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# A note on medium and short character sums

Matteo Bordignon  KTH Royal Institute of Technology, Stockholm
and
Charles University, Faculty of Mathematics and Physics, Department of Algebra,
Sokolovská 83, 186 00 Praha 8, Czech Republic Department of Mathematics
<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Following the work of Hildebrand we improve the Pólya-Vinogradov inequality in
a specific range, we also give a general result that shows its dependency on
Burgess bound and at last we improve the range of validity for a special case
of Burgess’ character sum estimate.

It is of high interest studying the possible upper bounds of the following
quantity

(1) $S(N,\chi):=\left|\sum_{n=1}^{N}\chi(n)\right|,$

with $N\in\mathbb{N}$ and $\chi$ a non-principal Dirichlet character modulo
$q$. The famous Pólya–Vinogradov inequality tells us that for any $\chi$ non-
principal

$S(N,\chi)\ll\sqrt{q}\log q,$

and aside for the implied constant, this is the best known result. Paley in
[19] proved that for infinitely many characters we have

$\max_{N}\left|\sum_{n=1}^{N}\chi(n)\right|\gg\sqrt{q}\log\log q.$

On the other hand Montgomery and Vaughan [18] showed, assuming the Generalized
Riemann Hypothesis (GRH), we have that

$\left|\sum_{n=1}^{N}\chi(n)\right|\ll\sqrt{q}\log\log q.$

The best known asymptotic constant in (1) for primitive characters is
$\frac{69}{70\pi 3\sqrt{3}}+o(1)$, if $\chi$ is even from [11] and
$\frac{1}{3\pi}+o(1)$ if $\chi$ is odd from [17]. For the best completely
explicit constant see [2], [3] and [8]. For primitive characters of odd order
$g$ Granville and Soundararajan improved the Pólya–Vinogradov inequality,
proving the following bound

$\left|\sum_{n=1}^{N}\chi(n)\right|\ll\sqrt{q}(\log q)^{1-\delta_{g}/2+o(1)},$

with $\delta=1-\frac{g}{\pi}\sin\frac{\pi}{g}$. Another interesting result is
Theorem 1.1 in [7] by E. Fouvry, E. Kowalski, P. Michel, C. S. Raju, J. Rivat,
and K. Soundararajan, where they extend the so called Pólya–Vinogradov range.
It is interesting to note that it appears that $S(N,\chi)$ assumes its maximum
for $N\approx q$, see the work by Bober et al. in [1] and the one by
Hildebrand, Corollary 3 of [17], which proves that for even characters we have
that $N=o(q)$ implies $S(N,\chi)=o(\sqrt{q}\log q)$.
In this paper we will first give an improved version of the Pólya–Vinogradov
inequality for a limited range, drawing inspiration from the work of
Hildebrand in [16] and [17]. The interesting aspect of this result is that it
does not use Burgess bound as it only relies on Montgomery and Vaughan [18,
Corollary 1] and that it allows to prove the best possible Pólya–Vinogradov
inequality in a certain range. Specifically, we prove the following result
that follows from Theorem 4 taking $a(q)=(\log q)^{c}$, with $c>4+2\epsilon$,
and $R(q)=(\log q)^{2+\epsilon}$ and going from primitive to non principal
characters as done by Hildebrand in [16].

######  Theorem 1.

Take $N$ and $q$ such that

$\frac{q}{(\log q)^{c+2}}<N<\frac{q}{2(\log q)^{2}},$

for any fixed $c>0$. Then, for $\chi$ an even non-principal character mod $q$,
we have

$\displaystyle\sum_{n=1}^{N}\chi(n)\ll_{c}\sqrt{q}\log\log q.$

Following the work of Hildebrand in [16], we show that the best possible
Burgess bound, see [12], comes slightly short from proving the best possible
Pólya–Vinogradov inequality, see [18]. Specifically, we assume the following
Burgess-like result, which is the one proven in [12], assuming GRH, but with a
stronger upper bound.

######  Conjecture 1.

For any non-principal character $\chi$ modulo $q$ and $x$ such that $\log
x/\log\log q\rightarrow\infty$ we have

$\sum_{n\leqslant x}\chi(n)\ll_{\epsilon}\frac{x}{(\log x)^{3+\epsilon}},$

for a fixed $\epsilon>0$.

This allows us to prove the following result, which comes $(\log\log
q)^{\epsilon}$ short from the optimal result proven, assuming GRH, in [18].

######  Theorem 2.

Assuming Conjecture 1 and taking $\chi$ any non-principal character modulo
$q$, then we have

$\sum_{n\leqslant x}\chi(n)\ll_{\epsilon}\sqrt{q}(\log\log q)^{1+\epsilon}.$

We then focus on short character sums. Let $\chi$ be a non-principal character
modulo a prime $p$. Estimates of the type

(2) $\left|\sum_{n\leqslant N}\chi(n)\right|\leqslant\epsilon
N\quad(N\geqslant N_{0}(\epsilon,p)),$

are of great importance in number theory. By the Pólya–Vinogradov inequality
(2) holds with $N_{0}(\epsilon,p)=(\log p)\sqrt{p}/\epsilon$, and Burgess’
character sum estimate [4, 5] yields (2) with
$N_{0}(\epsilon,p)=p^{1/4+\delta}$ for any $\epsilon,\delta>0$ and $p\geqslant
p_{0}(\epsilon,\delta)$. Hildebrand in [15] proved that given $\epsilon>0$
there exist $\delta=\delta(\epsilon)>0$ and $p_{0}(\epsilon)\geqslant 2$ such
that for any $p\geqslant p_{0}(\epsilon)$ (2) holds with
$N_{0}(\epsilon,p)=p^{1/4-\delta}$, with
$\delta(\epsilon)=\exp(-C(\epsilon^{-2}+1))$ with $C$ a sufficiently large
constant. Using the result by Elliot in [6] it is possible to take
$\delta(\epsilon)=C\epsilon^{19}$ and using Granville and Soundararajan [10,
Corollary 3] $\delta(\epsilon)=C\epsilon^{2.76}$, that is currently the state
of the art result. Also in [13, ${\mathbb{S}}$ 9] Granville and Soundararajan
show that [10, Corollary 3] is optimal for general multiplicative functions,
but it is worth to note that the “worst” function they give is not a Dirichlet
character. It is thus interesting to ask if it is possible to improve [10,
Corollary 3] for Dirichlet characters. In this paper we show that this can be
done for real multiplicative functions proving Lemma 11, we thus improve the
size of the $\delta(\epsilon)$ in the chase of non-principal real characters.
While this result follows easily from the current techniques, we were not able
to find it in the literature and believe it is of interest. We obtain the
following result that appears optimal with the current techniques.

######  Theorem 3.

Given $\epsilon>0$ there exist $\delta>0$ and $p_{0}(\epsilon)\geqslant 2$
such that for any non-principal real character $\chi$ modulo a prime
$p\geqslant p_{0}(\epsilon)$ (2) holds with $N_{0}(\epsilon,p)=p^{1/4-\delta}$
with

$\delta(\epsilon)=C\epsilon^{2},$

with $C$ a sufficiently large absolute constant.

In Section 1 we prove a general version of Theorem 1, Section 2 we prove
Theorem 2 and a more general version of it where we assume weaker versions of
Conjecture 1 and in Section 3 we prove Theorem 3.

## Acknowledgements

I would like to thank Bryce Kerr, Pär Kurlberg, Igor Shparlinski and Tim
Trudgian for the useful comments and suggestions.

The research was partially supported by OP RDE project
No. CZ.$02.2.69/0.0/0.0/18\\_053/0016976$ International mobility of research,
technical and administrative staff at the Charles University.

##  1\. Improving Pólya–Vinogradov for a limited range without using Burgess
bound

In this section we prove the following general version of Theorem 1.

######  Theorem 4.

Let $R:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ and
$a:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ and $x$ such that $2\leqslant
R<a(q)\leqslant q$. Take $N$ and $q$ and assume

$\frac{qR(q)}{a(q)}<N<\frac{q}{R(q)}\left(1-\frac{R(q)}{a(q)}\right).$

Then, for $\chi$ an even primitive character, we have

$\displaystyle\sum_{n=1}^{N}\chi(n)\leqslant$
$\displaystyle\sqrt{q}2\left(\frac{2}{\pi}\log a(q)+\frac{2}{\pi}\left(C+\log
2+\frac{3}{a(q)}\right)\right)$
$\displaystyle+O\left(\sqrt{q}\max\left(\log\frac{\log q}{\log
a(q)},\frac{(\log R(q))^{3/2}}{\sqrt{R(q)}}\log\frac{q}{a(q)}\right)\right).$

Let $f:\mathbb{Z}\rightarrow\mathbb{C}$ be a multiplicative function with
$|f(n)|\leqslant 1$. With $\alpha$ real and $e(\alpha)=\exp(2\pi i\alpha)$
write

$S(\alpha)=\sum^{N}_{n=1}f(n)e(n\alpha).$

We will need [18, Corollary 1].

######  Lemma 5.

Suppose that $|\alpha-a/q|\leqslant q^{-2}$, $(a,q)=1$ and $2\leqslant
R\leqslant q\leqslant N/R$. Then

$S(\alpha)\ll\frac{N}{\log N}+\frac{N\log^{\frac{3}{2}}(R)}{\sqrt{R}}.$

We now obtain a variation of [16, Lemma 3].

######  Lemma 6.

Let $R:\mathbb{N}\rightarrow\mathbb{N}$ and
$a:\mathbb{N}\rightarrow\mathbb{N}$ and $x$ such that $2\leqslant R\leqslant
a(q)$, $a(q)\leqslant x\leqslant q$ and $R/a<1$. Take $N$ and $q$ and assume

(3) $\frac{qR(q)}{a(q)}<N<\frac{q}{R(q)}\left(1-\frac{R(q)}{a(q)}\right).$

We have, uniformly for all primitive characters $\chi$ modulo $q$ as above,

$\left|\sum_{n\leqslant x}\chi(n)e(\alpha n)\right|\ll\max\left(\frac{x}{\log
x},x\frac{(\log R(q))^{3/2}}{\sqrt{R(q)}}\right).$

###### Proof.

Set $M=[x]$, by Dirichlet’s theorem there exist integers $r$ and $s$, where
$(r,s)=1$ and $1\leqslant s\leqslant M/R$, such that

(4) $\left|\frac{N}{q}-\frac{r}{s}\right|\leqslant\frac{1}{sM/R}.$

Assuming $r=0$ we obtain, from (4),

$\displaystyle N\leqslant\frac{qR}{sx},$

but this, together with the left-hand side of (3), gives

$\displaystyle\frac{1}{a(q)}<\frac{1}{sx},$

that is in contradiction with the assumption $a(q)\leqslant x$ and $1\leqslant
s$. We can thus assume $r\neq 0$ which, (4) together with $M=[x]$ and the
assumptions $a(q)\leqslant x\leqslant q$ and $R/a<1$, gives

(5) $\displaystyle
s\geqslant\frac{q}{N}\left(r-\frac{R}{M}\right)\geqslant\frac{q}{N}\left(1-\frac{R(q)}{a(q)}\right)>0.$

Now from the right-hand side of (3) we have

$\displaystyle\frac{q}{N}>R(q)\left(1-\frac{R(q)}{a(q)}\right)^{-1}.$

Using this together with (5) we obtain $s\gg R$. Using $s\gg R$, the result
follows from Lemma 5. ∎

We then need an explicit bound on a trigonometric sum, by Pomerance in [20,
Lemma 3].

######  Lemma 7.

Uniformly for $x\geqslant 1$ and with $\alpha$ real we have

$\sum_{n\leqslant x}\frac{|\sin(\alpha n)|}{n}\leqslant\frac{2}{\pi}\log
x+\frac{2}{\pi}\left(C+\log 2+\frac{3}{x}\right).$

We can now prove Theorem 4.

###### Proof.

We take $\chi$ primitive and start with

$\chi(n)=\frac{1}{d(\overline{\chi})}\sum_{a=1}^{q}\overline{\chi}(a)e\left(\frac{an}{q}\right)=\frac{1}{d(\overline{\chi})}\sum_{0<|a|<q/2}\overline{\chi}(a)e\left(\frac{an}{q}\right),$

where $d(\overline{\chi})$ is the Gaussian sum. Summing over $1\leqslant
n\leqslant N$, we obtain

$\displaystyle\sum_{n=1}^{N}\chi(n)$
$\displaystyle=\frac{1}{d(\overline{\chi})}\sum_{0<|a|<q/2}\overline{\chi}(a)\sum_{n=1}^{N}e\left(\frac{an}{q}\right)$
$\displaystyle=\frac{1}{d(\overline{\chi})}\sum_{0<|a|<q/2}\overline{\chi}(a)\frac{e\left(\frac{aN}{q}\right)-1}{1-e\left(\frac{-a}{q}\right)}.$

It follows that

(6)
$\sum_{n=1}^{N}\chi(n)\leqslant\frac{\sqrt{q}}{2\pi}\left|\sum_{0<|a|<q/2}\frac{\overline{\chi(a)}\left(e(\frac{aN}{q})-1\right)}{a}\right|+O(\sqrt{q}).$

Now we split the inner sum in two parts: $\Sigma_{1}$ with $0<|a|\leqslant
a(q)$ and $\Sigma_{2}$ with $a(q)<|a|<q/2$.

Now as $\chi$ is even, by partial summation and Lemma 6, we have

$\displaystyle\Sigma_{2}\ll\max\left(\log\frac{\log q}{\log a(q)},\frac{(\log
R(q))^{3/2}}{\sqrt{R(q)}}\log\frac{q}{a(q)}\right).$

We now note that

$\Sigma_{1}=2i\sum\limits_{1\leqslant a\leqslant
a(q)}\frac{\overline{\chi(a)}\sin(\frac{2\pi aN}{q})}{a},$

and thus from Lemma 7 we obtain

$\left|\Sigma_{1}\right|\leqslant 2\left(\frac{2}{\pi}\log
a(q)+\frac{2}{\pi}\left(C+\log 2+\frac{3}{a(q)}\right)\right).$

We thus obtain the desired result Theorem 4. ∎

##  2\. Improving Pólya–Vinogradov using a Burgess-like bound

In this section we aim to prove Theorem 2. We first prove the following
fundamental result.

######  Lemma 8.

Assuming Conjecture 1 holds, then we have the following result. Take
$a:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that $a(q)\leqslant
x\leqslant q$ and with $x$ such that $2\leqslant(\log x)^{2+\epsilon}\leqslant
a(q)$. We have, uniformly for all primitive characters $\chi$ modulo $q$ as
above and $\log a(q)/\log\log q\rightarrow\infty$,

$\left|\sum_{n\leqslant x}\chi(n)e(\alpha n)\right|\ll_{\epsilon}\frac{x}{\log
x}.$

###### Proof.

Set $N=\lfloor x\rfloor$, $R=(\log N)^{2+\epsilon}$. Taking $q$ big enough, we
easily obtain $2\leqslant R\leqslant N$. By Dirichlet’s theorem there exist
integers $r$ and $s$, where $(r,s)=1$ and $1\leqslant s\leqslant N/R$, such
that

$\left|\alpha-\frac{r}{s}\right|\leqslant\frac{1}{sN/R}.$

If $s\geqslant R$, the result follows from Lemma 5, since

$\displaystyle\frac{N}{\log N}+N\frac{(\log
R)^{\frac{3}{2}}}{\sqrt{R}}\ll_{\epsilon}\frac{x}{\log x}.$

Now suppose $s<R$. By partial summation follows

$\displaystyle\left|\sum_{n\leqslant x}\chi(n)e(\alpha
n)\right|\ll\left(1+\left|\alpha-\frac{r}{s}\right|x\right)\max_{u\leqslant
x}|T(u)|\ll R\max_{u\leqslant x}|T(u)|,$

where

$T(u)=\sum_{n\leqslant u}\chi(n)e\left(\frac{rn}{s}\right).$

By grouping the terms of the sum $T(u)$ according to the value of $(n,s)$, we
get

$\displaystyle T(u)$
$\displaystyle=\sum_{dt=s}\sum_{\begin{subarray}{c}dm\leqslant u\\\
(m,t)=1\end{subarray}}\chi(md)e\left(\frac{rm}{t}\right)$
$\displaystyle=\sum_{dt=s}\chi(d)\sum_{\begin{subarray}{c}1\leqslant
a\leqslant t\\\
(a,t)=1\end{subarray}}e\left(\frac{ra}{t}\right)\sum_{\begin{subarray}{c}m\leqslant
u/d\\\ m=a\pmod{t}\end{subarray}}\chi(m)$
$\displaystyle=\sum_{dt=s}\frac{\chi(d)}{\varphi(t)}\sum_{\psi\pmod{t}}\sum_{1\leqslant
a\leqslant t}e\left(\frac{ra}{t}\right)\overline{\psi}(a)\sum_{m\leqslant
u/d}\chi(m)\psi(m).$

Applying Conjecture 1 to the right-hand sum we obtain

$\displaystyle\sum_{n\leqslant x}\chi(n)e(\alpha
n)\ll_{\epsilon}R\frac{x}{(\log x)^{3+\epsilon}}\ll_{\epsilon}\frac{x}{\log
x}.$

∎

We can now prove Theorem 2.

###### Proof.

We again use (6) and split the inner sum in two parts: $\Sigma_{1}$ with
$0<|a|\leqslant a(q)$ and $\Sigma_{2}$ with $a(q)<|a|<q/2$.

Now using that $\chi$ is even, by partial summation and Lemma 8, we have

$\displaystyle\Sigma_{2}\ll\log\frac{\log q}{\log a(q)}.$

We then note that

$\Sigma_{1}=2i\sum\limits_{1\leqslant a\leqslant
a(q)}\frac{\overline{\chi(a)}\sin(\frac{2\pi aN}{q})}{a},$

and thus from Lemma 7 we obtain

$\left|\Sigma_{1}\right|\leqslant 2\left(\frac{2}{\pi}\log
a(q)+\frac{2}{\pi}\left(C+\log 2+\frac{3}{a(q)}\right)\right).$

We thus obtain the desired result raking $\log a(q)=(\log\log
q)^{1+\epsilon}$. ∎

In general, we can assume the following Burgess-like result.

######  Conjecture 2.

For any non-principal character $\chi$ modulo $q$ and $x$ such that $x\gg
l(q)$, for a fixed $l:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$, we have

$\sum_{n\leqslant x}\chi(n)\ll\frac{x}{c(x)},$

for a certain $c:\mathbb{N}\rightarrow\mathbb{N}$.

In the same way as in the proof of Lemma 8, assuming Conjecture 2, we prove
the following result.

######  Lemma 9.

Assuming Conjecture 2 holds for specific $l$ and $c$, then we have the
following result. Take $a:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$a(q)\leqslant x\leqslant q$ and with $x$ such that $2\leqslant R(x)\leqslant
a(q)$. We have, uniformly for all primitive characters $\chi$ modulo $q$ as
above and $x\gg l(q)$,

$\displaystyle\left|\sum_{n\leqslant x}\chi(n)e(\alpha n)\right|\ll
x\max\left(\frac{1}{\log x}+\frac{(\log
R(x))^{3/2}}{\sqrt{R(x)}},\frac{R(x)}{c(x)}\right)$

In the same way as in the proof of Theorem 2, using Lemma 9, we can prove the
following more general result.

######  Theorem 10.

Assuming Conjecture 2 holds for specific $l$ and $c$, then we have the
following result. We have, uniformly for all even primitive characters $\chi$
modulo $q$ as above, $2\leqslant R(x)\leqslant a(q)$ and $l(q)\leqslant a(q)$,

$\displaystyle\left|\sum_{n\leqslant x}\chi(n)\right|\ll\log
a(q)+\int_{a(q)}^{q}\frac{\max(x)}{x}dx,$

where

$\displaystyle\max(x)=\max\left(\frac{1}{\log x},\frac{(\log
R(x))^{3/2}}{\sqrt{R(x)}},\frac{R(x)}{c(x)}\right).$

##  3\. Hildebrand’s version of Burgess bound

Theorem 3 is made possible by the following improvement of Lemma 4 in [14].

######  Lemma 11.

Let $f$ be a real multiplicative function satisfying $-1\leqslant f\leqslant
1$, and let

$M(x)=M(x,f)=\frac{1}{x}\sum_{n\leqslant x}f(x).$

Then for $3\leqslant x\leqslant x^{\prime}\leqslant x^{\mathcal{O}(1)}$ we
have

$\left|M(x^{\prime})-M(x)\right|\ll\left(\frac{\log(2x^{\prime}/x)}{\log
x^{\prime}}\right)^{1/2-\epsilon_{1}},$

for any $\epsilon_{1}>0$ and where the implied constant is uniform.

Before focusing on Lemma 11 we will use it to prove Theorem 3.

###### Proof.

Applying Lemma 11 with $f=\chi$ (a non-principal real character modulo a prime
$p$), $x=N\geqslant p^{1/4-\delta}$ and $x^{\prime}=Np^{2\delta}(\geqslant
p^{1/4+\delta})$, we obtain

$M(N,\chi)=M(Np^{2\delta},\chi)+\mathcal{O}\left(\delta^{1/2-\epsilon_{1}}\right).$

The result now follows bounding $M(Np^{2\delta},\chi)$ using Burgess estimate.
∎

We are now left with proving Lemma 11. The proof is based on two results.
First we need the version of Halász result given by Tenenbaum in [21, pag.
343] which, defining

$S(x,T):=\min_{|\gamma|\leqslant 2T}\sum_{p\leqslant
x}\frac{1-\mathbf{R}\left(f(p)p^{-i\gamma}\right)}{p},$

gives for any multiplicative function $|f(n)|\leqslant 1$, $x\geqslant 3$ and
$T\geqslant 1$

(7) $M(x,f)\leqslant(1+S(x,T))e^{-S(x,T)}+\frac{1}{\sqrt{T}}.$

This gives the first of the two results that we need.

######  Theorem 12.

For all real multiplicative functions $f$ with $|f|\leqslant 1$, all
$x\geqslant 3$ and with

$S^{\prime}(x,t):=\min\left\\{\log\log x+O(1),\sum_{p\leqslant
x}\frac{1-f(p)}{p}\right\\}$

we have

$S(x,f)\ll(1+S^{\prime}(x,t))\exp\left(-S^{\prime}(x,t)\right).$

###### Proof.

The result follows from (7), observing that for $f(n)$ real we have

$S(x,T)=\min\left\\{\log\log x+O(1),\sum_{p\leqslant
x}\frac{1-f(p)}{p}\right\\}.$

∎

The second is Proposition 4.1. in [9].

######  Lemma 13.

For all real multiplicative functions $f$ with $-1\leqslant f\leqslant 1$. Let
$x$ be large, $1\leqslant x\leqslant x^{\prime}$. Then

$\left|M(x^{\prime})-M(x)\right|\ll\frac{\log 2x^{\prime}/x}{\log
x^{\prime}}\exp\left(\sum_{p\leqslant x^{\prime}}\frac{1-f(p)}{p}\right).$

We can now prove Lemma 11.

###### Proof.

Given $x^{\prime}\geqslant x\geqslant 3$, define $\delta$ by
$x^{\prime}=x^{1+\delta}$ and put

$R=\log\left(\frac{\log x^{\prime}}{\log
2x^{\prime}/x}\right)^{1/2+\epsilon}.$

If we assume

$\sum_{p\leqslant x^{\prime}}\frac{1-f(p)}{p}\geqslant R,$

we have that Lemma 11 follows from Theorem 12 and using that from $3\leqslant
x\leqslant x^{\prime}\leqslant x^{\mathcal{O}(1)}$ we have

$\sum_{x\leqslant p\leqslant x^{\prime}}\frac{1-f(p)}{p}\ll 1.$

On the other hand if we assume

$\sum_{p\leqslant x^{\prime}}\frac{1-f(p)}{p}\leqslant R,$

we have that Lemma 11 follows from Lemma 13. This concludes the proof. ∎

## References

  * [1] J. Bober, L. Goldmakher, A. Granville and D. Koukoulopoulos, The frequency and the structure of large character sums, J. Eur. Math. Soc. (JEMS), 20(7): 1759–1818, 2018.
  * [2] M. Bordignon, Partial Gaussian sums and the Pólya–Vinogradov inequality for primitive characters, Rev. Mat. Iberoam.,2021, DOI: 10.4171/RMI/1328
  * [3] M. Bordignon and B. Kerr, An explicit Pólya–Vinogradov inequality via Partial Gaussian sums, Trans. Amer. Math. Soc., 373(9):6503–6527, 2020.
  * [4] D. A. Burgess, The distribution of quadratic residues and non-residues Mathematika, 4:106–112, 1957.
  * [5] D. A. Burgess, On character sums and primitive roots, Proc. London Math. Soc.,12(3): 179–192, 1962.
  * [6] P. D. T. A. Elliott, Extrapolating the mean-values of multiplicative functions, Indag. Math., 51: 409–420, 1989.
  * [7] E. Fouvry, E. Kowalski, P. Michel, C. S. Raju, J. Rivat, and K. Soundararajan. On short sums of trace functions, Ann. Inst. Fourier (Grenoble), 67(1):423– 449, 2017.
  * [8] D. A. Frolenkov and K. Soundararajan, A generalization of the Pólya-Vinogradov inequality. Ramanujan J., 31(3): 271–279, 2013.
  * [9] A. Granville; K. Soundararajan, The spectrum of multiplicative functions, Ann. of Math. (2), 153(2):407–470, 2001.
  * [10] A. Granville; K. Soundararajan, Decay of mean values of multiplicative functions, Canad. J. Math., 55(6): 1191–1230, 2003.
  * [11] A. Granville and K. Soundararajan, Large character sums: pretentious characters and the Pólya-Vinogradov theorem, J. Amer. Math. Soc, 20(2): 357–384, 2007.
  * [12] A. Granville; K. Soundararajan, Large character sums: Burgess’s theorem and zeros of $L$-functions, J. Eur. Math. Soc., 20(1):1–14, 2018.
  * [13] A. Granville; K. Soundararajan, A new proof of Halász’s theorem, and its consequences, Compos. Math., 155(1): 126–163, 2019.
  * [14] A. Hildebrand, On Wirsing’s mean value theorem for multiplicative functions, Bull. London Math. Soc., 18(2):147–152, 1986.
  * [15] A. Hildebrand, A note on Burgess’ character sum estimate, C. R. Math. Rep. Acad. Sci. Canada, 8(1):35–37, 1986.
  * [16] A. Hildebrand, On the constant in the Pólya–Vinogradov inequality, Canad. Math. Bull., 31(3):347–352, 1988.
  * [17] A. Hildebrand, Large values of character sums, J. Number Theory, 29(3): 271–296, 1988.
  * [18] H. Montgomery; R. Vaughan, Exponential sums with multiplicative coefficients, Invent. Math., 43(1):69–82, 1977.
  * [19] R. E. A. C. Paley, A Theorem on Characters, J. London Math. Soc., 7(1):28–32, 1932.
  * [20] C. Pomerance, Remarks on the Pólya-Vinogradov inequality, Integers, 11(4):531–542, 2011.
  * [21] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge, 1995. xvi+448 pp.


# Radiation of fast positrons interacting with periodic microstructure on the
surface of a crystal

V. Epp<EMAIL_ADDRESS>Tomsk State Pedagogical University, ul. Kievskaya 60,
634061 Tomsk, Russia  Tomsk State University, pr. Lenina 36, 634050 Tomsk,
Russia  J. G. Janz<EMAIL_ADDRESS>V. V. Kaplin<EMAIL_ADDRESS>Tomsk Polytechnic
University, pr. Lenina 34, 634050 Tomsk, Russia

###### Abstract

Radiation of charged particles passing through a set of equidistant ridges on
the surface of a single crystal is calculated. The ridges are rectangular in
shape, each of thickness of half of the particle trajectory period at planar
channeling in a thick crystal. Positively charged particle entering the first
ridge with angle smaller than the critical channeling angle is captured into
channeling and changes the direction of its transversal velocity to reversed.
Between the half-wave ridges the particle moves along a straight line. Passing
through such set of half-wave crystal plates the particle moves on quasi-
undulator trajectories. Properties of the particle radiation emitted during
their passage through such “multicrystal undulator” are calculated. The
radiation spectrum in each direction is discrete, and the frequency of the
first harmonic and the number of harmonics in the spectrum depends on the
distance between the plates, on energy of the particles and on the averaged
potential energy of atomic planes of the crystal. The radiation is bound to a
narrow cone in the direction of the average particle velocity and polarized
essentially in a plane orthogonal to the atomic planes of the crystal.

###### keywords:

channeling, radiation, half-wave crystal, relativistic particles, spectrum

##  1 Introduction

Relativistic particles channeled in a single crystal have long been used as a
source of hard electromagnetic radiation. One of the drawbacks of such a
source is the limited ability of tuning the frequency of radiation. To address
this shortcoming various schemes were proposed, which combine the advantages
of channeling radiation and undulator radiation. The most common schemes
include periodically bent crystals. The crystallographic planes can be bent by
means of ultrasonic waves [1, 2] or periodic microscopic cuts of a single-
crystal plate, which in this case is bent due to internal stresses [3, 4, 5,
6]. In order to get high energy radiation the crystal undulators were offered
[7, 8] and then realized [9, 10] with the period much shorter than the period
of particle channeling, and the bending amplitude much less than the spacing
between the crystallographic planes.

In some cases, on the contrary, it is required to obtain more soft radiation,
however, it is desirable to use a beam of sufficiently high energy particles
to generate radiation of high intensity. For this purpose one can use either
periodically bent crystal, or as proposed in [11], a set of thin crystalline
plates, each changing the transversal velocity of the beam to reversal one, so
that the particle trajectory in the set of the crystals looks like a zigzag
line. Calculation of radiation in such a device referred to as a multicrystal
undulator, is the focus of this paper.

The trajectories of electrons and positrons in a crystal plate of thickness
equal to half of the particle trajectory period at planar channeling are
investigated by numerical methods in [12]. Experimental study of the passage
of electrons through such a half-wave plate is described in paper [13]. The
radiation emitted in a single half-wave plate was studied numerically and it
was shown that the basic properties of radiation are similar to those of
radiation from the arc of a circle [14, 15]. Coherent superposition of the
radiation fields generated in a series of half-wave plates results in the
emission spectrum with specific properties that are studied theoretically in
this paper.

##  2 The particle trajectory

Consider a multicrystal undulator constructed on the surface of a single
crystal as shown in Fig. 1. It consists of a set of equidistant ridges
rectangular in shape and each of half-wave thickness. The crystal planes which
enable the channeling of positively charged particles, are orthogonal to the
surfaces of the ridges. The height of the ridges is much greater than their
thickness, hence, the rest of the crystal can be neglected. Hereafter we
consider the ridges as a set of thin crystal plates. Similar design was used
in experiments [16, 17] for production of diffracted X-ray transition
radiation and parametric X-ray radiation at Bragg angle, but the thickness of
the ridges was much greater than half of the channeling period.

Figure 1: Diagram of multicrystal undulator. The dotted lines show the crystal
planes.

We choose a coordinate system in such a way that the $x$-axis is parallel to,
and the $y$-axis is orthogonal to the crystal planes. $Z$-axis is
perpendicular to the initial velocity of the incident particle. The thickness
of each plate is $d_{1}$, the spacing between the plates is $d_{2}$, and the
distance between crystal planes equals $2a$. We use harmonic approximation for
the average potential of the electric field between the crystal planes:

$U(y)=\frac{U_{0}}{a^{2}}y^{2},$ (1)

where $U_{0}$ is the depth of the potential valley. The relativistic equations
of motion of the particle in the plane $z=0$ have the form

$\frac{{d}p_{x}}{{d}t}=0,\quad\frac{{d}p_{y}}{{d}t}=-\frac{2eU_{0}}{a^{2}}y.$
(2)

Here $p_{x}$ and $p_{y}$ are the momentum components

$p_{i}=\gamma
m\dot{x}_{i},\,x_{i}=x,y,\,\gamma=(1-\beta^{2})^{1/2},\,\beta^{2}=\beta_{x}^{2}+\beta_{y}^{2},$

$m$ and $e$ are the mass and the charge of the particle,
$\beta_{i}=\dot{x}_{i}/c$, the dot denotes the time derivative, $c$ is the
speed of light.

Let the particle be ultrarelativistic ($\gamma\gg 1$) and the conditions of
the dipole approximation are fulfilled [18]

$\displaystyle\beta_{y}\ll\gamma^{-1},\quad\delta\beta_{x}\ll\gamma^{-2},$ (3)

where $\delta\beta_{x}$ is the amplitude of $\beta_{x}$ variation. In this
case the value of $\gamma$ can be considered as a constant. Integration of the
equations (2) in the adopted approximation gives

$\displaystyle x(t)=vt,\quad
y(t)=\frac{v_{0y}}{\omega_{0}}\sin{\omega_{0}t}+y_{0}\cos{\omega_{0}t},$ (4)

where $v$ and $v_{0y}$ are the initial particle velocity and its
$y$-component, $\omega_{0}$ is the frequency of the particle oscillations

$\omega_{0}^{2}=\frac{2U_{0}}{a^{2}m\gamma}.$ (5)

The condition of channeling is that the amplitude of oscillations
$y_{0}^{2}+(v_{0y}/\omega_{0})^{2}$ must be less than half of the distance
between the crystal planes. Hence, if a parallel beam of particles is incident
upon the surface of the crystal plate, only particles with initial coordinate
$y_{0}$ satisfying the condition

$y_{0}^{2}<y_{m}^{2}=a^{2}-\left(\frac{v_{0y}}{\omega_{0}}\right)^{2}.$ (6)

will be captured into the channeling mode. The remaining particles have above-
barrier transverse energy. It follows from the above inequality that the
initial transverse velocity $v_{0y}$ must satisfy condition
$|v_{0y}|<a\omega_{0}$. If we introduce the angle of incidence as
$\alpha=|v_{0y}|/c$, the last inequality can be written as a well known
Lindhard condition

$\alpha<\alpha_{c}=\sqrt{\frac{2U_{0}}{m\gamma c^{2}}}.$ (7)

The thickness of each crystal plate is equal to half of the trajectory period:
$d_{1}=\pi v/\omega_{0}$. Hence, the $y$-coordinate of the particle as it
leaves the first crystal plate is $y_{1}=-y_{0}$, and the $y$-component of the
velocity is equal to $v_{y1}=-v_{y0}$. Between the first and the second
crystal plates the particle moves along a stright line. If we require that all
the particles deflected by the first plate are captured into channeling by the
second plate, the particle coordinate $y$ at its incident on the second plate
must satisfy condition similar to (6). This means that the transversal shift
of the particle between the plates must make up a integer number of the
distance between the crystal planes. This imposes the following condition on
the distance $d_{2}$ between the plates

$d_{2}=2an\frac{v}{v_{0y}}=\frac{2an}{\alpha},\quad n=0,1,2\dots\,.$ (8)

Hereafter we suppose that this condition is fulfilled. Then the law of the
particle motion within the second crystal plate has the form

$\displaystyle x$ $\displaystyle=vt,$ (9) $\displaystyle y$
$\displaystyle=-\dfrac{v_{0y}}{\omega_{0}}\sin{\omega_{0}(t-t_{2})}-y_{0}\cos{\omega_{0}(t-t_{2})}-2an,$
(10)

where $t_{2}=(d_{1}+d_{2})/v$. If the angle of incidence $\alpha$ is zero,
then all the particles of the beam will be captured into channeling regardless
of the distance $d_{2}$.

The first two crystal plates and two subsequent gaps make up the spatial
period of the multi-crystal undulator. The time period of motion is equal to
$T=2(d_{1}+d_{2})/v$.

##  3 Radiation

Spectral and angular distribution of the energy radiated by the particle at
quasi-periodic motion along a plane trajectory is given by formula [18]

$\frac{d{\cal E}}{d\Omega
d\omega}=\frac{4e^{2}\gamma^{4}}{c\tilde{\omega}^{2}}\frac{\sin^{2}\pi\nu
N}{\sin^{2}\pi\nu}(\rho_{\sigma}+\rho_{\pi})|\dot{\boldsymbol{\beta}}(\nu)|^{2},$
(11)

where $N$ is the number of periods of the undulator, the angular distributions
of the polarization components $\rho_{\sigma}$ and $\rho_{\pi}$ are defined by
equations

$\displaystyle\rho_{\sigma}=\frac{(1-\psi^{2}\cos
2\varphi)^{2}}{(1+\psi^{2})^{4}}\>,\ \ \
\rho_{\pi}=\frac{\psi^{4}\sin^{2}2\varphi}{(1+\psi^{2})^{4}},$ (12)

and the Fourier component of acceleration is given by

$\displaystyle\dot{\boldsymbol{\beta}}(\nu)=\frac{1}{T}\int\limits_{0}^{T}\dot{\boldsymbol{\beta}}(t)e^{i\tilde{\omega}\nu
t}dt,\quad\nu=\frac{\omega}{2\gamma^{2}\tilde{\omega}}(1+\psi^{2}).$ (13)

Here $\tilde{\omega}=2\pi/T$ is the frequency of the particle oscillations,
$\psi=\gamma\theta$, $\theta$ is the angle between the direction of radiation
and the channeling axis ($x$-axis), $\varphi$ is the azimuthal angle in
$yz$-plane .

Calculating the acceleration by use of equations (4) and (10) and substituting
it in equations (11) and (13), we obtain

$\frac{d{\cal E}}{d\Omega
d\omega}=\frac{16e^{2}\gamma^{4}a^{2}\omega_{0}^{2}N^{2}}{\pi^{2}c^{3}}I_{1}(\nu)I_{2}(\nu)(\rho_{\sigma}+\rho_{\pi})\times\\\
\times\left[\left(\frac{y_{0}\nu\eta}{a}\right)^{2}+\phi^{2}\right],$ (14)

where $\phi=\alpha/\alpha_{c}$ is the ratio of the angle of incidence to the
critical angle of channeling, and $\eta=\tilde{\omega}/\omega_{0}<1$ is the
ratio of the frequency of particle oscillations to the frequency of
oscillation at channeling (similar to duty cycle in electronics). The spectral
functions

$\displaystyle I_{1}(\nu)=$
$\displaystyle\frac{\cos^{2}\pi\nu\eta/2}{(1-\eta^{2}\nu^{2})^{2}},$
$\displaystyle I_{2}(\nu)=$ $\displaystyle\frac{\sin^{2}\pi\nu
N}{4N^{2}\cos^{2}(\pi\nu/2)}$ (15)

define essential properties of radiation. In case $N\gg 1$ the function
$I_{2}(\nu)$ generates a narrow spectral lines of width $\Delta\nu=N^{-1}$ in
the neighborhood of odd values of $\nu$, while the function $I_{1}(\nu)$ is a
spectral envelope. Fig. 2 shows the function $I_{1}(\nu)$ (dashed line) and
the product $I_{1}(\nu)I_{2}(\nu)$ (solid line) for $N=10,\,\eta=0.5$.

Figure 2: The emission spectrum $I_{1}(\nu)I_{2}(\nu)$. $N=10,\,\eta=0.5$

Function $I_{1}(\nu)$ determines the width of the radiation spectrum (the
number of harmonics represented in the spectrum). It follows from definition
that function $I_{1}(\nu)$ has its first zero where $\nu=3/\eta$. If the
distance between the crystal plates is equal to zero, i.e. if the particle is
channeling in a solid crystal, then $\tilde{\omega}=\omega_{0}$. In this case
$\eta=1$ and only the first harmonic is presented in the emission spectrum.
The number of harmonics in the spectrum increases with the reduction of
$\eta$. The frequency of the first harmonic of radiation of ultra-relativistic
particles is determined by equation (13):

$\omega_{1}=\frac{2\tilde{\omega}\gamma^{2}}{(1+\psi^{2})}.$ (16)

Let us find the spectral density of radiation. We calculate it for crystal
undulator with large number of periods $N$. Using the limit [18]

$\lim\limits_{N\to\infty}\frac{\sin^{2}\pi\nu
N}{N\sin^{2}\pi\nu}=\sum\limits_{n=1}^{\infty}\delta(n-\nu)$ (17)

and integrating over the solid angle $d\Omega=\theta d\theta d\varphi$, we
obtain the integral spectrum of radiation

$\frac{d{\cal E}}{d\omega}=\frac{4e^{2}a^{2}\omega_{0}^{2}\gamma^{2}\xi N}{\pi
c^{3}}\times\\\ \times\sum\limits_{n=1}^{\infty}\frac{s_{n}\cos^{2}\pi
n\eta/2}{n^{4}(1-n^{2}\eta^{2})^{2}}\left[1-(-1)^{n}\right]G_{n}\Theta(n-\xi),$
(18)

where $s_{n}=s_{n\sigma}+s_{n\pi}$ is the sum of the components of linear
polarization

$s_{n\sigma}=3\xi^{2}-2\xi n+n^{2},\quad s_{n\pi}=(\xi-n)^{2},$

$\xi$ is a reduced frequency, and $G_{n}$ defines the influence of initial
coordinate and the incidence angle on the spectrum:

$\xi=\frac{\omega}{2\gamma^{2}\eta{\omega_{0}}},\quad
G_{n}=\left(\frac{y_{0}n\eta}{a}\right)^{2}+\phi^{2},$

$\Theta(n-\xi)$ is the Heaviside step function.

The emission spectrum (18) is represented as a sum of discrete harmonics as
shown in Fig, 3. The shape of each harmonic defined by the polarization
components $s_{n\sigma}$ and $s_{n\pi}$ coincides with the well-known profile
of undulator radiation in the dipole approximation [18, 19]. Distribution of
the radiation energy in harmonics, as can be seen from the form factor
$G_{n}$, essentially depends on the angle of incidence of the particle in the
crystal plate, its initial coordinate $y_{0}$ and the distance between the
plates.

Figure 3: Spectrum of radiation of parallel beam of particles. $\eta=0.5$,
$\alpha=0$

So far we investigated the properties of radiation of a single particle. In
order to find the spectral density emitted by a parallel beam of particles we
average the expression (18) over the initial coordinate $y_{0}$ through the
interval $2y_{m}$ given by equation (6). Since $y_{0}$ is represented only in
$G_{n}$, it will suffice to average only this multiple

${\overline{G}}_{n}=\frac{1}{2a}\int\limits_{-y_{m}}^{y_{m}}G_{n}dy_{0}=(1-\phi^{2})^{1/2}\left[\frac{n^{2}\eta^{2}}{3}(1-\phi^{2})+\phi^{2}\right]$
(19)

and use $\overline{G}_{n}$ instead of $G_{n}$ in equation (18). The spectrum
of radiation of parallel beam of particles for parameters $\eta=0.5$ and
$\alpha=0$ is plotted in Fig, 3. For the purpose of comparison with the
possible experimental results the same graph is shown in Fig. 4 as the number
of photons $dn$ per frequency interval $d\omega$. The spectrum consists in
this case mainly of the first and third harmonics.

Figure 4: Spectrum of radiation of parallel beam of particles for parameters
$\eta=0.5$ and $\alpha=0$

The number of harmonics, making the greatest contribution to the emission
spectrum is determined by the factor $\cos^{2}(\pi
n\eta/2)/(1-n^{2}\eta^{2})^{2}$, which is close to unity in the domain of low
$n\eta$ and decreases rapidly if $n\eta\gtrsim 3$. Thus, the main part of the
spectrum contains about $n\sim 3/\eta$ harmonics. Furthermore, the emission
spectrum depends on the angle of incidence of the particle beam on the
crystal. To explore this issue in more detail, let us find the energy of the
radiation at each harmonic by integration of the expression (18) over $\xi$
within a single harmonic. Taking into account the replacement of
$G_{n}\to{\overline{G}}_{n}$ we obtain the following

$\displaystyle{\cal E}=\sum\limits_{n=1}^{\infty}({\cal E}_{n\sigma}+{\cal
E}_{n\pi}),\quad{\cal E}_{n\sigma}=\frac{7}{8}{\cal E}_{n},\quad{\cal
E}_{n\pi}=\frac{1}{8}{\cal E}_{n},$ $\displaystyle{\cal
E}_{n}=\frac{16e^{2}a^{2}\omega_{0}^{3}\gamma^{4}N\eta}{3\pi c^{3}}S(n\eta).$
(20)

The distribution of emitted energy over the harmonics is given by the function

$S(n\eta)=\frac{\cos^{2}\pi
n\eta/2}{(1-n^{2}\eta^{2})^{2}}\left[1-(-1)^{n}\right]{\overline{G}}_{n}.$
(21)

This is a discrete function of odd integer values of $n$. The envelopes of
this function for different angles of incidence $\phi$ are shown in Fig.5

Figure 5: The envelopes of spectral harmonics for different angles of
incidence $\phi$. The harmonics for undulator with $\eta=0.4$ and the angle of
incidence $\phi=0.95$ are shown for example

The dependence of the emission spectrum of the relative angle $\phi$ is
presented only in the factor $\overline{G}_{n}(\phi)$. If $\phi=0$, this
factor takes the value $\overline{G}_{n}(0)=n^{2}\eta^{2}/3$. As the angle of
incidence increases, the value of $\overline{G}_{n}(\phi)$ increases in the
low-frequency part of the spectrum ($n^{2}\eta^{2}<2$) and reaches its maximum
at $\phi=\phi_{m}$

$\overline{G}_{n}(\phi_{m})=\frac{2}{3\sqrt{3-n^{2}\eta^{2}}},\quad\phi_{m}=\sqrt{\frac{2-n^{2}\eta^{2}}{3-n^{2}\eta^{2}}}.$
(22)

On further increasing of $\phi$ this function decreases and vanishes at
$\phi=1$. In the high-frequency part of the spectrum ($n^{2}\eta^{2}\geq 2$)
the function $\overline{G}_{n}(\phi)$ decreases monotonically to zero with
increasing of the angle of incidence.

Summing up equation (20) over the harmonics we obtain the total energy emitted
by the beam of particles, per particle

${\cal E}=\frac{2\pi
e^{2}a^{2}\omega_{0}^{3}\gamma^{4}N}{9c^{3}}\sqrt{1-\phi^{2}}\left(1+2\phi^{2}\right),$
(23)

which does not depend on $\eta$ and is distributed between the polarization
components according to the ratio of 1:7. As a function of the angle of
incidence, the emitted energy is a maximum at $\phi=1/\sqrt{2}$, i.e., when
the angle of incidence is $\sqrt{2}$ times less than the critical angle of
channeling.

##  4 Discussion

If a source of radiation with a frequency less than the frequency of radiation
at channeling is needed, the proposed method has certain advantages compared
with radiation at channeling emitted by less energetic particles. Indeed, if
you want to reduce the frequency of radiation of a channeling particles $k$
times, you have to use particles with energy $k^{2/3}$ times less. This will
cause the cone of radiation to increase $k^{2/3}$ times, and spectral-angular
density (14), which is proportional to $\gamma^{4}$, to decrease $k^{8/3}$
times. In the case of multi-crystal undulator the same decrease in frequency
can be achieved if the distance between the crystal plates is equal to
$d_{2}=(k-1)d_{1}$. Therewith the cone of radiation is not changed, and the
spectral-angular density of radiation can both increase or decrease dependent
on the angle of beam incidence.

There is an interesting peculiarity of the spectral-angular density of
radiation. If the distance between the crystal plates increases, then the
frequency of radiation on each harmonic decreases, and the number of harmonics
represented in the main part of the spectrum ($n\eta\lesssim 3$) increases in
the same ratio. According to equation (23) the total emitted energy does not
depend on $\eta$. Hence the energy emitted at each harmonic decreases as it is
indicated by factor $\eta$ in equation (20). However, the spectral-angular
density and the spectral density does not necessary decrease with $\eta$,
because there is not an $\eta$-factor in equations (14) and (18). This is also
seen from Fig. 3: as $\eta$ decreases, the scale-factor of $\xi$-axes changes
and the cut-off edge of each harmonic moves to the left in terms of $\omega$.
This definitely leads to decrease of the area under the curve representing the
spectrum. But the spectral density itself changes slowly according to the
terms of the sum in equation (18). If $\phi>2/\sqrt{3\pi^{2}-20}\approx
0.645$, then with the decrease of $\eta$ the spectral-angular and the spectral
density of radiation at each harmonic is growing, despite the emergence of
additional harmonics.

We have considered an idealized model in order to investigate the general
properties of radiation. For example, the averaged potential in the vicinity
of the crystal plane is not harmonic. As a result, the particles channeling
with high amplitude of oscillations will have a smaller spatial period and
will leave the ‘half-wave’ crystal plate at another angle. In other words, the
plate being half-wave for some particles will not be such for others. This
will cause slight scattering of the originally parallel beam of particles.
Indeed, modeling of the trajectories in a more realistic potential shows that
after the passage of a half-wave plate two side maxima in the angular
distribution of the beam appear[12]. This and some other factors, such as a
finite number of undulator periods, the energy spread of the particles in the
beam, errors in manufacturing, will lead to broadening of the spectral lines
and to blurring of the boundaries between the harmonics shown in Fig. 3. At
the same time, these factors do not change significantly the distribution of
radiation energy over the harmonics.

## Acknowledgments

This work was supported by a grant of the Ministry of Education and Science of
the Russian Federation under project no. 3.867.2014/K

## References

  * [1] V. Kaplin, S. V. Plotnikov, S. A. Vorobiev, Radiation by charged particles channeled in deformed crystals, Zh. Tekh. Fiz. 50 (1980) 1079.
  * [2] V. Baryshevsky, I. Dubovskaya, A. Grubich, Generation of $\gamma$-quanta by channeled particles in the presence of a variable external field, Phys. Let. A 77 (1) (1980) 61–64.  doi:10.1016/0375-9601(80)90637-4.
  * [3] V. Baryshevsky, V. Tikhomirov, Crystal undulators: from the prediction to the mature simulations, Nucl. Instrum. Methods Phys. Res., Sect. B 309 (2013) 30–36.  doi:10.1016/j.nimb.2013.03.013.
  * [4] E. Bagli, L. Bandiera, V. Bellucci, A. Berra, R. Camattari, D. D. Salvador, G. Germogli, V. Guidi, L. Lanzoni, D. Lietti, A. Mazzolari, M. Prest, V. Tikhomirov, E. Vallazza, Experimental evidence of planar channeling in a periodically bent crystal, Eur. Phys. J. C 14 (10) (2014) 1–7.  doi:10.1140/epjc/s10052-014-3114-x.
  * [5] S. Bellucci, S. Bini, V. M. Biryukov, Y. A. Chesnokov, S. Dabagov, G. Giannini, V. Guidi, Y. M. Ivanov, V. I. Kotov, V. A. Maisheev, C. Malagù, G. Martinelli, A. A. Petrunin, V. V. Skorobogatov, M. Stefancich, D. Vincenzi, Experimental study for the feasibility of a crystalline undulator., Phys. Rev. Lett. 90 (3) (2003) 034801.  doi:10.1103/PhysRevLett.90.034801.
  * [6] V. M. Biryukov, A. G. Afonin, V. T. Baranov, S. Baricordi, S. Bellucci, G. I. Britvich, V. N. Chepegin, Y. A. Chesnokov, C. Balasubramanian, G. Giannini, V. Guidi, Y. M. Ivanov, V. I. Kotov, A. Kushnirenko, V. A. Maisheev, C. Malagu, G. Martinelli, E. Milan, A. A. Petrunin, V. A. Pikalov, V. V. Skorobogatov, M. Stefancich, V. I. Terekhov, F. Tombolini, U. I. Uggerhoj, Accelerator Tests of Crystal Undulators, arXiv: physics/0412159 [physics.acc-ph]arXiv:0412159.
  * [7] A. Kostyuk, Crystalline undulator with a small amplitude and a short period., Phys. Rev. Lett. 110 (11) (2013) 115503.  doi:10.1103/PhysRevLett.110.115503.
  * [8] G. B. Sushko, A. V. Korol, A. V. Solov’yov, A small-amplitude crystalline undulator based on 20 gev electrons and positrons: Simulations, St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (3) (2015) 341 – 345.  doi:http://dx.doi.org/10.1016/j.spjpm.2015.09.002.
  * [9] T. N. Wistisen, K. K. Andersen, S. Yilmaz, R. Mikkelsen, J. L. Hansen, U. I. Uggerhøj, W. Lauth, H. Backe, Experimental realization of a new type of crystalline undulator., Phys. Rev. Lett. 112 (25) (2014) 254801.  doi:10.1103/PhysRevLett.112.254801.
  * [10] U. Uggerhøj, T. Wistisen, Intense and energetic radiation from crystalline undulators, Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. with Mater. Atoms 355 (2015) 35–38.  doi:10.1016/j.nimb.2015.02.069.
  * [11] S. Vorobiev, V. Kaplin, E. Rozum, Patent SU 876044 A1 (priority date 07.10.1982).
  * [12] Y. L. Pivovarov, T. A. Tukhfatullin, Flux dynamics and angular distributions of relativistic electrons and positrons passing through a half-wave crystals, Journal of Physics: Conference Series 517 (1) (2014) 012028\.
  * [13] Y. Takabayashi, V. Bagrov, O. Bogdanov, Y. Pivovarov, T. Tukhfatullin, Angular distributions of relativistic electrons under channeling in half-wavelength crystal and corresponding radiation, Nucl. Instruments Methods Phys. Res. Sect. B Beam Interact. with Mater. Atoms 355 (2015) 188–192.  doi:10.1016/j.nimb.2015.02.007.
  * [14] V. G. Bagrov, N. I. Fedosov, I. M. Ternov, Radiation of relativistic electrons moving in an arc of a circle, Phys. Rev. D 28 (1983) 2464–2472.  doi:10.1103/PhysRevD.28.2464.
  * [15] R. Polozkov, V. Ivanov, G. Sushko, A. Korol, A. Solov’yov, Channeling of ultra-relativistic positrons in bent diamond crystals, St. Petersburg Polytechnical University Journal: Physics and Mathematics 1 (2) (2015) 212 – 218\.  doi:http://dx.doi.org/10.1016/j.spjpm.2015.06.004.
  * [16] V. Kaplin, S. Uglov, V. Zabaev, V. Kanaev, S. Litvin, N. Timchenko, Generation of x-rays by 850 mev electrons in a novel periodic multicrystal structure on a gaas plate surface, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 448 (1–2) (2000) 66 – 69.  doi:http://dx.doi.org/10.1016/S0168-9002(99)00742-1.
  * [17] V. Kaplin, S. Kuznetsov, N. Timchenko, S. Uglov, V. Zabaev, X-ray production by 500 mev electron beam in a periodically structured monocrystalline target of gaas, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 173 (1–2) (2001) 238 – 240, radiation from Relativistic Electrons in Periodic Structures.  doi:http://dx.doi.org/10.1016/S0168-583X(00)00186-5.
  * [18] V. Bordovitsyn (Ed.), Synchrotron Radiation Theory and its Development, World Scientific, 1999.
  * [19] A. Hofmann, The Physics of Synchrotron Radiation, Cambridge Univ. Press, 2004.