantSums of four coprime squaresSums of two integer squares in arithmetic progressions
Sums of two squares in arithmetic progressions
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)
Announcing the arrival of Valued Associate #679: Cesar Manara
Unicorn Meta Zoo #1: Why another podcast?Sums of two squares in (certain) integral domainsIs there another simple formula for the sum-of-squares function?Exact formula for the number of integers in an interval which are the sum of two squares.Arithmetic Progressions of SquaresSums of two squares: What is known about the distribution of r(n)?Primes in arithmetic progressions in number fieldsJacobi's theorem on sums of two squares (reference request)Average class number formula for imaginary quadratic fields with prime discriminantSums of four coprime squaresSums of two integer squares in arithmetic progressions
$begingroup$
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$sum _nleq xatop nequiv a(q)r(n)$$ and in particular is there an asymptotic formula?
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
add a comment |
$begingroup$
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$sum _nleq xatop nequiv a(q)r(n)$$ and in particular is there an asymptotic formula?
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
$begingroup$
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$sum _nleq xatop nequiv a(q)r(n)$$ and in particular is there an asymptotic formula?
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
$endgroup$
Let $r(n)$ denote the number of representations of $n$ as the sum of two squares. Are there any known results on $$sum _nleq xatop nequiv a(q)r(n)$$ and in particular is there an asymptotic formula?
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
nt.number-theory reference-request analytic-number-theory arithmetic-progression sums-of-squares
edited Apr 2 at 19:33
GH from MO
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
edited Apr 2 at 19:33
GH from MO
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
edited Apr 2 at 19:33
GH from MO
59.3k5150227
edited Apr 2 at 19:33
GH from MO
59.3k5150227
edited Apr 2 at 19:33
GH from MO
59.3k5150227
59.3k5150227
asked Apr 2 at 12:30
cawscaws
734
asked Apr 2 at 12:30
cawscaws
734
asked Apr 2 at 12:30
cawscaws
734
734
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f326964%2fsums-of-two-squares-in-arithmetic-progressions%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
$endgroup$
The first result in this direction seems to be due to R. A. Smith, ``The Circle Problem in an Arithmetic Progression,'' Can. Math. Bull. 11 (2), 175–184 (1968). He showed that if we write
$$sum _nleq xatop nequiv a(q)r(n) =pi x cdot fraceta_a(q)q^2+ R_q,a(x)$$
where $eta_a(q) := (x_1,x_2) in (mathbbZ/qmathbbZ)^2 : x_1^2 +x_2^2 equiv a bmod q$, then
$$R_q,a(x) = Oleft( x^frac23 + xi q^-frac12(1+3xi)gcd(a,q)^1/2tau(q) right)$$
for any $xi in (0,1/3)$. This is non-trivial for $q le x^frac23-varepsilon$. In particular, as $x$ tends to infinity, your expression is asymptotic to $pi x$ times the probability that $x_1^2+x_2^2 equiv a bmod q$ (for uniformly drawn $x_1,x_2$ mod $q$). If you consider $a$ and $q$ as fixed, this answers your question.
The state-of-the-art result is due to D. I. Tolev, ``On the remainder term in the circle problem in an arithmetic progression,'' Tr. Mat. Inst. Steklova 276 (2012), Teoriya Chisel, Algebra i Analiz, 266--279; translation in Proc. Steklov Inst. Math. 276 (2012), no. 1, 261–274. He showed that
$$R_q,a(x) = Oleft( (q^frac12+x^frac13) gcd(a,q)^1/2tau^4(q)log^4 x right).$$
Interestingly, for $a=1$, there is a result which is superior in certain ranges of $x$ and $q$, see P. D. Varbanets, “Lattice Points in a Circle Whose Distances from the Center Are in an Arithmetic Progression,” Mat. Zametki 8 (6), 787–798 (1970) [Math. Notes 8, 917–923 (1970)].
All three results are explained in Tolev's paper.
In the 2002 PhD thesis of Michael J. Dancs under R. Vaughn, titled "On a Variance arising in the Gauss Circle Problem", he proves (independently of the above) that
$$f(q,a):=lim_x to infty fracsum _nleq xatop nequiv a(q)r(n) pi x$$
exists and can be written as
$$f(q,a)=q^-3 sum_k=1^q expleft( 2pi i frac-akq right) S(q,k)^2,$$
where $S(q,a)$ is a quadratic Gauss sum mod $q$. This gives a different expression for $eta_a(q)/q^2$, which was better suited for the applications given by Dancs. This is done in Theorem 2.1. He then proceeds to prove, in Theorem 2.2, the estimate
$$R_q,a(x) = Oleft( (sqrtx+q) log qright).$$
For small $q$ this is better than Smith's work. Still, it is superseded by Tolev's estimates.
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
edited Apr 2 at 14:08
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
answered Apr 2 at 13:27
Ofir GorodetskyOfir Gorodetsky
5,90312639
5,90312639
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
$begingroup$
Thanks for the very informative answer!
$endgroup$
– caws
Apr 2 at 21:42
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f326964%2fsums-of-two-squares-in-arithmetic-progressions%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
If we fix $a, q$ and increase $x$, isn't this essentially the Gauss circle problem with some shift in the lattice?
$endgroup$
– Dongryul Kim
Apr 2 at 12:50
$begingroup$
It would be nice to make clear whether you require uniformity in $a$ and $q$. If yes, then the answer of Ofir addresses this. If not, then the result is a simple application of Dirichlet series techniques.
$endgroup$
– Daniel Loughran
Apr 2 at 19:10