Open Access

A small improvement in the gaps between consecutive zeros of the Riemann zeta-function

Research in Number Theory20162:28

https://doi.org/10.1007/s40993-016-0053-7

Received: 27 May 2016

Accepted: 18 August 2016

Published: 8 December 2016

Abstract

Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko’s method for exhibiting irregularity in the gaps between consecutive zeros of \(\zeta (s)\) assuming the Riemann hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider a general sequence related to that of Feng and Wu, and introduce a somewhat less general sequence \(\{a_n\}\) for which we write the Montgomery–Odlyzko expressions explicitly. As an application, we give the following slight improvement of Feng and Wu’s result: infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.515396 times the average spacing and infinitely often they differ by at least 2.7328 times the average spacing.

Keywords

Riemann zeta function Zeros Critical line Gaps

Mathematics Subject Classification

Primary 11M26 Secondary 11M06

1 Background

It is well known that the Riemann zeta-function \(\zeta (s)\) has infinitely many nontrivial zeros \(s=\rho =\beta +i\gamma \), and all of them are in the critical strip \(0<\mathrm{Re}\,s=\sigma <1\), \(-\infty<\mathrm{Im}\, s=t<\infty \).

If N(T) denotes the number of zeros \(\rho =\beta +i\gamma \) (\(\beta \) and \(\gamma \) real), for which \(0<\gamma \le T\), then
$$\begin{aligned} N(T)=\frac{T}{2\pi }\log \left( \frac{T}{2\pi }\right) -\frac{T}{2\pi }+\frac{7}{8}+S(T)+O\left( \frac{1}{T}\right) , \end{aligned}$$
with
$$\begin{aligned} S(T)=\frac{1}{\pi }\arg \zeta \left( \frac{1}{2}+iT\right) \end{aligned}$$
and
$$\begin{aligned} S(T)=O(\log T). \end{aligned}$$
This is the Riemann–von Mangoldt formula for N(T). Hence, if we let \(0<\gamma \le \gamma '\) denote consecutive ordinates of non-trivial zeros of \(\zeta (s)\), the average size of \(\gamma '-\gamma \) is \(\gamma /N(\gamma )\sim 2\pi /\log \gamma \). Let
$$\begin{aligned} \lambda =\limsup \limits _{\gamma >0}(\gamma '-\gamma )\frac{\log \gamma }{2\pi } \end{aligned}$$
and
$$\begin{aligned} \mu =\liminf \limits _{\gamma >0}(\gamma '-\gamma )\frac{\log \gamma }{2\pi }. \end{aligned}$$
We note that \(\mu \le 1\le \lambda \) and it is expected that \(\mu =0\) and \(\lambda =+\infty \). The problem of studying \(\lambda \) and \(\mu \) is important for a number of reasons. Montgomery and Weinberger [11] discovered an effect of the exceptional zero of an L-function on the spacing of zeros, which led Montgomery [9] to the Pair Correlation Conjecture (see also Conrey and Iwaniec [4]): for any fixed \(0<\alpha <\beta \),
$$\begin{aligned} \sum _{\begin{array}{c} 0<\gamma ,\gamma '\le T\\ \frac{2\pi \alpha }{\log T}\le \gamma '-\gamma \le \frac{2\pi \beta }{\log T} \end{array}} \!\!\!\!\!\!\!\! 1\ \sim \ N(T)\ \int _\alpha ^\beta \Big (1-\Big (\frac{\sin \pi u}{\pi u}\Big )^2\Big )\,du. \end{aligned}$$
The paper [9] also contains explicit bound \(\mu \le 0{.}68\) conditionally on the Riemann hypothesis.

Let \(N_0(T)\) be the number of zeros of \(\zeta \left( \frac{1}{2}+it\right) \) when \(0<t\le T\), each zero counted with multiplicity. The Riemann hypothesis is the conjecture that \(N_0(T)=N(T)\).

In this note we prove the following theorem.

Theorem 1

Assume the Riemann hypothesis. Then we have
$$\begin{aligned} \mu <0{.}515396 \end{aligned}$$
and
$$\begin{aligned} \lambda >2.7328. \end{aligned}$$
We briefly describe the history of the problem, focusing mainly on \(\mu \).
  •  [12]: in 1946 Selberg remarked that \(\mu<1<\lambda \) unconditionally.

Now suppose that T is a large real number and \(K=T(\log T)^{-2}\). Let
$$\begin{aligned} h(c)=c-\frac{\text {Re}\left( \sum _{nk\le K}a_k\overline{a_{nk}}g_c(n)\Lambda (n)n^{-1/2}\right) }{\sum _{k\le K}\left| a_k\right| ^2}, \end{aligned}$$
(1)
where
$$\begin{aligned} g_c(n)=\frac{2\sin \left( \pi c\frac{\log n}{\log T}\right) }{\pi \log n} \end{aligned}$$
and \(\Lambda \) is the von Mangoldt’s function.
In the following results, the truth of the Riemann hypothesis is assumed.
  •  [10]: in 1981 by an argument using the Guinand–Weil explicit formula, Montgomery and Odlyzko showed that if \(h(c)<1\) for some choice of c and \(\{a_n\}\), then \(\lambda \ge c\), and if \(h(c)>1\) for some choice of c and \(\{a_n\}\), then \(\mu \le c\). They used the coefficients
    $$\begin{aligned} a_k=\frac{1}{k^{\frac{1}{2}}}f\left( \frac{\log k}{\log K}\right) \quad \text { and }\quad a_k=\frac{\lambda (k)}{k^{\frac{1}{2}}}f\left( \frac{\log k}{\log K}\right) , \end{aligned}$$
    where f is a continuous function of bounded variation, and \(\lambda (k)\) is the Liouville function. With this choice of the coefficients they obtained \(\lambda >1{.}9799\) and \(\mu <0{.}5179\) by optimizing over the functions f.
  •   [3]: in 1984 Conrey et al. chose the coefficients
    $$\begin{aligned} a_k=\frac{d_r(k)}{\sqrt{k}}\quad \text { and }\quad a_k=\frac{\lambda (k) d_r(k)}{\sqrt{k}}, \end{aligned}$$
    where \(d_r(k)\) is a multiplicative function defined on integral powers of a prime p by
    $$\begin{aligned} d_r(p^k)=\frac{\Gamma (k+r)}{\Gamma (r) k!}. \end{aligned}$$
    The choice \(r=1{.}1\) with the latter \(a_k\) yields \(\mu <0{.}5172\) and the choice \(r=2{.}2\) with the former \(a_k\) yields \(\lambda >2{.}337\).
  •   [7]: in 2005, by making use of Wirtinger’s inequality and the asymptotic formulae for the fourth mixed moments of the zeta-function and its derivative, Hall proved that \(\lambda >2{.}6306\).

  •   [2]: in 2010, Bui et al. considered the coefficients of the form
    $$\begin{aligned} a_k=\frac{d_r(k)}{\sqrt{k}} f\big (\tfrac{\log K/k}{\log K}\big )\quad \text { and }\quad a_k=\frac{\lambda (k) d_r(k)}{\sqrt{k}}f\big (\tfrac{\log K/k}{\log K}\big ) \end{aligned}$$
    for a polynomial f and obtained \(\lambda >2{.}69\) and \(\mu <0{.}5155\).
  •   [5]: in 2012, Feng and Wu introduced the coefficient
    $$\begin{aligned} a_k= & {} \frac{d_r(k)}{k^{\frac{1}{2}}}\left( f_1\left( \frac{\log K/k}{\log K}\right) +f_2\left( \frac{\log K/k}{\log K}\right) \sum _{p_1p_2\mid k}\frac{\log p_1\log p_2}{\log ^2K}\right. \\&+\,\,f_3\left( \frac{\log K/k}{\log K}\right) \sum _{p_1p_2p_3\mid k}\frac{\log p_1\log p_2\log p_3}{\log ^3K}+\cdots \\&\left. +\,\,f_I\left( \frac{\log K/k}{\log K}\right) \sum _{p_1p_2\cdots p_I\mid k}\frac{\log p_1\log p_2\cdots \log p_I}{\log ^IK}\right) , \end{aligned}$$
    for any integer \(I\ge 2\). Using \(I=2\) they obtained \(\lambda >2{.}7327\) and \(\mu <0{.}5154\), or, to higher precision, \(\lambda >2{.}73272\) and \(\mu <0{.}515398\).
We remark that for \(I=2\) the Feng and Wu coefficient is equivalent to
$$\begin{aligned} a_k= & {} \frac{d_r(k)}{k^{\frac{1}{2}}}\left( f_1\left( \frac{\log K/k}{\log K}\right) +f_2\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^2 p_1}{\log ^2K}\right. \\&\left. +\,\,f_3\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^3 p_1}{\log ^3K}+\cdots +f_I\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^I p_1}{\log ^IK}\right) , \end{aligned}$$
for which the calculations are simpler.
To prove Theorem 1, we choose the coefficients
$$\begin{aligned} a_k=\frac{\lambda (k)d_r(k)}{k^{\frac{1}{2}}}f_1\left( \frac{\log K/k}{\log K}\right) +\frac{\lambda (k)d_r(k)}{k^{\frac{1}{2}}}\sum _{p\mid k}P\left( \frac{\log p}{\log K}\right) \tilde{f}_1\left( \frac{\log K/k}{\log K}\right) , \end{aligned}$$
where \(f_1\), \(\tilde{f}_1\), P are some polynomials to be chosen later. These \(a_k\) are less general than
$$\begin{aligned} a_k= & {} \frac{\lambda (k)d_r(k)}{k^{\frac{1}{2}}}\left( f_1\left( \frac{\log K/k}{\log K}\right) +f_2\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^2 p_1}{\log ^2K}\right. \\&+\,f_3\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^3 p_1}{\log ^3K}+\cdots \\&\left. +\,f_I\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}\frac{\log ^I p_1}{\log ^IK}\right) , \end{aligned}$$
but the former sequence is simpler, so we are able to write the Montgomery–Odlyzko expressions for it explicitly.

As for the limitations of the employed method, in [3] it is shown that \(h(c)<1\) if \(c<\frac{1}{2}\) and \(h(c)>1\) if \(c\ge 6{.}2\) (the authors note that the latter bound can be improved to \(h(c)>1\) if \(c\ge 3{.}74\)) and the length K is \(\le T\). So the value of \(\mu =\frac{1}{2}\) is not attainable with any sequence \(\{a_k\}\) using this method. One may try to use the form of the coefficients \(\{a_k\}\) of the present paper in the method of [6] obtaining \(\lambda >3{.}072\) on the Generalized Riemann hypothesis, but this method is more technical and involved, and the numerical calculations seem to require more computational resources. The best known bound for \(\lambda \) just assuming the Riemann hypothesis is \(\lambda >2{.}9\) due to Bui [1]. It is possible that Feng and Wu’s result \(\lambda >3{.}072\) can also be obtained just assuming the Riemann hypothesis. For another application of Feng’s mollifier, see [8] and the references therein.

2 Lemmas

Lemma 1

Let \(a_i\) be integer for \(1\le i\le m\), \(D>1\) and f is a continuous function. Then
$$\begin{aligned}&\int _{1}^{D}\frac{\log ^{a_1-1}x_1}{x_1}dx_1\int _{1}^{\frac{D}{x_1}}\frac{\log ^{a_2-1}x_2}{x_2}dx_2\cdots \int _1^{\frac{D}{x_1x_2\cdots x_m}}\frac{f(x_1x_2\cdots x_m x)}{x}dx\\&\qquad =\frac{\prod _{i=1}^m(a_i-1)!}{\left( \sum _{i=1}^ma_i\right) !}\int _1^D\frac{f(x)\log ^{\sum _{i=1}^ma_i}x}{x}dx. \end{aligned}$$

Lemma 2

Let \(a_i\) be integer for \(1\le i\le m\), and g is a polynomial. Then we have
$$\begin{aligned} \sum _{k\le K}\frac{d_r(k)^2}{k}g\left( \frac{\log K/k}{\log K}\right)= & {} A_rr^2\int _1^Kg\left( \frac{\log K/x}{\log K}\right) (\log x)^{r^2-1}\frac{dx}{x}\\&+\,O\left( (\log K)^{r^2-1}\right) \end{aligned}$$
and
$$\begin{aligned}&\sum _{p_1p_2\cdots p_m\le K}\prod _{i=1}^{m}\frac{\log ^{a_i}p_i}{p_i}\mu ^2(p_1p_2\cdots p_m) \sum _{k_0\le K/(p_1p_2\dots p_m)}\frac{d_r(k_0)^2}{k_0}g \left( \frac{\log K/(p_1p_2\dots p_mk_0)}{\log K}\right) \\&\quad =(1+O(\log ^{-1}K))A_rr^2 \int _1^K\log ^{a_1-1}x_1\frac{dx_1}{x_1}\int _1^{\frac{K}{x_1}}\log ^{a_2-1}x_2\frac{dx_2}{x_2}\cdots \\&\qquad \times \int _1^{\frac{K}{x_1x_2\cdots x_{m-1}}}\log ^{a_m-1}x_m\frac{dx_m}{x_m}\int _1^{\frac{K}{x_1x_2\cdots x_m}}g\left( \frac{\log K/(x_1x_2\cdots x_mx)}{\log K}\right) (\log x)^{r^2-1}\frac{dx}{x}, \end{aligned}$$
where \(A_r\) is a constant that depends only on r.

For the proof of Lemmas 1 and 2, see [5].

3 Proof of Theorem 1

To give an upper bound for \(\mu \), we evaluate h(c) in (1) with the coefficients
$$\begin{aligned} a_k = \frac{\lambda (k)d_r(k)}{k^{\frac{1}{2}}}f_1\left( \frac{\log K/k}{\log K}\right) +\frac{\lambda (k)d_r(k)}{k^{\frac{1}{2}}}\sum _{p\mid k}P\left( \frac{\log p}{\log K}\right) \tilde{f}_1\left( \frac{\log K/k}{\log K}\right) , \end{aligned}$$
where \(r\ge 1\) and \(f_1\), \(\tilde{f}_1\), P are polynomials.
First, we evaluate the denominator in the ratio in the definition of h(c).
$$\begin{aligned} \sum _{k\le K}\left| a_k\right| ^2= & {} \sum _{k\le K}\frac{d_r(k)^2}{k}f_1\left( \frac{\log K/k}{\log K}\right) ^2\\&+2\sum _{k\le K}\frac{d_r(k)^2}{k}f_1\left( \frac{\log K/k}{\log K}\right) \tilde{f}_1\left( \frac{\log K/k}{\log K}\right) \sum _{p\mid k}P\left( \frac{\log p}{\log K}\right) \\&+\sum _{k\le K}\frac{d_r(k)^2}{k}\tilde{f}_1\left( \frac{\log K/k}{\log K}\right) ^2\sum _{p\mid k}P\left( \frac{\log p}{\log K}\right) \sum _{q\mid k}P\left( \frac{\log q}{\log K}\right) \\= & {} \tilde{D}_1+\tilde{D}_2+\tilde{D}_3. \end{aligned}$$
Using Lemma 2 and recalling that \(K=T(\log T)^{-2}\), we have
$$\begin{aligned} \tilde{D}_1= & {} A_rr^2\int _1^Kf_1\left( \frac{\log K/x}{\log K}\right) ^2(\log x)^{r^2-1}\frac{dx}{x}+O\left( (\log T)^{r^2-1}\right) \nonumber \\= & {} A_rr^2(\log K)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)^2du+O\left( (\log T)^{r^2-1}\right) \nonumber \\= & {} A_rr^2(\log T)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)^2du+O\left( (\log T)^{r^2-1+\epsilon }\right) , \end{aligned}$$
(2)
where \(\varepsilon >0\) is arbitrarily small and the constant in the O-term depends on r, \(\varepsilon \) and \(f_1\). By Lemma 2 we obtain that
$$\begin{aligned} \tilde{D}_2= & {} \frac{2A_rr^4}{\log K}\int _1^K\frac{P_1(\log x_1)}{x_1}\int _1^{\frac{K}{x_1}}f_1\left( \frac{\log K/x_1x}{\log K}\right) \\&\times \,\,\tilde{f}_1\left( \frac{\log K/x_1x}{\log K}\right) (\log x)^{r^2-1}\frac{dx}{x}dx_1+O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
where \(P_1(y)=\frac{P(y)}{y}\). By the variable changes \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_1x}{\log K}\), we have
$$\begin{aligned} \tilde{D}_2= & {} 2A_rr^4(\log K)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}f_1(v)\tilde{f}_1(v)dvdu\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) \nonumber \\= & {} 2A_rr^4(\log T)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}f_1(v)\tilde{f}_1(v)dvdu\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(3)
where the constant in the O-term depends on r, \(\varepsilon \) and \(f_1\), \(\tilde{f}_1\).
We have
$$\begin{aligned} \tilde{D}_3= & {} \sum _{k\le K}\frac{d_r(k)^2}{k}\tilde{f}_1\left( \frac{\log K/k}{\log K}\right) ^2 \sum _{p_1p_2\mid k}\mu ^2(p_1p_2)P\left( \frac{\log p_1}{\log K}\right) P\left( \frac{\log p_2}{\log K}\right) \\&+\sum _{k\le K}\frac{d_r(k)^2}{k}\tilde{f}_1\left( \frac{\log K/k}{\log K}\right) ^2\sum _{p\mid k}P^2\left( \frac{\log p}{\log K}\right) \\= & {} \tilde{D}_{31}+\tilde{D}_{32}. \end{aligned}$$
Again by Lemma 2 we obtain that
$$\begin{aligned} \tilde{D}_{31}= & {} \frac{A_rr^6}{\log ^2K}\int _1^K\frac{P_1(\log x_1)}{x_1}\int _1^{\frac{K}{x_1}}\frac{P_1(\log x_2)}{x_2} \int _1^{\frac{K}{x_1x_2}}\tilde{f}_1\left( \frac{\log K/x_1x_2x}{\log K}\right) ^2\\&\times \,(\log x)^{r^2-1}\frac{dx}{x}dx_2dx_1+O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
where \(P_1(y)=\frac{P(y)}{y}\). We remark that by Lemma 1 we can reduce the number of the repeated integrations in the above expression. By the change of variables \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_2}{\log K}\), \(w=1-\frac{\log x_1x_2x}{\log K}\),
$$\begin{aligned} \tilde{D}_{31}= & {} A_rr^6(\log K)^{r^2}\int _0^1P_1(1-u)\int _{1-u}^1P_1(1-v)\nonumber \\&\times \,\int _0^{u+v-1}(u+v-w-1)^{r^2-1}\tilde{f}_1(w)^2dw\,dv\,du\nonumber \\&+\,O\left( (\log T)^{r^2-1+\epsilon }\right) \nonumber \\= & {} A_rr^6(\log T)^{r^2}\int _0^1P_1(1-u)\int _{1-u}^1P_1(1-v)\nonumber \\&\times \int _0^{u+v-1}(u+v-w-1)^{r^2-1}\tilde{f}_1(w)^2dw\,dv\,du\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(4)
where the constant in the O-term depends on r, \(\varepsilon \) and \(\tilde{f}_1\). Similarly,
$$\begin{aligned} \tilde{D}_{32}=&A_rr^4(\log K)^{r^2}\int _0^1P_2(1-u)\int _0^u(u-v)^{r^2-1}\tilde{f}_1(v)^2dv\,du+O\left( (\log T)^{r^2-1+\varepsilon }\right) \nonumber \\ =&A_rr^4(\log T)^{r^2}\int _0^1P_2(1-u)\int _0^u(u-v)^{r^2-1}\tilde{f}_1(v)^2dv\,du+O\left( (\log T)^{r^2-1+\varepsilon }\right) ,\nonumber \\ \end{aligned}$$
(5)
where \(P_2(y)=\frac{P(y)^2}{y}\) and the constant in the O-term depends on r, \(\varepsilon \) and \(\tilde{f}_1\).
We now proceed to evaluation of the numerator in the ratio in (1). If we let
$$\begin{aligned} N(c)=\sum _{nk\le K}a_ka_{nk}g_c(n)\Lambda (n)n^{-1/2}, \end{aligned}$$
then
$$\begin{aligned} N(c)= & {} \frac{2}{\pi }\sum _{nk\le K}\frac{\lambda (k)d_r(k)\lambda (nk)d_r(nk)\Lambda (n)}{kn\log n}\sin \left( \pi c\frac{\log n}{\log T}\right) \\&\times \left( f_1\left( \frac{\log K/k}{\log K}\right) f_1\left( \frac{\log K/nk}{\log K}\right) \right. \\&+\,\,f_1\left( \frac{\log K/nk}{\log K}\right) \tilde{f}_1\left( \frac{\log K/k}{\log K}\right) \sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) \\&+\,\,f_1\left( \frac{\log K/k}{\log K}\right) \tilde{f}_1\left( \frac{\log K/nk}{\log K}\right) \sum _{p_1\mid nk}P\left( \frac{\log p_1}{\log K}\right) \\&\left. {}+\,\tilde{f}_1\left( \frac{\log K/k}{\log K}\right) \tilde{f}_1\left( \frac{\log K/nk}{\log K}\right) \sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) \sum _{q_1\mid nk}P\left( \frac{\log q_1}{\log K}\right) \right) , \end{aligned}$$
so we can write
$$\begin{aligned} N(c)=N_1+N_2+N_3+N_4. \end{aligned}$$
Using the distribution of \(\Lambda (n)\), we obtain
$$\begin{aligned} N_1= & {} -\frac{2}{\pi }\sum _{pk\le K}\frac{d_r(k)d_r(pk)}{kp}\sin \left( \pi c\frac{\log p}{\log T}\right) f_1\left( \frac{\log K/k}{\log K}\right) f_1\left( \frac{\log K/pk}{\log K}\right) \\&+\;O\left( (\log T)^{r^2-1}\right) \\= & {} -\frac{2r}{\pi }\sum _{p\le K}\frac{\sin \left( \pi c\frac{\log p}{\log T}\right) }{p}\sum _{k\le K/p}\frac{d_r(k)^2}{k}f_1\left( \frac{\log K/k}{\log K}\right) f_1\left( \frac{\log K/pk}{\log K}\right) \\&+\;O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
By Lemma 2 we have
$$\begin{aligned} N_1= & {} -\frac{2A_rr^3}{\pi }\sum _{p\le K}\frac{\sin \left( \pi c\frac{\log p}{\log T}\right) }{p} \int _1^{\frac{K}{p}}f_1\left( \frac{\log K/x}{\log K}\right) f_1\left( \frac{\log K/px}{\log K}\right) (\log x)^{r^2-1}\frac{dx}{x}\\&+\;O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
From the Mertens theorem
$$\begin{aligned} \sum _{p\le y}\frac{\log p}{p}=\log y+O(1) \end{aligned}$$
(6)
and Abel’s summation, we get
$$\begin{aligned} N_1= & {} -\frac{2A_rr^3}{\pi }\int _1^K\frac{\sin \left( \pi c\frac{\log x_1}{\log T}\right) }{x_1\log x_1}\int _1^{\frac{K}{x_1}} f_1\left( \frac{\log K/x}{\log K}\right) f_1\left( \frac{\log K/xx_1}{\log K}\right) (\log x)^{r^2-1}\frac{dx}{x}dx_1\\&+\,O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
Interchanging the order of integration and the names of the variables x and \(x_1\), we find
$$\begin{aligned} N_1= & {} -\frac{2A_rr^3}{\pi }\int _1^Kf_1\left( \frac{\log K/x_1}{\log K}\right) \frac{(\log x_1)^{r^2-1}}{x_1}\int _1^{\frac{K}{x_1}} \frac{\sin \left( \pi c\frac{\log x}{\log T}\right) }{\log x}f_1\left( \frac{\log K/xx_1}{\log K}\right) \frac{dx}{x}dx_1\\&+\;O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
Let \(u=1-\frac{\log x_1}{\log K}\), \(v=\frac{\log x}{\log K}\). Then
$$\begin{aligned} N_1= & {} -\frac{2A_rr^3}{\pi }(\log K)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)\int _0^u\frac{\sin \left( \pi cv\frac{\log K}{\log T}\right) }{v}f_1(u-v)\,dv\,du\nonumber \\&+\;O\left( (\log T)^{r^2-1}\right) \nonumber \\= & {} -\frac{2A_rr^3}{\pi }(\log T)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)\int _0^u\frac{\sin (\pi cv)}{v}f_1(u-v)\,dv\,du\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(7)
where the constant in the O-term depends on r, \(\varepsilon \) and \(f_1\).
In \(N_2\) we can replace the product of the summation variables nk by \(pp_1k_0\) to get
$$\begin{aligned} N_2= & {} -\frac{2r^3}{\pi }\sum _{p_1\le K}\frac{P\left( \frac{\log p_1}{\log K}\right) }{p_1}\sum _{pk_0\le K/p_1}\frac{\sin \left( \pi c\frac{\log p}{\log T}\right) d_r(k_0)^2}{pk_0}\\&\times \, f_1\left( \frac{\log K/(pp_1k_0)}{\log K}\right) \tilde{f}_1\left( \frac{\log K/(p_1k_0)}{\log K}\right) +O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
The inner sum \(\sum _{pk_0\le K/p_1}\) in the expression above is the sum \(\sum _{k_0\le K/p_1}\sum _{p\le K/(p_1k_0)}\). As in the calculation of \(N_1\), we can show that this double sum is
$$\begin{aligned}&A_rr^2\int _1^{\frac{K}{p_1}}\tilde{f}_1\left( \frac{\log K/(p_1x_2)}{\log K}\right) \frac{(\log x_2)^{r^2-1}}{x_2}\\&\times \int _1^{\frac{K}{p_1x_2}} \frac{\sin \left( \pi c\frac{\log x}{\log T}\right) }{\log x}f_1\left( \frac{\log K/(p_1xx_2)}{\log K}\right) \frac{dx}{x}dx_2 +O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
By (6) we obtain
$$\begin{aligned} N_2= & {} -\frac{2A_rr^5}{\pi (\log K)}\int _1^K\frac{P_1\left( \frac{\log x_1}{\log K}\right) }{x_1} \int _1^{\frac{K}{x_1}}\tilde{f}_1\left( \frac{\log K/(x_1x_2)}{\log K}\right) \frac{(\log x_2)^{r^2-1}}{x_2}\\&\times \int _1^{\frac{K}{x_1x_2}}\frac{\sin \left( \pi c\frac{\log x}{\log T}\right) }{\log x}f_1\left( \frac{\log K/(xx_1x_2)}{\log K}\right) \frac{dx}{x}dx_2\,dx_1+O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
Making the variable changes \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_1x_2}{\log K}\), \(w=\frac{\log x}{\log K}\), we get
$$\begin{aligned} N_2= & {} -\frac{2A_rr^5}{\pi }(\log T)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}\tilde{f}_1(v) \nonumber \\&\times \int _0^v\frac{\sin (\pi cw)}{w}f_1(v-w)\,dw\,dv\,du +O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(8)
where the constant in the O-term depends on r, \(\varepsilon \) and \(f_1\), \(\tilde{f}_1\).
As in \(N_1\) and \(N_2\), the terms with \(n=p\) for the primes p give the main contribution to \(N_3\):
$$\begin{aligned} N_3= & {} -\frac{2}{\pi }\sum _{pk\le K}\sin \left( \pi c\frac{\log p}{\log T}\right) \frac{d_r(k)d_r(kp)}{kp}f_1\left( \frac{\log K/k}{\log K}\right) \tilde{f}_1\left( \frac{\log K/(pk)}{\log K}\right) \\&\times \sum _{p_1\mid pk}P\left( \frac{\log p_1}{\log K}\right) +O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
For \((p,k)=1\) it follows that
$$\begin{aligned} \sum _{p_1\mid pk}P\left( \frac{\log p_1}{\log K}\right) =\sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) +P\left( \frac{\log p}{\log K}\right) . \end{aligned}$$
(9)
Since the contribution of the terms with \((p,k)\ne 1\) in \(N_3\) is \(O\left( (\log T)^{r^2-1}\right) \), then, according to decomposition (9), we can write
$$\begin{aligned} N_3=N_{31}+N_{32}+O\left( (\log T)^{r^2-1}\right) , \end{aligned}$$
where
$$\begin{aligned} N_{31}= & {} -\frac{2r^3}{\pi }\sum _{p_1\le K}\frac{P\left( \frac{\log p_1}{\log K}\right) }{p_1} \sum _{pk_0\le K/p_1}\frac{\sin \left( \pi c\frac{\log p}{\log T}\right) d_r(k_0)^2}{pk_0}\\&\times \,\tilde{f}_1\left( \frac{\log K/(pp_1k_0)}{\log K}\right) f_1\left( \frac{\log K/(p_1k_0)}{\log K}\right) +O\left( (\log T)^{r^2-1}\right) \end{aligned}$$
and
$$\begin{aligned} N_{32}= & {} -\frac{2r}{\pi }\sum _{pk\le K}\frac{\sin \left( \pi c\frac{\log p}{\log T}\right) d_r(k)^2P\left( \frac{\log p}{\log K}\right) }{pk}\tilde{f}_1\left( \frac{\log K/(pk)}{\log K}\right) \\&\times \, f_1\left( \frac{\log K/k}{\log K}\right) +O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
As in the calculation of \(N_2\) we get
$$\begin{aligned} N_{31}= & {} -\frac{2A_rr^5}{\pi }(\log T)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}f_1(v)\\&\times \,\int _0^v\frac{\sin (\pi cw)}{w}\tilde{f}_1(v-w)\,dw\,dv\,du+O\left( (\log T)^{r^2-1+\epsilon }\right) , \end{aligned}$$
and as in the calculation of \(N_1\),
$$\begin{aligned} N_{32}= & {} -\frac{2A_rr^3}{\pi }(\log T)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)\int _0^u\sin (\pi cv)P_1(v)\tilde{f}_1(u-v)\,dv\,du\\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
where \(P_1(y)=\frac{P(y)}{y}\).
Thus,
$$\begin{aligned} N_3= & {} -\frac{2A_rr^5}{\pi }(\log T)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}f_1(v)\nonumber \\&\times \int _0^v\frac{\sin (\pi cw)}{w}\tilde{f}_1(v-w)\,dw\,dv\,du\nonumber \\&-\,\frac{2A_rr^3}{\pi }(\log T)^{r^2}\int _0^1(1-u)^{r^2-1}f_1(u)\int _0^u\sin (\pi cv)P_1(v)\tilde{f}_1(u-v)\,dv\,du\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(10)
where the constant in the O-term depends on r, \(\varepsilon \) and \(f_1\), \(\tilde{f}_1\), P.
Again, in the sum defining \(N_4\) we can replace the integers \(n\ge 2\) with the primes p:
$$\begin{aligned} N_4= & {} -\frac{2}{\pi }\sum _{pk\le K}\sin \left( \pi c\frac{\log p}{\log T}\right) \frac{d_r(k)d_r(kp)}{kp}\tilde{f}_1\left( \frac{\log K/k}{\log K}\right) \tilde{f}_1\left( \frac{\log K/(pk)}{\log K}\right) \\&\times \sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) \sum _{q_1\mid pk}P\left( \frac{\log q_1}{\log K}\right) +O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
For the two innermost sums, if \((k,p)=1\), we have
$$\begin{aligned} \sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) \sum _{q_1\mid pk}P\left( \frac{\log q_1}{\log K}\right)= & {} \sum _{p_1q_1\mid k}\mu ^2(p_1q_1)P\left( \frac{\log p_1}{\log K}\right) P\left( \frac{\log q_1}{\log K}\right) \\&+\sum _{p_1\mid k}P^2\left( \frac{\log p_1}{\log K}\right) +P\left( \frac{\log p}{\log K}\right) \sum _{p_1\mid k}P\left( \frac{\log p_1}{\log K}\right) . \end{aligned}$$
According to this decomposition, we write
$$\begin{aligned} N_4=N_{41}+N_{42}+N_{43}. \end{aligned}$$
As before, by Lemma 2 we find
$$\begin{aligned} N_{41}= & {} -\frac{2A_rr^7}{\pi (\log K)^2}\int _1^K\frac{P_1\left( \frac{\log x_1}{\log K}\right) }{x_1} \int _1^{\frac{K}{x_1}}\frac{P_1\left( \frac{\log x_2}{\log K}\right) }{x_2} \\&\times \int _1^{\frac{K}{x_1x_2}}\tilde{f}_1\left( \frac{\log K/(x_1x_2x_3)}{\log K}\right) \frac{(\log x_3)^{r^2-1}}{x_3}\\&\times \int _1^{\frac{K}{x_1x_2x_3}}\frac{\sin \left( \pi c\frac{\log x}{\log T}\right) }{\log x}\tilde{f}_1\left( \frac{\log K/(xx_1x_2x_3)}{\log K}\right) \frac{dx}{x}dx_3\,dx_2\,dx_1\\&+\,O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
Making the variable changes \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_2}{\log K}\), \(w=1-\frac{\log x_1x_2x_3}{\log K}\), \(z=\frac{\log x}{\log K}\), we get
$$\begin{aligned} N_{41}= & {} -\frac{2A_rr^7}{\pi }(\log T)^{r^2}\int _0^1P_1(1-u)\int _{1-u}^1P_1(1-v)\nonumber \\&\times \int _0^{u+v-1}(u+v-w-1)^{r^2-1}\tilde{f}_1(w)\int _0^w\frac{\sin (\pi cz)}{z}\tilde{f}_1(w-z)\,dz\,dw\,dv\,du\nonumber \\&+\,O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(11)
where the constant in the O-term depends on r, \(\varepsilon \) and \(\tilde{f}_1\), P.
Next,
$$\begin{aligned} N_{42}= & {} -\frac{2A_rr^5}{\pi (\log K)}\int _1^K\frac{P_2\left( \frac{\log x_1}{\log K}\right) }{x_1} \int _1^{\frac{K}{x_1}}\tilde{f}_1\left( \frac{\log K/(x_1x_2)}{\log K}\right) \frac{(\log x_2)^{r^2-1}}{x_2}\\&\times \int _1^{\frac{K}{x_1x_2}}\frac{\sin \left( \pi c\frac{\log x}{\log T}\right) }{\log x}\tilde{f}_1\left( \frac{\log K/(xx_1x_2)}{\log K}\right) \frac{dx}{x}dx_2\,dx_1+\,O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
By the variable changes \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_1x_2}{\log K}\), \(w=\frac{\log x}{\log K}\), we get
$$\begin{aligned} N_{42}= & {} -\frac{2A_rr^5}{\pi }(\log T)^{r^2}\int _0^1P_2(1-u)\int _0^u(u-v)^{r^2-1}\tilde{f}_1(v)\nonumber \\&\times \int _0^v\frac{\sin (\pi cw)}{w}\tilde{f}_1(v-w)\,dw\,dv\,du+O\left( (\log T)^{r^2-1+\varepsilon }\right) , \end{aligned}$$
(12)
where the constant in the O-term depends on r, \(\epsilon \) and \(\tilde{f}_1\), P.
Finally,
$$\begin{aligned} N_{43}= & {} -\frac{2A_rr^5}{\pi (\log K)^2}\int _1^K\frac{P_1\left( \frac{\log x_1}{\log K}\right) }{x_1} \int _1^{\frac{K}{x_1}}\tilde{f}_1\left( \frac{\log K/(x_1x_2)}{\log K}\right) \frac{(\log x_2)^{r^2-1}}{x_2}\\&\times \int _1^{\frac{K}{x_1x_2}}\sin \left( \pi c\frac{\log x}{\log T}\right) P_1\left( \frac{\log x}{\log K}\right) \tilde{f}_1\left( \frac{\log K/(xx_1x_2)}{\log K}\right) \frac{dx}{x}dx_2\,dx_1\\&+\,O\left( (\log T)^{r^2-1}\right) . \end{aligned}$$
By the variable changes \(u=1-\frac{\log x_1}{\log K}\), \(v=1-\frac{\log x_1x_2}{\log K}\), \(w=\frac{\log x}{\log K}\), we get
$$\begin{aligned} N_{43}= & {} -\frac{2A_rr^5}{\pi }(\log T)^{r^2}\int _0^1P_1(1-u)\int _0^u(u-v)^{r^2-1}\tilde{f}_1(v)\nonumber \\&\times \int _0^v\sin (\pi cw)P_1(w)\tilde{f}_1(v-w)\,dw\,dv\,du+O\left( (\log T)^{r^2-1+\epsilon }\right) , \end{aligned}$$
(13)
where the constant in the O-term depends on r, \(\epsilon \) and \(\tilde{f}_1\), P.
Using \(D_i\), \(N_i\) given by (2)–(13) we can evaluate (note the sign change in comparison with h(c) for \(\mu \) in [5])
$$\begin{aligned} h(c)=c-\frac{N_1+N_2+N_3+N_4}{D_1+D_2+D_3}. \end{aligned}$$
The results of our numerical calculations are summarized in Tables 1 and 2. To find the coefficients of the numerically optimal polynomials, we perform one iteration of Newton’s method for multidimensional optimization, using in the initial vector the coefficients found by Feng and Wu [5]. To obtain the bound for \(\lambda \) in Theorem 1, we use the relaxed Newton’s method with the step size multiplier equal to 0.01.
Table 1

Numerically optimal polynomials in the coefficients \(\{a_k\}\), for which \(h(c)>1\)

Degrees

Value of \(\varvec{c}\)

Value of \(\varvec{r}\)

Polynomials

\(\varvec{f}_{\mathbf {1}}\)

\(\tilde{\varvec{f}}_{\mathbf {1}}\)

\(\varvec{P}\)

\(\varvec{f}_{\mathbf {1}}\)

\(\tilde{\varvec{f}}_{\mathbf {1}}\)

\(\varvec{P}\)

3

1

2

0.515398

1.18

\(1{.}95+1{.}47x-1{.}07x^2-0{.}29x^3\)

\(-0{.}7-1{.}92x\)

\(x^2\)

3

1

3

0.515397

1.18

\(1{.}655+1{.}25x-0{.}886x^2-0{.}25x^3\)

\(-0{.}57-1{.}6x\)

\(x^2+0{.}036x^3\)

6

2

3

0.515396

1.18

\(1{.}78+1{.}017x+0{.}2x^2-1{.}56x^3+0{.}45x^4-0{.}06x^5+0{.}05x^6\)

\(-0{.}629-0{.}88x-1{.}799x^2\)

\(x^2+0{.}083x^3\)

Table 2

Numerically optimal polynomials in the coefficients \(\{a_k/\lambda (k)\}\), for which \(h(c)<1\) [(see (1)]

Degrees

Value of \(\varvec{c}\)

Value of \(\varvec{r}\)

Polynomials

\(\varvec{f}_{\mathbf {1}}\)

\(\tilde{\varvec{f}}_{\mathbf {1}}\)

\(\varvec{P}\)

\(\varvec{f}_{\mathbf {1}}\)

\(\tilde{\varvec{f}}_{\mathbf {1}}\)

\(\varvec{P}\)

5

3

2

2.73272

2.6

\(1{.}02+10{.}96x+9{.}29x^2-22{.}3x^3-26{.}18x^4+34{.}45x^5\)

\(-4{.}56-63{.}02x-42{.}72x^2-34{.}45x^3\)

\(x^2\)

6

3

3

2.7328

2.6

\(1{.}08+11{.}79x+8{.}61x^2-21{.}95x^3-24{.}58x^4+27{.}79x^5+5{.}03x^6\)

\(-4{.}89-69{.}11x-36{.}7x^2-51{.}28x^3\)

\(x^2-0{.}044x^3\)

Declarations

Open Access

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Acknowledgements

The author wishes to thank the anonymous referee for careful reading of this paper and helpful suggestions, and Professor Ken Ono for his encouragement.

Authors’ Affiliations

(1)
Department of Mathematical Analysis, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University

References

  1. Bui, H.M.: Large gaps between consecutive zeros of the Riemann zeta-function II. Acta Arith. 165, 101–122 (2014)MathSciNetView ArticleMATHGoogle Scholar
  2. Bui, H.M., Milinovich, M.B., Ng, N.: A note on the gaps between consecutive zeros of the Riemann zeta-function. Proc. Amer. Math. Soc. 138(12), 4167–4175 (2010)MathSciNetView ArticleMATHGoogle Scholar
  3. Conrey, J.B., Ghosh, A., Gonek, S.M.: A note on gaps between zeros of the zeta function. Bull. Lond. Math. Soc. 16, 421–424 (1984)MathSciNetView ArticleMATHGoogle Scholar
  4. Conrey, J.B., Iwaniec, H.: Spacing of zeros of Hecke \(L\)-functions and the class number problem. Acta Arith. 103(3), 259–312 (2002)MathSciNetView ArticleMATHGoogle Scholar
  5. Feng, S., Wu, X.: On gaps between zeros of the Riemann zeta-function. J. Number Theor. 132, 1385–1397 (2012)MathSciNetView ArticleMATHGoogle Scholar
  6. Feng, S., Wu, X.: On large spacing of the zeros of the Riemann zeta-function. J. Number Theor. 133, 2538–2566 (2013)MathSciNetView ArticleMATHGoogle Scholar
  7. Hall, R.R.: A new unconditional result about large spaces between zeta zeros. Mathematika 52, 101–113 (2005)MathSciNetView ArticleMATHGoogle Scholar
  8. Kühn, P., Robles, N., Zeindler, D.: On a mollifier of the perturbed Riemann zeta-function, preprint, arXiv:1605.02604 [math.NT]
  9. Montgomery, H. L.: The pair correlation of the zeros of the zeta function. In: Proceedings Symposium Pure Mathematics 24, A.M.S., Providence 1973, 181–193Google Scholar
  10. Montgomery, H.L., Odlyzko, A.M.: Gaps between zeros of the zeta function. In: Topics in Classical Number Theory, Colloq. Math. Soc. Jãnos Bolyai, Budapest 1981, vol. 34, pp. 1079–1106. North-Holland, Amsterdam (1984)Google Scholar
  11. Montgomery, H.L., Weinberger, P.J.: Notes on small class numbers. Acta Arith. 24, 529–542 (1974)MathSciNetMATHGoogle Scholar
  12. Selberg, A.: The zeta-function and the Riemann Hypothesis. Skandinaviske Matematikerkongres 10, 187–200 (1946)MathSciNetGoogle Scholar

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