A note on Fourier coefficients of Hecke eigenforms in short intervals

Abstract

In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length $\frac{x}{(\log x)^A}$ for any $A>0$, modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length $x^{1/2 + \epsilon }$ for any $\epsilon >0$.

1 Introduction and statements of results

Let x, y be real numbers, p, q be prime numbers, $N \ge 1$ be an integer and f be a non-CM normalized cuspidal Hecke eigenform of weight $k \ge 2$ for $\Gamma _0(N)$ with integer Fourier coefficients $a_f(m)$ for $m \ge 1$. In this article, we investigate large prime factors of Fourier coefficients of f in short intervals. We note that even the existence of a prime p in short intervals with $a_f(p) \ne 0$ in itself a difficult question. It follows from a recent work of Lemke Oliver and Thorner [16, Theorem 1.6] that there exists an absolute constant $\delta >0$ and a prime $p \in (x, x + y]$ such that $a_f(p) \ne 0$ when $y \ge x^{1- \delta }$.

In this work, we find prime factors of size at least $(\log x)^{1/8}$ in intervals of length $\frac{x}{{(\log x})^A}$ for any positive A. This begs the question about the possible/expected order of such prime factors in such short or even shorter intervals of size/length, say, a small power of x. We show that under the generalized Riemann hypothesis for all symmetric power L-functions of f and all Artin L-series, one can find prime factors of size at least $x^{\epsilon /7}$ in intervals of length $x^{1/2 + \epsilon }$ for any $\epsilon < 1/10$.

In an earlier work [4], the present authors along with Bilu investigated lower bounds for the largest prime factor of $a_f(p)$. However finding such large prime factors in short intervals is a different ball game. We need to establish a explicit version of a result of Balog-Ono [2]. Further, for the conditional result on the generalized Riemann hypothesis (as specified above), we need to strengthen a conditional result of Rouse-Thorner [20] (see also Thorner [25]) in short intervals.

Before proceeding further, let us fix a notation. For any integer n, let P(n) denote the largest prime factor of n with the convention that $P(0)= P(\pm 1) = 1$. Throughout the article, by GRH, we mean the generalized Riemann hypothesis for all symmetric power L-functions of f and all Artin L-series, unless otherwise specified. In this set up, we prove the following results.

Theorem 1

Let f be a non-CM normalized cuspidal Hecke eigenform of weight k for $\Gamma _0(N)$ having integer Fourier coefficients $a_f(m)$ for $m \ge 1$. For positive real numbers $A, \epsilon$ and natural numbers $n \ge 1$, there exists a prime $p \in (x, ~x + \frac{x}{(\log x)^A}]$ such that

$$\begin{aligned} P(a_f(p^{n})) ~>~ (\log x^n)^{1/8} (\log \log x^n)^{3/8 -\epsilon } \end{aligned}$$

for all sufficiently large x depending on $A, \epsilon , n$ and f.

Remark 1.1

The lower bound in Theorem 1 can be replaced by $(\log x^n)^{1/8}(\log \log x^n)^{3/8}u(x^n)$ for any real valued non-negative function u with $u(x) \rightarrow 0$ as $x \rightarrow \infty$.

Theorem 2

Suppose that GRH is true, f is as in Theorem 1 and $\epsilon \in (0, \frac{1}{10})$. For any natural number $n >1$, there exists a positive real constant c (depending on $\epsilon , n, f$), a positive constant b (depending on n) and a prime number $p \in (x, x+ x^{\frac{1}{2}+\epsilon }]$ such that

$$\begin{aligned} P\left( a_f(p^{n})\right) ~>~ c x^{\epsilon b} \end{aligned}$$

for all sufficiently large x depending on $\epsilon , n, f$. When $n=1$, there exists a positive real constant c (depending on $\epsilon , f$) and a prime number $p \in (x, x+ x^{\frac{1}{2}+\epsilon }]$

$$\begin{aligned} P\left( a_f(p) \right) ~>~ c x^{\epsilon /7} (\log x)^{2/7} \end{aligned}$$

for all sufficiently large x depending on $\epsilon , f$.

If we are allowed to go up to a little longer than $x^{\frac{3}{4}}$, then GRH ensures even larger prime factors. More precisely, we have the following:

Theorem 3

Suppose that GRH is true and let $\eta (x) = x^{3/4} \log x \cdot \log \log x$. Then for all sufficiently large x (depending on n and f), there exists a prime $p \in (x, x + \eta (x) ]$ such that

$$\begin{aligned} P(a_f(p^{n})) ~>~ c x^{1/28} (\log x)^{3/7} (\log \log x)^{1/7} \end{aligned}$$

for some positive real number c depending on f.

Remark 1.2

Suitable modifications of the proofs of Theorem 1, Theorem 2 and Theorem 3 will show that these theorems are true for a set of primes of positive density. More precisely, it follows that the number of primes $p \in (x, ~x + \frac{x}{(\log x)^A}]$ for which Theorem 1 is true is at least $\frac{a_1x}{(\log x)^{A+1}}$ for some positive constant $a_1$ and for all sufficiently large x. If $\epsilon > 0$ is sufficiently small, then the number of primes $p \in (x, x+ x^{\frac{1}{2}+\epsilon }]$ for which Theorem 2 is true is at least $a_2\frac{x^{1/2+ \epsilon }}{\log x}$ for some $a_2>0$ and for all sufficiently large x. Further, the number of primes $p \in (x, x + \eta (x) ]$ for which Theorem 3 is true is at least $\frac{a_3\eta (x)}{\log x}$ for some positive constant $a_3$ and for all sufficiently large x.

2 Preliminaries

2.1 Distribution of zeros of Dedekind zeta functions

Let L/K be an abelian extension of number fields with Galois group G. Then we have

$$\begin{aligned} \zeta _L(s) ~=~ \prod _{\chi } L(s, \chi , L/K), \end{aligned}$$

where $\chi$ runs over the irreducible characters of G (see [15, Ch. XII], [19, VII] for more details). Let $\mathfrak {f}_\chi$ denote the conductor of $\chi$ and set

$$\begin{aligned} \mathcal {Q} ~=~ \mathcal {Q}(L/K)~=~ \max _{\chi } N_K\left( \mathfrak {f}_\chi \right) , \end{aligned}$$

where $N_K$ denotes the absolute norm on K. Also let

$$\begin{aligned} Q ~=~ Q(L/K) ~=~ D_K \mathcal {Q} n_K^{n_K}, \end{aligned}$$

where $D_K$ is the absolute discriminant of K and $n_K = [K:\mathbb {Q}]$. We write $s \in \mathbb {C}$ as $s= \sigma +it$, where $\sigma = \Re (s)$ and $t= \Im (s)$. A zero-free region of $\zeta _L(s)$ is given by the following theorem ( [24, Theorem 3.1], see also [26, Theorem 1.9]).

Theorem 4

There exists an absolute positive constant $c_1$ such that the Dedekind zeta function $\zeta _L(s)$ has atmost one zero in the region

$$\begin{aligned} \sigma ~>~ 1- \frac{c_1}{\log \left( Q (|t|+3)^{n_K}\right) }. \end{aligned}$$

Suppose such a zero $\beta _1$ exists, then it is real, simple and is a zero of the L-function corresponding to a real Hecke character $\chi _1$ of G.

Remark 2.1

The above exceptional zero $\beta _1$ (if it exists) is usually known as Landau-Siegel zero.

For $0 \le \sigma \le 1$ and $T \ge 1$, let

$$\begin{aligned} N(\sigma , T, \chi ) ~=~ \#\{ \rho = \beta +i \gamma ~:~ L(\rho , \chi , L/K)= 0, ~ \sigma< \beta< 1 \text { and } -T<\gamma < T\}, \end{aligned}$$

where the zeros $\rho$ are counted with multiplicity. Set

$$\begin{aligned} N(\sigma , T) ~=~ \sum _{\chi } N(\sigma , T, \chi ), \end{aligned}$$

where $\chi$ runs over the irreducible characters of G. In this set up, we have the following theorem (see [24, Theorem 3.2], [26, Theorem 4.3]).

Theorem 5

There exists an absolute constant $c_2 \ge 1$ such that

$$\begin{aligned} N(\sigma , T) ~\ll ~ B_1 \left( QT^{n_K}\right) ^{c_2 (1-\sigma )} \end{aligned}$$

uniformly for any $0< \sigma < 1$ and $T \ge 1$. Here

$$\begin{aligned} B_1 ~=~ B_1(T) ~=~ \min \{1, (1-\beta _1) \log (QT^{n_K})\}. \end{aligned}$$

2.2 Chebotarev density theorem in short intervals

Let L/K be a Galois extension of number fields with Galois group G. Let $n_L= [L: \mathbb {Q}]$ and $n_K = [K:\mathbb {Q}]$. Also let $D_L$ (resp. $D_K$) denote the absolute discriminant of L (resp. K). For a conjugacy class $C \subseteq G$, define

$$\begin{aligned} \pi _C(x, L/K) ~=~ \#\{\mathfrak {p} \subseteq \mathcal {O}_K ~:~ N_K(\mathfrak {p}) \le x,~ \mathfrak {p} \ \text {is unramified in } \ L \ \text { and } \ [\sigma _\mathfrak {p}] = C\}, \end{aligned}$$

where $\sigma _\mathfrak {p}$ is a Frobenius element of $\mathfrak {p}$ in G and $[\sigma _\mathfrak {p}]$ denotes the conjugacy class of $\sigma _\mathfrak {p}$ in G. In [2], Balog and Ono proved the following theorem.

Theorem 6

Let $\epsilon >0$ be a real number and $x^{1 -1/c(L) + \epsilon } \le y \le x$, then we have

$$\begin{aligned} (1-\epsilon ) \frac{\#C}{\#G} \frac{y}{\log x} ~<~ \pi _C\left( x+y, L/K\right) - \pi _C\left( x, L/K\right) ~<~ (1+\epsilon ) \frac{\#C}{\#G} \frac{y}{\log x} \end{aligned}$$

for all sufficiently large x depending on $\epsilon$ and L. Here

$$\begin{aligned} c(L) ~=~ {\left\{ \begin{array}{ll} &{} n_L \text {if } n_L \ge 3, \\ &{} \frac{8}{3} \text {if } n_L = 2, \\ &{} \frac{12}{5} \text {if } n_L = 1. \end{array}\right. } \end{aligned}$$

For our application, we need a version of Theorem 6 which is uniform in L. In Sect. 3, we prove the following explicit version of the Chebotarev density theorem in short intervals.

Theorem 7

There exists a positive absolute constant $c_3$ such that if $y \ge x^{1-c_3/n_L}$ and $\log x \gg _{c_3} \log \left( D_L n_L^{n_L})\right)$, then we have

$$\begin{aligned} \bigg | \pi _C\left( x+y, L/K\right) - \pi _C\left( x, L/K\right) ~-~ \frac{\#C}{\#G} \left( \frac{y}{\log x} - \theta _1 \frac{(x+y)^{\beta _1} - x^{\beta _1}}{\beta _1 \log x}\right) \bigg | ~\le ~ \frac{1}{4} \frac{\#C}{\#G} \frac{y}{\log x}. \end{aligned}$$

Here $\theta _1 \in \{-1, 1\}$ if the Landau-Siegel zero of the Dedekind zeta function $\zeta _L(s)$ exists and $\theta _1 = 0$ otherwise.

Remark 2.2

The constant $\frac{1}{4}$ in Theorem 7 can be replaced with any small positive real number (see Section 3.1).

2.3 Hecke eigenforms and $\ell$-adic Galois representation

Let f be as in section 1 and m be a positive integer. For any integer $d > 1$ and real number $x >0$, let

$$\begin{aligned} \begin{aligned}&\pi _{f,m}(x,d) ~=~ \#\{p \le x: a_f(p^m) \equiv 0 ~({\textrm{mod}}\; d) \}. \end{aligned} \end{aligned}$$

Let $\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})$ be the Galois group of ${\overline{\mathbb {Q}}}/\mathbb {Q}$ and for a prime $\ell$, let $\mathbb {Z}_\ell$ denote the ring of $\ell$-adic integers. By the work of Deligne [9], there exists a continuous representation

$$\begin{aligned} \rho _{d} ~:~ {\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})} ~\rightarrow ~ {\textrm{GL}}_2\left( \prod _{\ell | d} \mathbb {Z}_\ell \right) \end{aligned}$$

which is unramified outside the primes dividing dN. Further, if $p \not \mid dN$, then we have

$$\begin{aligned} \textrm{tr}\rho _{d}(\sigma _p) ~=~ a_f(p) {\text {and}} {\text {det}}\rho _{d}(\sigma _p) ~=~ p^{k-1}, \end{aligned}$$

where $\sigma _p$ is a Frobenius element of p in $\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})$. Here $\mathbb {Z}$ is embedded diagonally in $\prod _{\ell | d} \mathbb {Z}_\ell$. Let ${\tilde{\rho }}_{d}$ denote the reduction of $\rho _{d}$ modulo d:

$$\begin{aligned} {\tilde{\rho }}_{d} ~:~ {\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})} ~\xrightarrow []{\rho _{d}}~ {\textrm{GL}}_2\left( \prod _{\ell | d} \mathbb {Z}_\ell \right) ~\twoheadrightarrow ~ {\textrm{GL}}_2(\mathbb {Z}/ d\mathbb {Z}). \end{aligned}$$

Also denote by ${\tilde{\rho }}_{d,m}$, the composition of ${\tilde{\rho }}_{d}$ with $Sym^m$, where $Sym^m$ denotes the symmetric m-th power map:

$$\begin{aligned} {\tilde{\rho }}_{d,m} ~:~ {\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})} ~\xrightarrow []{\rho _{d}}~ {\textrm{GL}}_2\left( \prod _{\ell | d} \mathbb {Z}_\ell \right) ~\rightarrow {}~ {\textrm{GL}}_2(\mathbb {Z}/ d\mathbb {Z}) ~\xrightarrow []{Sym^m}~ {\textrm{GL}}_{m+1}(\mathbb {Z}/ d\mathbb {Z}). \end{aligned}$$

For $p \not \mid dN$, we have

$$\begin{aligned} \textrm{tr}{\tilde{\rho }}_{d,m}(\sigma _{p}) ~=~ a_f(p^m)~ (\textrm{mod }\,\,d). \end{aligned}$$

Let $H_{d,m}$ be the kernel of ${\tilde{\rho }}_{d,m}$, $K_{d,m}$ be the subfield of ${\overline{\mathbb {Q}}}$ fixed by $H_{d,m}$ and

Suppose that $C_{d,m}$ is the subset of ${\tilde{\rho }}_{d,m}(\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q}))$ consisting of elements of trace zero. Let us set $\delta _{m}(d) = \frac{|C_{d,m}|}{|{\textrm{G}}_{d,m}|}$. For any prime $p\not \mid dN$, the condition $a_f(p^m) \equiv 0~ (\textrm{mod }\,\,d)$ is equivalent to the fact that ${\tilde{\rho }}_{d,m}(\sigma _p) \in C_{d,m}$, where $\sigma _{p}$ is a Frobenius element of p in $\textrm{Gal}({\overline{\mathbb {Q}}}/\mathbb {Q})$. Hence by the Chebotarev density theorem applied to $K_{d,m} / \mathbb {Q}$, we have

$$\begin{aligned} \lim _{x \rightarrow \infty } \frac{\pi _{f,m}(x,d)}{\pi (x)} ~=~ \frac{|C_{d,m}|}{|{\textrm{G}}_{d,m}|} ~=~ \delta _{m}(d). \end{aligned}$$

Applying Theorem 7, we can now deduce the following result.

Theorem 8

Let f be a non-CM normalized cuspidal Hecke eigenform of weight k and level N with integer Fourier coefficients $a_f(n)$ for $n \ge 1$. Then there exists a positive absolute constant $c_3$ such that if $y \ge x^{1- \frac{c_3}{d^4}}$ and $\log x \gg _{c_3} d^4 \log (dN)$, then

$$\begin{aligned} \pi _{f, m}\left( x+y, d\right) - \pi _{f, m}\left( x, d\right) ~\ll ~ \delta _m(d) \frac{y}{\log x}. \end{aligned}$$

When $m=1$, we have the following result (see [10, Proof of Theorem 3], [18, Lemma 5.4], [21, Section 4]) [5, 17, 23].

Lemma 9

For any prime $\ell$, we have

$$\begin{aligned} \delta (\ell ) ~=~ \frac{1}{\ell } +O\left( \frac{1}{\ell ^2}\right) \text {and} \delta (\ell ^n) ~=~O\left( \frac{1}{\ell ^n}\right) \end{aligned}$$

for any $n \in {\mathbb {N}}$. Here $\delta (\ell ) = \delta _1(\ell )$.

When $m+1$ is an odd prime q, the present authors in an earlier work (see [11, Lemma 17, Lemma 18]) proved the following results.

Lemma 10

Let $q, \ell$ be primes with q odd. Then $\delta _{q-1}(\ell ) = 0$ unless $\ell \equiv 0, \pm 1 ~(\textrm{mod }\,\,q)$ and

$$\begin{aligned} \delta _{q-1}(\ell ) ~\ll ~ \frac{q}{\ell }, \end{aligned}$$

where the implied constant depends only on f. Also we have

$$\begin{aligned} \delta _{q-1}(\ell ) ~=~ {\left\{ \begin{array}{ll} &{} \frac{q-1}{2} \frac{1}{\ell -1}, \textrm{if} \ell \equiv 1~ (\textrm{ mod }\,\, q) \\ &{} \frac{q-1}{2} \frac{1}{\ell +1}, \textrm{if} \ell \equiv -1~ (\textrm{ mod }\,\, q) \\ &{} \frac{q}{q^2 -1}, \textrm{if} \ell = q \end{array}\right. } \end{aligned}$$

for all sufficiently large $\ell$.

Lemma 11

For any integer $n \ge 2$ and primes $\ell , q$ with q odd, we have

$$\begin{aligned} \delta _{q-1}(\ell ^n) ~\ll ~ \frac{1}{\ell ^{n-1}} \delta _{q-1}(\ell ), \end{aligned}$$

where the implied constant depends only on f. We also have

$$\begin{aligned} \delta _{q-1}(\ell ^n) ~=~ \frac{1}{\ell ^{n-1}} ~\delta _{q-1}(\ell ) \end{aligned}$$

if $\ell \ne q$ and $\ell$ is sufficiently large. Further $\delta _{q-1}(q^n) = 0$ for $q \ge 5$.

Conditionally under GRH, i.e. assuming the generalized Riemann hypothesis for all Artin L-series, we can deduce the following theorem by applying a result of Lagarias and Odlyzko [14, Theorem 1.1] (see also [18, Lemma 5.3]).

Theorem 12

Suppose that GRH is true and f is a non-CM form. Then we have

$$\begin{aligned} \pi _{f, m}(x, d) ~=~ \delta _m(d) \left( \pi (x) ~+~ O\left( x^{1/2} d^4 \log (dNx)\right) \right) ~+~ O\left( d^4 \log (dN)\right) . \end{aligned}$$

2.4 Sato-Tate conjecture in short intervals

Let f be as before and

$$\begin{aligned} \lambda _f(p) ~=~ \frac{ a_f(p)}{2p^{(k-1)/2}}. \end{aligned}$$

The Sato-Tate conjecture states that the numbers $\lambda _f(p)$ are equidistributed in the interval ${[-1, 1]}$ with respect to the Sato-Tate measure

$$\begin{aligned} d\mu _{ST} = (2/\pi )\sqrt{1- t^2}~dt. \end{aligned}$$

This means that for any $-1 \le a \le b \le 1$, the density of the set of primes p satisfying ${\lambda _f(p) \in [a, b]}$ is

$$\begin{aligned} \frac{2}{\pi }\int _a^b\sqrt{1-t^2}~ dt. \end{aligned}$$

It is now a theorem due to the works of Barnet-Lamb, Clozel, Geraghty, Harris, Shepherd-Barron and Taylor ( [3, Theorem B], [7, 13]).

We will need Sato-Tate conjecture in short intervals due to Lemke Oliver and Thorner. For this, we need to introduce Chebyshev polynomials. The Chebyshev polynomials of second kind are defined by

$$\begin{aligned} U_0(x) ~=~ 1,~~ U_1(x) ~=~ 2x \textrm{and} U_n(x) ~=~ 2x U_{n-1}(x) - U_{n-2}(x) ~ \textrm{ for }\,\,n \ge 2. \end{aligned}$$

The generating function of $U_n$ is given by

$$\begin{aligned} \sum _{n=0}^{\infty } U_n(x) t^n ~=~ \frac{1}{1-2tx+t^2}. \end{aligned}$$

Note that if $p \not \mid N$, then $U_n(\lambda _f(p))$ is the Dirichlet coefficient of $L \left( s, \textrm{Sym}^n \pi _f \right)$ at p, where $\pi _f$ denotes the cuspidal representation of $\textrm{GL}_2\left( {\mathbb {A}}_\mathbb {Q}\right)$ attached to f. Let M be a natural number. A subset $I \subseteq [-1, 1]$ is said to be $\textrm{Sym}^M$-minorized if there exist constants $b_0, b_1, \cdots , b_M \in {\mathbb {R}}$ with $b_0 > 0$ such that

$$\begin{aligned} \mathbb {1}_{I}(t) ~\ge ~ \sum _{n=0}^{M} b_n U_n(t) ~~\mathrm { for~all~} t \in [-1,1]. \end{aligned}$$

Here $\mathbb {1}_I$ denotes the indicator function of I.

Remark 2.3

Let $B_0 = \frac{1+\sqrt{7}}{6} = 0.6076 \cdots$ and $B_1 =\frac{-1+\sqrt{7}}{6} = 0.2742 \cdots$. Then the interval $[-1, b]$ can be $\textrm{Sym}^4$-minorized if $b > -B_0$ and [a, 1] can be $\textrm{Sym}^4$-minorized if $a \in [B_1, B_0)$. It can be shown that the interval $I= [-1, -0.1]$ is $\textrm{Sym}^4$-minorized with $b_0 > 0.08$ (see [16, Lemma A.1]). Further, any interval $I \subseteq [-1, 1]$ can be $\textrm{Sym}^M$-minorized if M is sufficiently large (see [16, Page 6997, Remark 1]).

In this context, Lemke Oliver and Thorner proved the following version of the Sato-Tate conjecture in short intervals (see [16, Thorem 1.6]).

Theorem 13

Let f be a non-CM normalized Hecke eigenform of weight k and level N. Also let $I \subseteq [-1, 1]$ be a subset which can be $\textrm{Sym}^M$-minorized. Then there exists a constant $c_4 \in (0, 1)$ depending on I and N such that if $y \ge x^{1-c_4}$, then

$$\begin{aligned} \sum _{\begin{array}{c} x< p < x+y \\ p \not \mid N \end{array}} \mathbb {1}_I\left( \lambda _f(p)\right) \log p ~\asymp ~ y \end{aligned}$$

for all sufficiently large x depending on f and M. Here the implied constant depends on I and M.

Conditionally under GRH, i.e, assuming the generalized Riemann hypothesis for all symmetric power L-functions $L(s, \textrm{Sym}^m \pi _f)$, we have the following theorem due to Rouse and Thorner (see [20, 25]).

Theorem 14

Suppose that GRH is true and f is a non-CM form. Also let $I \subseteq [-1, 1]$ be an interval. Then we have

$$\begin{aligned} \#\{p \le x ~:~ p \not \mid N,~ \lambda _f(p) \in I\} ~=~ \mu _{ST}(I) \pi (x) ~+~ O\left( x^{3/4} \frac{\log \left( kNx\right) }{\log x}\right) . \end{aligned}$$

(1)

Remark 2.4

As remarked by Thorner in [25], it is expected that the error term in (1) can be replaced by $O(x^{1/2+\epsilon })$ for any $\epsilon > 0$, where the implied constant will depend on $\epsilon$ and f.

Let $\epsilon >0$ be a real number. From Theorem 14, it follows that if $y \ge x^{3/4} \log x \log \log x$, then

$$\begin{aligned} \sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} \mathbb {1}_I \left( \lambda _f(p)\right) \log p ~\ge ~ \left( \mu _{ST}(I) -\epsilon \right) y \end{aligned}$$

for all sufficiently large x depending on $\epsilon , I$ and f. In section 4, we will prove the following theorem conditionally under the generalized Riemann hypothesis for all symmetric power L-functions of f.

Theorem 15

Suppose that GRH is true, f is a non-CM form and $\epsilon > 0$ is a real number. Let $I \subseteq [-1, 1]$ be a subset which can be $\textrm{Sym}^M$-minorized and $b_0$ be as before. Then for $y \ge x^{1/2} (\log x)^3$, we have

$$\begin{aligned} \sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} \mathbb {1}_I \left( \lambda _f(p)\right) \log p ~\ge ~ \left( b_0 -\epsilon \right) y \end{aligned}$$

for all sufficiently large x depending on $\epsilon , I, M$ and f.

3 Chebotarev density theorem in short intervals

3.1 Proof of Theorem 7

Let the notations be as in section 2 and define

$$\begin{aligned} \Psi _C(x, L/K) = \sum _{\begin{array}{c} {N_K({\mathfrak {p}})^m \le x} \\ {{\mathfrak {p}} {\text { unramified}}} \\ {[\sigma _{{\mathfrak {p}}}]^m = C} \end{array}} \log N_K({\mathfrak {p}}). \end{aligned}$$

Let g be a non-identity element of G, $H = <g>$ and $E = L^H$. Also let $x \ge 2$, $T \ge 2$ and $1 \le y \le x$. Then from [14, Theorem 7.1], we get

$$\begin{aligned} \Psi _C(x+y, L/K) - \Psi _C(x, L/K)= & {} \frac{\#C}{\#G} \left( y - \sum _{\chi } {\overline{\chi }}(g) \sum _{\begin{array}{c} \rho \\ |\gamma | < T \end{array}} \frac{(x+y)^{\rho } - x^\rho }{\rho } \right) ~\nonumber \\{} & {} +~ \mathcal {E}_1 ~+~ \mathcal {E}_2, \end{aligned}$$

(2)

where $\chi$ runs over irreducible characters of H and $\rho$ runs over non-trivial zeros of $L(s, \chi , L/E)$. Further, we have

$$\begin{aligned} \mathcal {E}_1 ~\ll ~ \frac{\#C}{\#G} \left( \frac{x \log x + T}{T} \log D_L ~+~ n_L \log x ~+~ \frac{n_L x \log x \log T}{T}\right) \end{aligned}$$

(3)

and

$$\begin{aligned} \mathcal {E}_2 ~\ll ~ \log x \log D_L ~+~ n_K\frac{x \log ^2 x}{T}. \end{aligned}$$

(4)

Let us set $\mathcal {L}= \log \left( Q T^{n_E}\right)$, where $Q = D_E \mathcal {Q}(L/E) n_E^{n_E}$ (see Sect. 2.1).

We estimate the above double sum over $\chi$ and $\rho$ as follows:

$$\begin{aligned} \begin{aligned} \Bigg | \sum _{\chi } {\overline{\chi }}(g) \sum _{\begin{array}{c} \rho \ne \beta _1 \\ |\gamma |< T \end{array}} \frac{(x+y)^\rho - x^\rho }{\rho } \Bigg |&~\le ~ \sum _{\begin{array}{c} \chi , ~~\rho \ne \beta _1 \\ |\gamma |< T \\ 0< \beta< 1- {\tilde{c}}_1/\mathcal {L} \end{array}} yx^{\beta -1}\\&~\le ~ 3 \sum _{\begin{array}{c} \chi , ~\rho \ne \beta _1 \\ |\gamma |< T \\ 1/2 \le \beta < 1- {\tilde{c}}_1/\mathcal {L} \end{array}} yx^{\beta -1} ~\le ~ -3y \int _{1/2}^{1-{\tilde{c}}_1/\mathcal {L}} x^{\sigma -1 } dN^*(\sigma , T), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} N^*(\sigma , T) ~=~ \sum _{\chi }\sum _{\begin{array}{c} \rho \ne \beta _1 \\ \sigma< \beta< 1 \\ |\gamma | < T \end{array}} 1 \end{aligned}$$

and ${\tilde{c}}_1$ is a positive constant (see Theorem 4). Let $c_2$ be a positive constant which is sufficiently large and $x \ge 2 Q^{4c_2}$. Also choose $T = Q^{-\frac{1}{n_E}} x^{\frac{1}{4c_2 n_E}}$. Applying Theorem 5, we obtain

$$\begin{aligned} \begin{aligned} -\int _{1/2}^{1-{\tilde{c}}_1/\mathcal {L}} x^{\sigma -1 } dN^*(\sigma , T)&~=~ x^{-1/2} N^*\left( 1/2, T\right) ~+~ \log x \int _{1/2}^{1-{\tilde{c}}_1/\mathcal {L}} x^{\sigma -1 } N^*(\sigma , T) ~ d\sigma \\&~\ll ~ x^{-3/8} ~+~ e^{-3{\tilde{c}}_1 c_2}. \end{aligned} \end{aligned}$$

(5)

We note that $D_L \ge D_E \mathcal {Q}$ (see [1, Lemma 4.2]) and hence $Q = Q(L/E) \le D_L n_E^{n_E}\le D_L n_L^{n_L}$. Now we suppose that $x \ge \left( D_L n_L^{n_L}\right) ^B$, where $B= B(c_2)$ is a sufficiently large positive constant which depends on $c_2$. Then we can check that

$$\begin{aligned} \mathcal {E}_1 ~\ll ~ \frac{\#C}{\#G} \cdot x^{1- \frac{1}{8c_2 n_E}} \textrm{and} \mathcal {E}_2 ~\ll ~ \frac{\#C}{\#G} \cdot x^{1- \frac{1}{8c_2 n_E}}. \end{aligned}$$

(6)

We suppose that $y \ge x^{1- \frac{1}{16 c_2 n_E}}$. Now Theorem 7 follows from (2), (5) and (6).

4 Sato-Tate conjecture in short intervals

4.1 Proof of Theorem 15

Suppose that GRH is true. Let $M \ge 1$ be an integer and $I \subseteq [-1, 1]$ be a subset which can be $\textrm{Sym}^M$-minorized. Then there exist $b_0, ~b_1, \cdots , b_M \in {\mathbb {R}}$ with $b_0 > 0$ such that

$$\begin{aligned} \mathbb {1}_{I}(t) ~\ge ~ \sum _{n=0}^{M} b_n U_n(t) ~~\mathrm { for~all~} t \in [-1,1]. \end{aligned}$$

Hence we get

$$\begin{aligned} \sum _{\begin{array}{c} x< p \le x+y \\ p \not \mid N \end{array}} \mathbb {1}_{I}(\lambda _f(p)) \log p ~\ge ~ \sum _{n=0}^{M} b_n \sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} U_n(\lambda _f(p)) \log p. \end{aligned}$$

(7)

From [20, Page 3596], we have

$$\begin{aligned} \bigg |\sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} U_n(\lambda _f(p)) \log p \bigg | ~\ll ~ x^{1/2} (\log x)^2 \end{aligned}$$

for any $n \ge 1$. Here the implied constant depends on M and f. Note that the proof in [20] is given for non-CM newforms of square-free level but it goes through also for non-CM forms of arbitrary level. If $n=0$, we have (see [8, page 113])

$$\begin{aligned} \sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} \log p ~=~ y + O\left( x^{1/2} (\log x)^2\right) . \end{aligned}$$

Hence from (7), we get

$$\begin{aligned} \sum _{\begin{array}{c} x < p \le x+y \\ p \not \mid N \end{array}} \mathbb {1}_{I}(\lambda _f(p)) \log p ~\ge ~ b_0 y + O\left( x^{1/2} (\log x)^2\right) , \end{aligned}$$

where the implied constant depends on M, $\max _{0 \le i \le M} |b_i|$ and f. This completes the proof of Theorem 15.$\square$

5 Large prime factors of Fourier coefficients in short intervals

In this section, we detail the proofs of Theorem 1, Theorem 2, Theorem 3 and Remark 1.1. We need the following lemmas to prove them.

Lemma 16

Let $n \ge 1$ be a natural number and $p \not \mid N$ be a prime number. Then for $d | (n+1)$, we have

$$\begin{aligned} P\left( a_f(p^n)\right) ~\ge ~ P\left( a_f(p^{d-1})\right) \end{aligned}$$

provided $a_f(p^{n}) \ne 0$.

Proof

For any prime $p \not \mid N$ and integer $n \ge 1$, we have

$$\begin{aligned} a_f(p^{n+1}) ~=~ a_f(p) a_f(p^n) - p^{k-1} a_f(p^{n-1}). \end{aligned}$$

Hence for natural numbers $n\ge 2$, we get

$$\begin{aligned} a_f(p^{n-1}) ~=~ \frac{\alpha _p^n - \beta _p^n}{\alpha _p - \beta _p}, \end{aligned}$$

(8)

where $\alpha _p, \beta _p$ are the roots of the polynomial $x^2-a_f(p)x + p^{k-1}$. Since $a_f(p)$’s are assumed to be integers, it follows from (8) that

$$\begin{aligned} a_f(p^{d-1}) \mid a_f(p^n) \textrm{whenever }\,\,d \mid n+1 \end{aligned}$$

provided $a_f(p^{d-1}) \ne 0$ (see [6, page 37, Theorem IV] and [22, page 434, Eq. 14]). Hence if $a_f(p^{n}) \ne 0$, we obtain

$$\begin{aligned} P\left( a_f(p^n)\right) ~\ge ~ P\left( a_f(p^{d-1})\right) \end{aligned}$$

whenever $d \mid (n+1)$. $\square$

Lemma 17

Let h(x) be a real valued non-negative function of x. Also let $q \ge 2$ be a prime number, $V_q(x)~=~ \left\{ p \in (x, ~x + h(x)] ~:~ p \not \mid N,~ a_f(p^{q-1}) \ne 0 \right\}$ and

$$\begin{aligned} \prod _{p \in V_q(x)} |a_f(p^{q-1})| ~=~ \prod _{\ell ~\textrm{prime}} {\ell }^{\nu _{x, \ell }}. \end{aligned}$$

Then we have

$$\begin{aligned} \nu _{x, \ell } ~\le ~ \sum _{1 \le m \le \frac{\log (q x^{qk})}{\log \ell }} \Big (\pi _{f,~q-1}(x + h(x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ). \end{aligned}$$

Proof

Note that, using Deligne’s bound, we have

$$\begin{aligned} \nu _{x,\ell } ~=~ \sum _{\begin{array}{c} p \in V_q(x) \end{array}} \nu _\ell (a_f(p^{q-1}))\ {}= & {} \sum _{\begin{array}{c} p \in V_q (x) \end{array}} \sum _{\begin{array}{c} m \ge 1 \\ \ell ^m | a_f(p^{q-1}) \end{array}} 1 \\= & {} \sum _{1 \le m \le \frac{\log (q x^{(q-1)(k-1)/2})}{\log \ell }} \sum _{\begin{array}{c} p \in V_q(x) \\ a_f(p^{q-1}) \equiv 0 (\textrm{ mod }\,\, \ell ^m) \end{array}} 1 \nonumber \\\le & {} \sum _{1 \le m \le \frac{\log (q x^{qk/2})}{\log \ell }} \bigg ( \pi _{f, ~q -1}( x + h(x), ~\ell ^m) - \pi _{f, ~q-1}(x,~ \ell ^m)\bigg ). \end{aligned}$$

$\square$

5.1 Proof of Theorem 1

Let f be as in Theorem 1 and $\epsilon > 0$ be a real number.

Applying Lemma 16, we see that to prove Theorem 1, it is sufficient to consider $n=q-1$, where q is a prime number. The case $q=2$ corresponds to $n=1$ whereas when $n>1$, we can assume that q is an odd prime.

For any real number $A > 0$, set $\eta _1(x) = \frac{x}{(\log x)^A}$. Let $V_q(x)$ be as in Lemma 17 for $h(x) = \eta _1(x)$ and

$$\begin{aligned} \prod _{p \in V_q(x)} |a_f(p^{q-1})| ~=~ \prod _{\ell ~\textrm{prime}} {\ell }^{\nu _{x, \ell }}. \end{aligned}$$

(9)

Then by Lemma 17, we have

$$\begin{aligned} \nu _{x,\ell } ~\le ~ \sum _{1 \le m \le \frac{\log (q x^{ qk} )}{\log \ell }} \Big (\pi _{f,~q-1}(x + \eta _1(x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ). \end{aligned}$$

(10)

From Theorem 8, there exists a constant $c > 0$ depending on f and A such that whenever $1< \ell ^m \le c \frac{(\log x)^{1/4}}{(\log \log x)^{1/4}}$, we have

$$\begin{aligned} \pi _{f,~q-1}(x + \eta _1(x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) ~\ll ~ \delta _{q-1}(\ell ^m)~ \pi (\eta _1(x)). \end{aligned}$$

(11)

Suppose that

$$\begin{aligned} P\left( a_f(p^{q-1})\right) ~\le ~ (\log x^{q})^{1/8} (\log \log x^{q})^{3/8 -\epsilon } \end{aligned}$$

(12)

for all $p \in V_q(x)$. Set

$$\begin{aligned} z ~=~ c \frac{(\log x)^{1/4}}{(\log \log x)^{1/4}} \textrm{and} w ~=~ (\log x^q)^{1/8} (\log \log x^q)^{3/8-\epsilon }. \end{aligned}$$

(13)

From now on, assume that $\ell \le w$ and x be sufficiently large. For any prime $\ell \le w$, set

$$\begin{aligned} m_0 = m_0(x, \ell ) = \Big [\frac{\log z}{\log \ell }\Big ]. \end{aligned}$$

When $n=1$ or equivalently $q=2$. Using (11) and Lemma 9, we obtain

$$\begin{aligned} \sum _{1 \le m \le m_0} \left( \pi _{f, ~1}(x + \eta _1(x), ~\ell ^m) - \pi _{f, ~1}(x, ~\ell ^m)\right){} & {} \ll ~ \sum _{1 \le m \le m_0} \delta (\ell ^m) \pi (\eta _1(x)) \nonumber \\{} & {} \ll \sum _{1\le m \le m_0} \frac{\pi (\eta _1(x))}{\ell ^m} \nonumber \\{} & {} \ll \frac{\pi (\eta _1(x))}{\ell } \end{aligned}$$

(14)

and

$$\begin{aligned} \sum _{m_0 < m \le \frac{\log (2 x^{2k} )}{\log \ell }} \left( \pi _{f, ~1}(x+ \eta _1(x), ~\ell ^m) - \pi _{f, ~1}(x,~ \ell ^m)\right){} & {} \le \delta (\ell ^{m_0}) \pi (\eta _1(x)) \sum _{m \le \frac{\log (2 x^{ 2k} )}{\log \ell }} 1 \nonumber \\{} & {} \ll \frac{\pi (\eta _1(x))\log x}{\ell ^{m_0} \log \ell } \nonumber \\{} & {} \ll ~ \frac{\eta _1(x)}{z} \cdot ~\frac{\ell }{\log \ell }. \end{aligned}$$

(15)

From (14) and (15), we deduce that

$$\begin{aligned} \nu _{x,\ell } ~\ll ~ \frac{\eta _1(x)}{z} \cdot ~\frac{\ell }{\log \ell }. \end{aligned}$$

(16)

It follows from (9), (12), (13) and (16) that

$$\begin{aligned} \begin{aligned} \sum _{p \in V_2(x)} \log |a_f(p)| ~=~ \sum _{\ell \le w} \nu _{x, \ell } \log \ell ~\ll ~ \frac{\eta _1(x)}{z} \cdot \sum _{\ell \le w} \ell ~\ll ~ \frac{\eta _1(x)}{z} \cdot \frac{w^2}{\log w} ~\ll ~ \frac{\eta _1(x)}{(\log \log x)^{\epsilon }} \end{aligned}\nonumber \\ \end{aligned}$$

(17)

for all sufficiently large x. Applying Theorem 13 with $I = [-1, -1/2]$ and $M=4$, we get

$$\begin{aligned} \sum _{\begin{array}{c} p \in (x, ~x + \eta _1(x)] \\ \lambda _f(p) \in I \end{array}} \log p ~\gg ~ \eta _1(x)\\ \end{aligned}$$

for all sufficiently large x. Hence we deduce that

$$\begin{aligned} \sum _{p \in V_2(x)} \log |a_f(p)| ~\ge ~ \sum _{\begin{array}{c} p \in (x, x + \eta _1(x)] \\ \lambda _f(p) \in I \end{array}} \log |a_f(p)| ~\gg ~ \sum _{\begin{array}{c} p \in (x, x + \eta _1(x)] \\ \lambda _f(p) \in I \end{array}} \log p ~\gg ~ \eta _1(x)\nonumber \\ \end{aligned}$$

(18)

for all sufficiently large x. This is a contradiction to (17) when x is sufficiently large and completes the proof when $n=1$.

When $n >1$ or equivalently q is an odd prime. Let $\ell \le w$ be a prime such that $\ell \equiv 0, \pm 1 ~(\textrm{mod }\,\, q)$. Then for such an $\ell$, it follows from Lemma 10, Lemma 11 and (11) that

$$\begin{aligned} \sum _{1 \le m \le m_0} \Big (\pi _{f,~q-1}(x + \eta _1(x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ){} & {} \ll ~ q\sum _{1\le m \le m_0} \frac{ \pi (\eta _1(x))}{\ell ^m} \nonumber \\{} & {} \ll q \cdot \frac{\pi (\eta _1(x))}{\ell } \end{aligned}$$

(19)

and

$$\begin{aligned}{} & {} \sum _{m_0 < m \le \frac{\log (q x^{q k} )}{\log \ell }} \Big (\pi _{f,~q-1}(x + \eta _1(x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ) \nonumber \\{} & {} \quad \le \sum _{m \le \frac{\log (q x^{q k} )}{\log \ell }} \Big (\pi _{f,~q-1}(x + \eta _1(x), \ell ^{m_0}) - \pi _{f,~q-1}(x, \ell ^{m_0}) \Big ) \nonumber \\{} & {} \quad \ll \frac{q}{\ell ^{m_0}} \pi (\eta _1(x)) \cdot \frac{q\log x}{\log \ell } ~\ll ~ \frac{q^2 \eta _1(x)}{\ell ^{m_0} \log \ell } ~\ll ~ \frac{q^2 \eta _1(x)}{z} \cdot ~\frac{ \ell }{\log \ell }. \end{aligned}$$

(20)

From (19) and (20), we get

$$\begin{aligned} \nu _{x,\ell } ~\ll ~ \frac{q^2 \eta _1(x)}{z} \cdot \frac{\ell }{\log \ell }. \end{aligned}$$

(21)

Note that if $\ell \not \equiv 0, \pm 1 ~(\textrm{mod }\,\, q)$, we have $C_{\ell ^m, ~q-1} = \emptyset$ (see Lemma 10 and Lemma 11). Hence if $\ell ^m \mid a_f(p^{q-1})$, then we must have $p \mid \ell N$ (see Sect. 2.3). Since $p \in V_q(x)$, we obtain $p = \ell$. Hence we have $\nu _{x, \ell } \le \nu _{\ell }(a_f(\ell ^{q-1})) \ll kq$ if $\ell \not \equiv 0, \pm 1 ~(\textrm{ mod }\,\, q)$. It follows from (9), (12) and (13) that

$$\begin{aligned} \sum _{p \in V_q(x)} \log |a_f(p^{q-1})| ~=~ \sum _{\ell \le w} \nu _{x, \ell } \log \ell . \end{aligned}$$

(22)

Now applying (21) and Brun-Titchmarsh inequality (see [12, Theorem 3.8]), we obtain

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \ell \le w \\ \ell \equiv 0, \pm 1 (\textrm{ mod }\,\, q) \end{array}} \nu _{x,\ell } \log \ell&~\le ~ \nu _{x,q} \log q ~+~ \sum _{\begin{array}{c} \ell \le w \\ \ell \equiv \pm 1 (\textrm{ mod }\,\, q) \end{array}} \nu _{x,\ell } \log \ell \\&~\ll ~ \frac{q^3 \eta _1(x)}{z} ~~+~ \frac{q^2 \eta _1(x)}{z} \sum _{\begin{array}{c} \ell \le w \\ \ell \equiv \pm 1 (\textrm{ mod }\,\, q) \end{array}} \ell ~~\ll ~~ \frac{q^3 \eta _1(x)}{z} \\&\quad + \frac{q^2 \eta _1(x)}{z} \frac{w^2}{q \log (w/q)} \end{aligned} \end{aligned}$$

for all sufficiently large x depending on $A, \epsilon , q$ and f. Also we have

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \ell \le w \\ \ell \not \equiv 0, \pm 1(\textrm{ mod }\,\, q) \end{array}} \nu _{x,\ell } ~\log \ell ~~\ll ~~ q \sum _{\ell \le w} \log \ell ~\ll ~ q w. \end{aligned} \end{aligned}$$

Hence we conclude that

$$\begin{aligned} \sum _{\ell \le w} \nu _{x, \ell } \log \ell ~\ll ~ \frac{q^3 \eta _1(x)}{z} ~+~ \frac{q^2 \eta _1(x)}{z} \frac{w^2}{q \log (w/q)} \end{aligned}$$

(23)

for all sufficiently large x depending on $A, \epsilon , q, f$ and the implied constant depends only on f.

Using Deligne’s bound, we can write

$$\begin{aligned} a_f(p) ~=~ 2 p^{\frac{k-1}{2}} \lambda _f(p)~, \lambda _f(p) \in [-1, 1]. \end{aligned}$$

For any prime $p \not \mid N$, we can deduce from (8) that

$$\begin{aligned} a_f(p^{q-1})= & {} \prod _{j=1}^{\frac{q-1}{2}} \left( a_f(p)^2 - 4 \cos ^2(\pi j/q) p^{k-1}\right) \nonumber \\= & {} (4 p^{k-1})^{\frac{q-1}{2}} \prod _{j=1}^{\frac{q-1}{2}} \left( \lambda _f(p)^2 - \cos ^2(\pi j/q)\right) . \end{aligned}$$

(24)

Set

$$\begin{aligned} \mathcal {I}_{q} ~=~ \bigg \{ t \in [-1, 1] ~:~ \bigg |t-\cos \left( \frac{\pi j}{q}\right) \bigg | \ge \frac{1}{q^2} ~\textrm{ and }~ \bigg |t+\cos \left( \frac{\pi j}{q}\right) \bigg | \ge \frac{1}{q^2} ~ \forall ~ 1 \le j \le \frac{q-1}{2} \bigg \}.\nonumber \\ \end{aligned}$$

(25)

From Remark 2.3, the set $\mathcal {I}_q$ can be $\textrm{Sym}^M$-minorized if M is sufficiently large (depending on q) and hence from Theorem 13, we deduce that

$$\begin{aligned} \sum _{\begin{array}{c} p \in (x, x+\eta _1(x)] \\ \lambda _f(p) \in \mathcal {I}_q \end{array}} \log p ~\gg ~ \eta _1(x), \end{aligned}$$

(26)

where the implied constant depends on q and f. For any prime $p \in V_q(x)$ with $\lambda _f(p) \in \mathcal {I}_q$, we have $|a_f(p^{q-1})| \ge (4p^{k-1})^{\frac{q-1}{2}} {q}^{-2(q-1)}$. Thus from (26), we get

$$\begin{aligned} \begin{aligned} \sum _{p \in V_{q}(x)} \log |a_f(p^{q-1})| ~\ge ~ \sum _{\begin{array}{c} p \in V_q(x)\\ \lambda _f(p) ~\in ~ \mathcal {I}_{q} \end{array}} \log |a_f(p^{q-1})|&~\gg ~ \sum _{\begin{array}{c} p \in V_q(x)\\ \lambda _f(p) ~\in ~ \mathcal {I}_{q} \end{array}} \log p ~+~ O\left( \pi (\eta _1(x))\right) \\&~\gg ~ \eta _1(x) \end{aligned} \end{aligned}$$

(27)

for all sufficiently large x depending on A, q and f. Here the implied constant depends on q and f. This gives a contradiction to (23) if x is sufficiently large depending on $A, \epsilon , q$ and f. This completes the proof of Theorem 1.

5.2 Proof of Theorem 2

Suppose that GRH is true. The proof now follows along the lines of the proof of Theorem 1. As in the proof of Theorem 1, it is sufficient to investigate large prime factors $a_f(p^{q-1})$, where q is a prime number. For any real number $\epsilon \in (0, 1/10)$ and for any prime q, let $V_q(x)$ be as in Lemma 17 for $h(x)~=~x^{1/2+\epsilon }$.

When q is an odd prime, set

$$\begin{aligned} \mathcal {J}_{q}= & {} \bigg \{ t \in [-1, 1] ~:~ \bigg |t-\cos \left( \frac{\pi j}{q}\right) \bigg | \ge \frac{1}{ Cq^2} ~\textrm{ and }~ \bigg |t+\cos \left( \frac{\pi j}{q}\right) \bigg | \\\ge & {} \frac{1}{Cq^2} \forall ~ 1 \le j \le (q-1)/2\bigg \}, \end{aligned}$$

where $C > 0$ is a constant such that $\mu _{ST}\left( \mathcal {J}_q\right) > 1/2$. From Remark 2.3, we know that $\mathcal {J}_q$ is $\textrm{Sym}^M$-minorized (with $b_0=b_0 (q) > 0$) if M is sufficiently large (depending on q). Let $0< b < \min \{b_0, \frac{1}{7}\}$ and c be a positive constant which will be chosen later. When $q=2$, $c_1$ is a positive constant which will be chosen later.

Suppose that for any $p \in V_q(x)$,

$$\begin{aligned} P(a_f(p^{q-1})) ~\le ~ w, \end{aligned}$$

where

$$\begin{aligned} w = {\left\{ \begin{array}{ll} c_1x^{\epsilon /7} (\log x)^{2/7} &{} \text { when}\ q=2; \\ c x^{\epsilon b} &{} {\hbox {when}}\; q\; \text {is an odd prime}. \end{array}\right. } \end{aligned}$$

Write

$$\begin{aligned} \prod _{p \in V_q(x)} |a_f(p^{q-1})| ~=~ \prod _{\ell \textrm{ prime }} \ell ^{\nu _{x,\ell }}. \end{aligned}$$

This implies that

$$\begin{aligned} \sum _{p \in V_q(x)} \log |a_f(p^{q-1})| ~=~ \sum _{\ell \le w} \nu _{x,\ell } \log \ell , \end{aligned}$$

(28)

Using Lemma 17, we know that

$$\begin{aligned} \nu _{x,\ell } ~\le ~ \sum _{1 \le m \le \frac{\log (q x^{q k })}{\log \ell }} \bigg (\pi _{f, q-1}(x+ x^{1/2+\epsilon }, \ell ^m) - \pi _{f, q-1}(x, \ell ^m)\bigg ). \end{aligned}$$

(29)

Set

$$\begin{aligned} z = {\left\{ \begin{array}{ll} c_1\frac{x^{2\epsilon /7}}{(\log x)^{3/7}} &{} \hbox { when}\ q=2; \\ c \frac{x^{2\epsilon b}}{\log x} &{} \text {when } q \text { is an odd prime}~~. \end{array}\right. } \end{aligned}$$

From now on, assume that $\ell \le w$, x be sufficiently large and $m_0 = m_0(x, \ell ) = \Big [\frac{\log z}{\log \ell }\Big ]$.

When $n=1$ or equivalently $q=2$. Applying Theorem 12 and Lemma 9, we have

$$\begin{aligned} \sum _{1 \le m \le m_0} \left( \pi _{f, ~1}(x+ x^{1/2+\epsilon }, \ell ^m) - \pi _{f, ~1}(x, \ell ^m)\right)= & {} \sum _{1 \le m \le m_0} \delta (\ell ^m) \bigg \{\frac{x^{1/2+\epsilon }}{\log x}\nonumber \\{} & {} +O\left( \ell ^{4m} x^{1/2} \log x \right) \bigg \}\nonumber \\= & {} \frac{1}{\ell } \cdot \frac{x^{1/2+\epsilon }}{\log x} ~+~ O\left( \frac{x^{1/2+\epsilon }}{\ell ^2 \log x}\right) \nonumber \\{} & {} +O\left( z^3 x^{1/2} \log x\right) . \end{aligned}$$

(30)

Further $\sum _{m_0 < m \le \frac{\log (2 x^{2k})}{\log \ell }} \left( \pi _{f, ~1}(x+ x^{1/2+\epsilon }, \ell ^m) - \pi _{f, ~1}(x, \ell ^m)\right)$ is less than or equal to

$$\begin{aligned}{} & {} \bigg (\pi _{f, ~1}(x+ x^{1/2+\epsilon }, \ell ^{m_0}) - \pi _{f, ~1}(x, \ell ^{m_0})\bigg ) \sum _{m \le \frac{\log (2 x^{2k})}{\log \ell }} 1\nonumber \\{} & {} \qquad \ll \left( \frac{x^{1/2+\epsilon }}{\ell ^{m_0} \log x} ~+~ \ell ^{3m_0} x^{1/2} \log x \right) \frac{\log x}{\log \ell } \nonumber \\{} & {} \qquad \ll \frac{x^{1/2+\epsilon }}{z}\frac{\ell }{\log \ell } ~+~ z^3 x^{1/2}\frac{(\log x)^2}{\log \ell }. \end{aligned}$$

(31)

From (30) and (31), we get

$$\begin{aligned} \nu _{x,\ell } ~\le ~ \frac{1}{\ell } \cdot \frac{x^{1/2+\epsilon }}{\log x} ~+~ O\left( \frac{x^{1/2+\epsilon }}{\ell ^2 \log x} ~+~ \frac{x^{1/2+\epsilon }}{z}\frac{\ell }{\log \ell } ~+~ z^3 x^{1/2}\frac{(\log x)^2}{\log \ell } \right) . \qquad \end{aligned}$$

(32)

It follows from (32) that

$$\begin{aligned} \begin{aligned} \sum _{\ell \le w} \nu _{x,\ell } \log \ell&~\le ~ \frac{x^{1/2+\epsilon }}{\log x} \log w ~+~ c_5\left( \frac{x^{1/2+\epsilon }}{\log x} ~+~ \frac{x^{1/2+\epsilon }}{z}\frac{w^2}{\log w} ~+~ z^3 x^{1/2} (\log x)^2 \frac{w}{\log w}\right) , \end{aligned} \end{aligned}$$

where $c_5>0$ is a constant depending on f. Now we choose $c_1$ such that $2000 c_5 c_1(1 + c_1^3) < \epsilon$. Then by substituting the values of w and z, we obtain

$$\begin{aligned} \sum _{\ell \le w} \nu _{x,\ell } \log \ell ~<~ \frac{x^{1/2+\epsilon }}{30} \end{aligned}$$

(33)

for all sufficiently large x depending on $\epsilon$ and f. On the other hand, from Theorem 15 with $I = [-1, -0.1]$ and Remark 2.3, we get

$$\begin{aligned} \sum _{\begin{array}{c} x < p \le x+ x^{1/2+\epsilon } \\ \lambda _f(p) \in I \end{array}} \log p ~>~ \frac{2}{25} x^{1/2+\epsilon } \end{aligned}$$

for all sufficiently large x depending on $\epsilon$ and f. Hence we deduce that

$$\begin{aligned} \sum _{p \in V_2(x)} \log |a_f(p)| ~>~ \frac{3}{40}\frac{k-1}{2} \cdot x^{1/2+\epsilon } \end{aligned}$$

(34)

for all sufficiently large x depending on $\epsilon$ and f. This is a contradiction to (33).

When $n >1$ or equivalently q is an odd prime. Arguing as before and applying Theorem 15 to the interval $\mathcal {J}_q$, we can show that

$$\begin{aligned} \sum _{p \in V_q(x)} \log |a_f(p^{q-1})| ~~\ge ~~ \frac{k q b}{16} x^{1/2 + \epsilon } \end{aligned}$$

(35)

for all sufficiently large x depending on $\epsilon , q$ and f.

For any prime $\ell \le w$ with $\ell \not \equiv 0, \pm 1 ~(\textrm{ mod }\,\,q)$, we can deduce as in Theorem 1 that

$$\begin{aligned} \nu _{x, \ell } ~\le ~ \nu _{\ell } (a_f(\ell ^{q-1})) ~=~ O(kq). \end{aligned}$$

(36)

When $\ell \equiv 0, \pm 1 (\textrm{mod }\,\, q)$, applying Lemma 9, Lemma 10, Lemma 11 and Theorem 12, we have $\sum _{1 \le m \le m_0} \left( \pi _{f,~q-1}(x+ x^{1/2+\epsilon }, \ell ^m) - \pi _{f,~q-1}(x, \ell ^m)\right)$ is less than or equal to

$$\begin{aligned} \frac{q-1}{2\ell } \cdot \frac{x^{1/2+\epsilon }}{\log x} ~~+~~O\left( \frac{qx^{1/2+\epsilon }}{\ell ^2 \log x}\right) ~+~ O\left( q z^3 x^{1/2} \log x\right) . \end{aligned}$$

(37)

Further $\sum _{m_0 < m \le \frac{\log (q x^{qk})}{\log \ell }} \left( \pi _{f, q-1}(x+ x^{1/2+\epsilon }, \ell ^m) - \pi _{f, q-1}(x, \ell ^m)\right)$ is less than or equal to

$$\begin{aligned}&\bigg (\pi _{f, q-1}(x+ x^{1/2+\epsilon }, \ell ^{m_0}) - \pi _{f, q-1}(x, \ell ^{m_0})\bigg ) \sum _{m \le \frac{\log (q x^{ q k} )}{\log \ell }}1\nonumber \\&\qquad \ll \left( \frac{ q x^{1/2 + \epsilon } }{\ell ^{m_0} \log x} + q \ell ^{3m_0} x^{1/2} \log x \right) \frac{q\log x}{\log \ell } \nonumber \\&\qquad \ll \frac{q^2 x^{1/2+\epsilon }}{z}\frac{\ell }{\log \ell } ~+~ q^2 z^3 x^{1/2}\frac{(\log x)^2}{\log \ell }. \end{aligned}$$

(38)

From (36), (37) and (38), we get

$$\begin{aligned} \nu _{x,\ell } \le \frac{q-1}{2\ell } \cdot \frac{x^{1/2+\epsilon }}{\log x} + O\left( \frac{qx^{1/2+\epsilon }}{\ell ^2 \log x} + \frac{q^2x^{1/2+\epsilon }}{z}\frac{\ell }{\log \ell } + q^2z^3 x^{1/2}\frac{(\log x)^2}{\log \ell } \right) . \end{aligned}$$

(39)

It follows from (28) and (39) that

$$\begin{aligned} \begin{aligned} \sum _{\ell \le w} \nu _{x,\ell } \log \ell&~\le ~ \frac{x^{1/2+\epsilon }}{\log x} \log w ~+~ c_6\left( \frac{qx^{1/2+\epsilon }}{\log x} ~+~ \frac{qx^{1/2+\epsilon }}{z}\frac{w^2}{\log w} ~+~ q z^3 x^{1/2} (\log x)^2 \frac{w}{\log w}\right) \end{aligned} \end{aligned}$$

for all sufficiently large x depending on $\epsilon , q$ and f and where $c_6>0$ is a constant depending on $\epsilon , q$ and f. Substituting the values of z, w and by choosing c such that $2000 \cdot c_6 c < \epsilon b^2$, we get a contradiction to (35) for all sufficiently large x depending on $\epsilon , q$ and f. Hence there exists a prime $p \in (x, x+x^{1/2+\epsilon }]$ with $p \not \mid N$ such that

$$\begin{aligned} P\left( a_f(p^{q-1})\right) ~>~ c x^{\epsilon b} \end{aligned}$$

for some positive constant c depending on $\epsilon , q, f$ and for all sufficiently large x depending on $\epsilon , q$ and f. This completes the proof of Theorem 2. $\square$

5.3 Proof of Theorem 3

Proof of this theorem follows along the lines of the proof of Theorem 1 or Theorem 2. Let $V_{q}(x)$ be as in Lemma 17 with $h(x)=\eta (x) = x^{3/4} \log x \cdot \log \log x$. Set

$$\begin{aligned} z ~=~ c x^{1/14}\frac{(\log \log x)^{2/7}}{(\log x)^{1/7}} \textrm{and} w ~=~ c x^{1/28} (\log x)^{3/7} (\log \log x)^{1/7}, \end{aligned}$$

where $c> 0$ is a constant which will be chosen later. Suppose that

$$\begin{aligned} P(a_f(p^{q-1})) ~\le ~ c x^{1/28} (\log x)^{3/7} (\log \log x)^{1/7} \end{aligned}$$

for any $p \in V_q(x)$. Write

$$\begin{aligned} \prod _{p \in V_q(x)} |a_f(p^{q-1})| ~=~ \prod _{\ell \le w} \ell ^{\nu _{x,\ell }}. \end{aligned}$$

Then

$$\begin{aligned} \sum _{p \in V_q(x)} \log |a_f(p^{q-1})| ~=~ \sum _{\ell \le w} \nu _{x,\ell } \log \ell , \end{aligned}$$

(40)

where, using (16), we see that

$$\begin{aligned} \nu _{x, \ell } ~\le ~ \sum _{1 \le m \le \frac{\log (q x^{qk})}{\log \ell }} \Big (\pi _{f,~q-1}(x + \eta (x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ). \end{aligned}$$

Fix a prime $\ell \le w$ such that $\nu _{x, \ell } \ne 0$. If $\ell \not \equiv 0, \pm 1 ~(\textrm{ mod }\,\, q)$, then as before, we have

$$\begin{aligned} \nu _{x, \ell } \le \nu _{\ell } (a_f(\ell ^{q-1})) ~=~ O(kq). \end{aligned}$$

Now suppose that $\ell \equiv 0, \pm 1 (\textrm{mod }\,\, q)$ and set $m_0 = \Big [\frac{\log z}{\log \ell }\Big ]$. Let x be sufficiently large from now on. Then applying Theorem 12, Lemma 10 and Lemma 11, we get

$$\begin{aligned} \sum _{1 \le m \le m_0} \Big (\pi _{f,~q-1}(x + \eta (x), \ell ^m) - \pi _{f,~q-1}(x, \ell ^m) \Big )\le & {} \frac{q-1}{2 \ell }\frac{\eta (x)}{\log x} ~+~ O\left( \frac{ q}{\ell ^2} \frac{\eta (x)}{\log x}\right) \nonumber \\{} & {} + O\left( q z^3 x^{1/2} \log x\right) \end{aligned}$$

(41)

and

$$\begin{aligned} \sum _{m_0 < m \le \frac{\log (q x^{qk})}{\log \ell }} \Big (\pi _{f,~q-1}(x + \eta (x), \ell ^m) ~-~ \pi _{f,~q-1}(x, \ell ^m) \Big ){} & {} \ll ~ \frac{q^2 \eta (x)}{z}\frac{\ell }{\log \ell } \nonumber \\{} & {} \quad + q^2 z^3 x^{1/2}\frac{(\log x)^2}{\log \ell }.\nonumber \\ \end{aligned}$$

(42)

From (41) and (42), we get

$$\begin{aligned} \nu _{x,\ell } ~\le ~ \frac{q-1}{2 \ell }\frac{\eta (x)}{\log x} ~+~ O\left( \frac{ q}{\ell ^2} \frac{\eta (x)}{\log x} ~+~ \frac{q^2 \eta (x)}{z}\frac{\ell }{\log \ell } ~+~ q^2 z^3 x^{1/2}\frac{(\log x)^2}{\log \ell }\right) \nonumber \\ \end{aligned}$$

(43)

when $\ell \equiv 0, \pm 1 (\textrm{mod }\,\, q)$. It follows from (43) and Brun-Titchmarsh inequality that

$$\begin{aligned} \begin{aligned} \sum _{\begin{array}{c} \ell \le w \\ \ell \equiv \pm 1(\textrm{ mod }\,\, q) \end{array}} \nu _{x,\ell } \log \ell&\le \frac{\eta (x)}{\log x} \log w ~+~ c_7\left( q \frac{\eta (x)}{\log x} + \frac{q \eta (x)}{z}\frac{w^2}{\log (w/q)} \right. \\&\quad \left. ~+~ q z^3 x^{1/2} (\log x)^2 \frac{w}{\log (w/q)}\right) \end{aligned} \end{aligned}$$

(44)

for all sufficiently large x (depending on q and f). Here $c_7$ is a positive constant depending only on f. We also have

$$\begin{aligned} \begin{aligned} \nu _{x,q} \log q&\ll \frac{\eta (x)}{\log x} \log q ~+~ \frac{q^3 \eta (x)}{z} + q^2 z^3 x^{1/2} (\log x)^2 ~~~\textrm{ and }\\&\quad \sum _{\begin{array}{c} \ell \le w \\ \ell \not \equiv 0, \pm 1 (\textrm{mod }\,\, q) \end{array}} \nu _{x, \ell }\log \ell ~\ll ~ q w. \end{aligned} \end{aligned}$$

(45)

Let c be such that $2000 \cdot c_7 (c+c^4) < 1$. Then by substituting the values for w and z in (44) and (45), we deduce that

$$\begin{aligned} \sum _{\ell \le w} \nu _{x, \ell } \log \ell ~~<~ \frac{q}{20}~ \eta (x) \end{aligned}$$

(46)

for all sufficiently large x (depending on q and f). Set $\mathcal {J}_q$ as in Sect. 5.2. As before, by applying Theorem 14, we can show that

$$\begin{aligned} \sum _{p \in V_q(x)} \log |a_f(p^{q-1})| ~>~ \frac{kq}{17} \eta (x) \end{aligned}$$

(47)

for all sufficiently large x depending on q and f. This gives a contradiction to (46) and completes the proof for large prime factor of $a_f(p^{q-1})$ in the interval $(x, x+ \eta (x)]$ under GRH.

5.4 Proof of Remark 1.2

In Theorem 1, instead of working with

$$\begin{aligned} V_q(x)~=~ \left\{ p \in (x, x+\eta _1(x)] ~:~ p \not \mid N,~ a_f(p^{q-1}) \ne 0 \right\} , \end{aligned}$$

one has to consider

$$\begin{aligned} S_q(x)&=~ \left\{ p \in (x, x+\eta _1(x)] ~:~ p \not \mid N,~ a_f(p) \ne 0, ~ P(a_f(p^{q-1})) ~\le ~ (\log x^q)^{1/8} (\log \log x^q)^{3/8 -\epsilon } \right\} . \end{aligned}$$

Arguing as in the proof of Theorem 1 (see (17), (18), (23) and (27)), we can deduce that

$$\begin{aligned} \sum \limits _{\begin{array}{c} p \in S_q(x) \\ \lambda _f(p) \in \mathcal {I}_q \end{array}} \log p ~+~ O\left( \pi (\eta _1(x))\right) ~\ll ~\sum _{p \in S_q(x)} \log |a_f(p^{q-1})| ~=~ \sum _{\ell \le w} \nu _{x, \ell } \log \ell ~\ll ~\frac{\eta _1(x)}{(\log \log x)^{\epsilon }}. \end{aligned}$$

Let $T_q(x) = \{p \in (x, x+\eta _1(x)]: \lambda _f(p) \in \mathcal {I}_q\}$, where $\mathcal {I}_2 = [-1, -1/2]$ and $\mathcal {I}_q$ is as in (25) for $q \ge 3$. Thus we get

$$\begin{aligned} \# \left( S_q(x) \cap T_q(x) \right) ~\ll ~ \frac{\pi ( \eta _1(x) )}{ (\log \log x)^{\epsilon }} \end{aligned}$$

(48)

for all sufficiently large x depending on $A, \epsilon , q$ and f. As observed earlier, from Theorem 13, there exists a positive constant $0< b_1 < 1$ (depending on A, q, f) such that

$$\begin{aligned} \#T_q(x) ~\ge ~ b_1 \pi ( \eta _1(x) ) \end{aligned}$$

(49)

for all sufficiently large x (depending on A, q and f). From (48) and (49), we deduce that

$$\begin{aligned} \limsup _{x \rightarrow \infty } \frac{\#S_q(x)}{\pi ( \eta _1(x) )} ~\le ~ 1-b_1 ~<~ 1. \end{aligned}$$

Thus there exists a positive constant $a_1$ such that for all sufficiently large x, there are at least $a_1\pi ( \eta _1(x) )$ many primes $p \in (x, ~x + \frac{x}{(\log x)^A}]$ for which Theorem 1 is true.

In Theorem 2, we suppose that $\epsilon > 0$ is sufficiently small and let $S_q(x)$ be the set of primes $p \in ( x, x+x^{\frac{1}{2}+\epsilon }]$ such that $a_f(p^{q-1}) \ne 0$ and

$$\begin{aligned} P(a_f(p^{q-1})) ~\le ~ {\left\{ \begin{array}{ll} c_1x^{\epsilon /7} (\log x)^{2/7} &{} \hbox { when}\ q=2; \\ c x^{\epsilon b} &{} \text {when } \,\, q\,\, \text { is an odd prime}, \end{array}\right. } \end{aligned}$$

where $c, c_1$ are as in the proof of Theorem 2 and $T_q(x) = \{p \in (x, x+x^{\frac{1}{2}+\epsilon }]: \lambda _f(p) \in \mathcal {I}_q\}$. As before, there exists a positive constant $0< b_2 < 1$ (depending on $\epsilon , f$) such that

$$\begin{aligned} \#T_q(x) ~\ge ~ b_2 \frac{x^{1/2+\epsilon }}{\log x} \end{aligned}$$

(50)

for all sufficiently large x (depending on $\epsilon , q$ and f). In the case when $n=1$ or $q=2$, by choosing the constant $c_5> 0$ sufficiently small, we can deduce that

$$\begin{aligned} \sum _{\ell \le w} \nu _{x,\ell } \log \ell ~<~ \frac{b_2}{2} \cdot x^{1/2+\epsilon } \end{aligned}$$

for all sufficiently large x (depending on $\epsilon , q$ and f). Arguing as before, we can deduce that

$$\begin{aligned} \limsup _{x \rightarrow \infty } \frac{\#S_q(x)}{x^{1/2+\epsilon }/ \log x} ~\le ~ 1-\frac{b_2}{2} ~<~ 1. \end{aligned}$$

One can deduce a similar conclusion in the remaining case of Theorem 2. To deduce a similar conclusion for Theorem 3, we proceed as follows. Let

$$\begin{aligned} S_q(x) ~= & {} ~ \{ p \in (x, x+ \eta (x)] ~:~ a_f(p^{q-1}) \ne 0,~~ P(a_f(p^{q-1})) ~\\ {}{} & {} \quad \le ~ c x^{1/28} (\log x)^{3/7} (\log \log x)^{1/7} \}, \end{aligned}$$

where $\eta (x) = x^{3/4} \log x \cdot \log \log x$ and c is as in the proof of Theorem 3. Arguing as in the proof of Theorem 3, we deduce that

$$\begin{aligned} \sum _{\ell \le w} \nu _{x, \ell } \log \ell ~<~ \frac{q}{20}~ \eta (x) \end{aligned}$$

(51)

for all sufficiently large x (depending on q and f). Set $\mathcal {J}_q$ as in Sect. 5.2 and $C>0$ is a constant which we will choose later. Then we can deduce that

$$\begin{aligned} \sum _{p \in S_q(x)} \log |a_f(p^{q-1})|&\ge \sum \limits _{\begin{array}{c} p \in S_q(x) \\ \lambda _f(p) \in \mathcal {J}_q \end{array}} \log |a_f(p^{q-1})| \nonumber \\&\ge \frac{kq}{17} \cdot \#\left( S_q(x) \cap T_q(x)\right) \log x +O\left( \pi (\eta (x))\right) \end{aligned}$$

(52)

for all sufficiently large x depending on q and f. Here $T_q(x) = \{p \in (x, x+\eta (x)]: \lambda _f(p) \in \mathcal {J}_q\}$. Hence we get

$$\begin{aligned} \limsup _{x \rightarrow \infty } \frac{ \#\left( S_q(x) \cap T_q(x)\right) }{\pi (\eta (x))} ~\le ~ \frac{17}{20} < 1. \end{aligned}$$

We choose $C>0$ sufficiently large such that $\mu _{ST}(\mathcal {J}_{q}) > 1- \delta$ for some $0<\delta < 17/2000$. Then we get

$$\begin{aligned} \#T_q(x) ~\ge ~ \left( 1- 2\delta \right) \pi (\eta (x)) \end{aligned}$$

for all sufficiently large x depending on q and f. Note that

$$\begin{aligned} \begin{aligned} \#\left( S_q(x) \cap T_q(x)\right)&~=~ \#S_q(x) ~+~ \#T_q(x) ~-~ \#\left( S_q(x) \cup T_q(x)\right) \\&~\ge ~ \#S_q(x) ~+~ \left( 1- 2 \delta \right) \pi (\eta (x)) ~-~ \pi (\eta (x)) \\&~\ge ~ \#S_q(x) ~-~ 2 \delta \pi (\eta (x)). \end{aligned} \end{aligned}$$

(53)

Hence we deduce that

$$\begin{aligned} \limsup _{x \rightarrow \infty } \frac{\#S_q(x)}{\pi (\eta (x))} ~\le ~ \frac{17}{20} ~+~ 2 \delta ~<~ 1. \end{aligned}$$