Nonreal zeros of integrated Eisenstein Fourier coefficients

1. The question and the result

Lagarias and Suzuki [1, Section 5, equations (64)–(65)] ask whether integration in the cusp preserves the critical-line property of the nonconstant Fourier coefficients of the completed Eisenstein series for $\operatorname{PSL}_2(\mathbb Z)$. Their special-function question is whether, for every $a>0$, the entire function

$$ G_a(z)=\int_a^\infty u^{-3/2}K_{iz}(u)\,du \tag{1.1} $$

has only real zeros. Here

$$ K_\nu(u)=\int_0^\infty e^{-u\cosh v}\cosh(\nu v)\,dv, \qquad u>0. $$

We give a negative answer for every $a>2/\pi$. The proof identifies a nonoscillatory term in the incomplete Mellin integral which is of the same size as its oscillatory endpoint term. Their balance also gives the asymptotic locations of nonreal zeros and a transition at $a=2/\pi$.

For $\operatorname{Re}z>0$, use the principal logarithm and put

$$ \begin{aligned} A_a&=\sqrt{\frac{2\pi}{a}}, & \delta_a&=\frac{a^2}{4}+\frac5{12},\\ \theta_a(z)&=z\log\frac{2z}{a}-z+\frac\pi4, & \psi_a(z)&=\theta_a(z)+\frac{\delta_a}{z}. \end{aligned} \tag{1.2} $$

For each sufficiently large positive integer $m$, let $b_m$ be the large positive solution of $\psi_a(b_m)=2\pi m$, and write

$$ L_m=\psi_a'(b_m)=\log\frac{2b_m}{a}-\frac{\delta_a}{b_m^2}. $$

The phase is strictly increasing for all sufficiently large arguments, so these definitions are unambiguous. Also $b_m\sim2\pi m/\log m$.

Theorem 1. Fix $a>2/\pi$ and set

$$ \eta_a=\operatorname{arcosh}\sqrt{\frac{\pi a}{2}}>0. $$

Then $G_a(t)>0$ for all sufficiently large real $|t|$. Nevertheless, $G_a$ has infinitely many nonreal zeros. More precisely, for every sufficiently large $m$ there are simple conjugate zeros $z_m^\pm$ with

$$ z_m^\pm=b_m\pm\frac{i\eta_a}{L_m} +O_a\!\left(\frac1{b_mL_m^3}+\frac1{b_m^2L_m}\right). \tag{1.3} $$

If $w_m^\pm$ is the local inverse-phase solution $\psi_a(w_m^\pm)=2\pi m\pm i\eta_a$ near $b_m$, the sharper statement is

$$ z_m^\pm=w_m^\pm+O_a\!\left(\frac1{b_m^2L_m}\right). \tag{1.4} $$

In particular, the distance of these zeros from the real axis is asymptotic to $\eta_a/\log(2b_m/a)$.

The change at the threshold is visible without any numerical zero search.

Theorem 2. If $0<a<2/\pi$, put $\beta_a=\arccos\sqrt{\pi a/2}$. For each sufficiently large $m$, $G_a$ has a simple real zero within $O_a((b_m^2L_m)^{-1})$ of each of the two real solutions

$$ \psi_a(t)=2\pi m\pm\beta_a. $$

At $a=2/\pi$, let $c_0=3/16+3/\pi^2$. For each sufficiently large $m$, there are simple real zeros

$$ t_m^\pm=b_m\pm\frac{\sqrt{2c_0}}{b_mL_m} +O\!\left(\frac1{b_m^2L_m}\right). \tag{1.5} $$

Theorem 2 asserts the existence and location of these real zeros; it does not classify every zero for $a\leq2/\pi$. All constants concern fixed $a$. No uniformity as $a\to2/\pi$ or as $a$ grows with $m$ is asserted.

2. Consequence for the Eisenstein coefficients and prior work

In the normalization of [1], the nonconstant coefficient with integer $\ell\ne0$ is

$$ a_\ell(y,s)=2|\ell|^{s-1/2}\sigma_{1-2s}(|\ell|) \sqrt y\,K_{s-1/2}(2\pi|\ell|y), \qquad \sigma_w(n)=\sum_{d\mid n}d^w. $$

The twisted cusp integral in [1, equation (64)] is therefore

$$ \begin{aligned} J_\ell(T,s) &=\int_T^\infty a_\ell(y,s)\,\frac{dy}{y^2}\\ &=2|\ell|^{s-1/2}\sigma_{1-2s}(|\ell|)\sqrt{2\pi|\ell|} G_{2\pi|\ell|T}\bigl(-i(s-1/2)\bigr). \tag{2.1} \end{aligned} $$

The divisor factor has zeros only on $\operatorname{Re}s=1/2$. Indeed, for $p^e\Vert|\ell|$, its local factor is a finite geometric sum in $p^{1-2s}$, whose zeros have modulus one. The factor $|\ell|^{s-1/2}$ never vanishes. Thus every off-line zero supplied by Theorem 1 remains an off-line zero in (2.1).

Corollary 3. For every integer $\ell\ne0$ and every $T\geq1$, $J_\ell(T,s)$ has infinitely many zeros off $\operatorname{Re}s=1/2$. There are conjugate phase families approaching that line with distance asymptotic to

$$ \frac{\operatorname{arcosh}(\pi\sqrt{|\ell|T})} {\log(b_m/(\pi|\ell|T))}. $$

For sufficiently large $m$, the zeros in these families lie in $0<\operatorname{Re}s<1$.

The unintegrated $K_{iz}(a)$ has only real zeros, a classical theorem reviewed in [1]. Lagarias and Suzuki prove critical-line results for the constant coefficient and its cusp integral. Their Section 5 asks whether such a conclusion extends to the integrated nonconstant coefficients. The constant coefficient and the coefficients in (2.1) are different families, and Corollary 3 does not contradict their theorems.

Suzuki's additional note [2] treats polynomially modified constant terms $p(s)\zeta^*(2s)y^s+p(1-s)\zeta^*(2-2s)y^{1-s}$. It gives bounds on their off-line zeros and does not treat the incomplete Bessel integral (1.1). The contribution here is an answer to [1, equation (65)], together with the cutoff threshold and the asymptotic displacement of the zeros. The special-function identities used to prove it are classical [3]; no novelty is claimed for the complete Mellin transform or Stirling's formula.

This result limits one proposed transfer of positivity from individual Bessel coefficients to automorphic integrals. It gives no off-line zero of the Riemann zeta function and no new zero-free region for that function.

3. Entire continuation and the incomplete Mellin identity

Absolute convergence permits changing the order of integration in (1.1), uniformly for $z$ in compact sets. Consequently

$$ G_a(z)=\int_0^\infty q_a(v)\cos(zv)\,dv, \qquad q_a(v)=\int_a^\infty u^{-3/2}e^{-u\cosh v}\,du>0. \tag{3.1} $$

Since

$$ q_a(v)\leq a^{-3/2}\frac{e^{-a\cosh v}}{\cosh v}, $$

$G_a$ is an even real entire function. For $R\geq1$, using $\cosh v\geq e^v/2$ and substituting $w=ae^v/2$ gives

$$ \max_{|z|\leq R}|G_a(z)| \leq a^{-3/2}\left(\frac2a\right)^R\Gamma(R) \leq C_a\left(\frac2a\right)^R\Gamma(R+1). \tag{3.2} $$

Thus $G_a$ has order at most one. We will use this fact for an independent existence proof of nonreal zeros.

Introduce a complex Mellin exponent $\mu$ and write $\nu=iz$. Initially for $\operatorname{Re}\mu>|\operatorname{Re}\nu|$,

$$ \int_0^\infty u^{\mu-1}K_\nu(u)\,du =2^{\mu-2}\Gamma\!\left(\frac{\mu+\nu}{2}\right) \Gamma\!\left(\frac{\mu-\nu}{2}\right). \tag{3.3} $$

For completeness, integrating in $u$ first gives $\Gamma(\mu)\int_0^\infty\cosh(\nu v)(\cosh v)^{-\mu}\,dv$. Extend to the real line and put $r=e^{2v}$; the resulting beta integral proves (3.3). This is also [3, equation 10.43.19].

The connection formula between $K_\nu$ and $I_{\pm\nu}$, together with the power series for $I_\nu$ [3, equations 10.27.4 and 10.25.2], gives for $\nu\notin\mathbb Z$

$$ \begin{aligned} K_\nu(u)={}&\frac{\Gamma(\nu)}2(u/2)^{-\nu} \sum_{k\geq0}\frac{(u^2/4)^k}{k!(1-\nu)_k}\\ &+\frac{\Gamma(-\nu)}2(u/2)^\nu \sum_{k\geq0}\frac{(u^2/4)^k}{k!(1+\nu)_k}. \tag{3.4} \end{aligned} $$

Integrate this normally convergent expansion on $(0,a)$ in the initial convergence region and subtract from (3.3). Both sides then continue meromorphically in $\mu$. The upper incomplete integral is entire in $\mu$ and $\nu$, since $a>0$ removes the small-argument singularity. At $\mu=-1/2$ this proves the exact identity

$$ \begin{aligned} G_a(z)={}&2^{-5/2}\Gamma(-1/4+\nu/2)\Gamma(-1/4-\nu/2)\\ &-\frac1{2\sqrt a}\sum_{\epsilon=\pm1} \Gamma(-\epsilon\nu)(a/2)^{\epsilon\nu}S_\epsilon(z), \tag{3.5}\\ S_\epsilon(z)&=\sum_{k\geq0} \frac{v_a^k}{k!(1+\epsilon\nu)_k(2k-1/2+\epsilon\nu)}, \qquad v_a=\frac{a^2}{4}. \tag{3.6} \end{aligned} $$

Apparent poles at exceptional parameters are removable in the complete right side. Our asymptotic use will have $\operatorname{Re}z>0$, so none of the displayed denominators vanishes.

An explicit tail estimate will be useful. If $z=x+iy$ with $x>0$, then $|j+\epsilon iz|\geq x$ and $|2k-1/2+\epsilon iz|\geq x$. For every integer $r\geq1$,

$$ \left|\sum_{k\geq r} \frac{v_a^k}{k!(1+\epsilon iz)_k(2k-1/2+\epsilon iz)}\right| \leq\frac1x\sum_{k\geq r}\frac{(v_a/x)^k}{k!} \leq\frac{e^{v_a/x}v_a^r}{r!x^{r+1}}. \tag{3.7} $$

This bound controls the entire incomplete series before any large-order approximation.

4. The uniform asymptotic expansion

Set

$$ F_a(z)=e^{\pi z/2}z^{3/2}G_a(z),\qquad B_a=3v_a-\frac18. $$

Lemma 4. Fix $a>0$ and $B>0$. Uniformly as $x=\operatorname{Re}z\to\infty$ in $|\operatorname{Im}z|\leq B$,

$$ \begin{aligned} F_a(z)={}&\pi\left(1-\frac5{16z^2}\right) -A_a\left(1+\frac{B_a}{z^2}\right)\cos\psi_a(z)\\ &+O_{a,B}\!\left( \frac{1+|e^{i\theta_a(z)}|+|e^{-i\theta_a(z)}|}{x^3} \right). \tag{4.1} \end{aligned} $$

In particular,

$$ F_a(z)=\pi-A_a\cos\psi_a(z) +O_{a,B}\!\left( \frac{1+|e^{i\theta_a(z)}|+|e^{-i\theta_a(z)}|}{x^2} \right). \tag{4.2} $$

Proof. Keeping the first three terms of $S_-$ and using (3.7) with $r=3$, elementary expansion of the finite rational expressions gives

$$ S_-(z)=\frac{i}{z}\left[ 1+\frac{i(v_a+1/2)}z +\frac{-1/4+5v_a/2-v_a^2/2}{z^2}+O_{a,B}(x^{-3})\right]. \tag{4.3} $$

The plus branch is its analytic conjugate counterpart: replace $i$ by $-i$ in the coefficients and the leading factor. All errors are uniform in the stated strip.

Stirling's expansion, uniformly in closed sectors avoiding the negative real axis [3, Section 5.11], gives

$$ \Gamma(iz)=\sqrt{2\pi}\,e^{-\pi z/2}z^{-1/2} e^{i(z\log z-z-\pi/4)} \left[1-\frac{i}{12z}-\frac1{288z^2}+O_B(x^{-3})\right]. \tag{4.4} $$

The counterpart for $\Gamma(-iz)$ again reverses the signs of $i$. The arguments $\pm iz$ remain in such closed sectors. Multiplying (4.3) and (4.4), the linear correction is $i\delta_a/z$ and the quadratic correction is

$$ c_a=-\frac{61}{288}+\frac{31v_a}{12}-\frac{v_a^2}{2}. $$

Since $c_a+\delta_a^2/2=3v_a-1/8=B_a$, the sum of the two endpoint branches in (3.5), after scaling, is

$$ A_a\left(1+\frac{B_a}{z^2}\right)\cos\psi_a(z) +O_{a,B}\!\left( \frac{|e^{i\theta_a(z)}|+|e^{-i\theta_a(z)}|}{x^3}\right). \tag{4.5} $$

For the complete Mellin term, the same Stirling expansion yields

$$ e^{\pi z/2}z^{3/2}2^{-5/2} \Gamma(-1/4+iz/2)\Gamma(-1/4-iz/2) =\pi\left(1-\frac5{16z^2}+O_B(x^{-3})\right). \tag{4.6} $$

For example, the coefficient for $\Gamma(h+iw)\Gamma(h-iw)/(2\pi e^{-\pi w}w^{2h-1})$ is $h(h-1)(2h-1)/(6w^2)$. Substituting $h=-1/4$ and $w=z/2$ gives $-5/(16z^2)$. Combining (4.5) and (4.6) proves (4.1). Removing the explicit quadratic corrections gives (4.2). $\square$

On the real axis, (4.2) also implies

$$ G_a(t)=e^{-\pi t/2}t^{-3/2} \left[\pi-\sqrt{\frac{2\pi}{a}}\cos\theta_a(t)+O_a(t^{-1}) \right],\qquad t\to+\infty. \tag{4.7} $$

The $\pi$ in this formula is the complete-Mellin contribution (4.6). Estimating only the lower endpoint in the integral would miss it.

5. Nonreal zero pairs above the threshold

Assume $a>2/\pi$. Then $\pi>A_a$, and (4.7) proves eventual strict positivity on the real axis. Evenness gives the same conclusion at negative arguments.

There is a short independent proof that nonreal zeros must exist infinitely often. By (3.1), $G_a(0)>0$ and the Riemann–Lebesgue lemma gives $G_a(t)\to0$ for real $|t|\to\infty$. Hence $G_a$ is not a polynomial. If it had finitely many zeros, its order bound (3.2) and Hadamard factorization would give $G_a(z)=e^{Az+C}P(z)$ for a polynomial $P$. Evenness forces $A=0$: otherwise $e^{2Az}=P(-z)/P(z)$ would equate a nonconstant exponential with a rational function. Thus $G_a$ would be a polynomial, a contradiction. It has infinitely many zeros, while eventual positivity leaves only finitely many real zeros.

We next prove the locations and simplicity in Theorem 1. The analytic function $\psi_a$ has derivative $L_m\sim\log b_m$ at $b_m$ and second derivative $O_a(b_m^{-1})$ in a fixed neighborhood of that point. For either sign, the equation

$$ \psi_a(w)=2\pi m\pm i\eta_a $$

has a unique solution near $b_m\pm i\eta_a/L_m$, by the analytic inverse function theorem with the preceding derivative bounds (or by Rouché on the linear Taylor approximation). Taylor's formula gives

$$ w_m^\pm=b_m\pm\frac{i\eta_a}{L_m} +O_a\!\left(\frac1{b_mL_m^3}\right). \tag{5.1} $$

These are simple zeros of $H_a(z)=\pi-A_a\cos\psi_a(z)$, since

$$ H_a'(w_m^\pm)=\pm iA_a\sinh(\eta_a)\psi_a'(w_m^\pm), \qquad |H_a'(w_m^\pm)|\sim A_a\sinh(\eta_a)L_m. \tag{5.2} $$

On a circle of radius $r_m=C/(b_m^2L_m)$ around $w_m^\pm$, Taylor's formula and (5.2) give

$$ |H_a(z)|\geq\frac{A_a\sinh\eta_a}{2}\frac{C}{b_m^2} $$

for all sufficiently large $m$. The quadratic Taylor error is $O_a(L_m^2r_m^2)=O_a(C^2b_m^{-4})$. On these circles, $|\operatorname{Im}\theta_a(z)|$ remains bounded, so (4.2) bounds $|F_a-H_a|$ by $C_a/b_m^2$. Choose $C$ large enough, then $m$ large enough. Rouché's theorem gives exactly one zero of $F_a$ in each disk, counted with multiplicity. It is therefore simple. The scaling from $G_a$ to $F_a$ is nonzero there, so this is a zero of $G_a$.

The disks are disjoint, avoid the real axis, and are interchanged by complex conjugation. The reality of $G_a$ pairs their zeros. This proves (1.4), and (5.1) proves (1.3). $\square$

6. Real zero pairs at and below the threshold

For $0<a<2/\pi$, the two phases $2\pi m\pm\beta_a$ give simple real zeros of the model $\pi-A_a\cos\psi_a$. Their derivatives have magnitude asymptotic to $A_a\sin\beta_a\,L_m>0$. The Rouché argument in Section 5, now with disks centered on the real axis, gives exactly one zero of $G_a$ in each disk of radius $C_a/(b_m^2L_m)$. Conjugation preserves each disk; uniqueness forces its zero to be real. This proves the first assertion of Theorem 2.

At $a=2/\pi$, the leading model has double zeros, so its quadratic correction is essential. Here $A_a=\pi$, and

$$ \frac5{16}+B_a=\frac3{16}+\frac3{\pi^2}=c_0>0. $$

For $w$ in a fixed bounded complex set, put $z=b_m+w/(b_mL_m)$. Then

$$ \psi_a(z)=2\pi m+\frac{w}{b_m}+O\!\left(\frac1{b_m^3L_m^2}\right). $$

Substituting in (4.1) gives, uniformly on such sets,

$$ \frac{b_m^2}{\pi} F_a\!\left(b_m+\frac{w}{b_mL_m}\right) =\frac{w^2}{2}-c_0+O(b_m^{-1}). \tag{6.1} $$

The limiting polynomial has two distinct simple real roots $\pm\sqrt{2c_0}$. Rouché on disks of radius $C/b_m$ around those roots gives one simple zero in each, for a sufficiently large fixed $C$ and then sufficiently large $m$. The corresponding $z$ disks are centered on the real axis and have radius $C/(b_m^2L_m)$. Conjugation again forces their unique zeros to be real. Formula (1.5) follows. $\square$

7. Verification, reproducibility, and limits

The proofs are analytic and do not depend on a numerical zero search. The supplement provides three CPU scripts and fixed diagnostic outputs:

• an evaluation of (3.5) and independent defining-integral quadrature at $a=2\pi$, $z=12$, agreeing to the 75-digit working accuracy;
• four fixed real-frequency comparisons with (4.7), and two nonreal zeros located from the asymptotic seeds, near $40.0114696042+0.7059245510i$ and $82.6443480354+0.5528570334i$;
• one critical-cutoff check at $a=2/\pi$, $m=40$, confirming the sign and separation predicted by (6.1);
• an exact symbolic check of the endpoint-series coefficients in Lemma 4.

These floating calculations check normalization and formulas. They are not interval certificates and are not used to prove the existence of a particular numerical zero. The supplement's README gives dependencies, commands, precision, and expected tolerances; its code is MIT licensed.

Theorem 1 establishes infinitely many nonreal zeros for fixed $a>2/\pi$, with a quantitative approach to the real axis. Theorem 2 describes specified real-zero families for fixed $a\leq2/\pi$. Neither theorem classifies zeros outside these asymptotic families or controls a joint limit in which $a$ varies with the order. The Eisenstein consequence is the negative answer in Corollary 3, not a conclusion about the Riemann zeta function's nontrivial zeros.

References

[1] J. C. Lagarias and M. Suzuki. The Riemann hypothesis for certain integrals of Eisenstein series. Journal of Number Theory 118 (2006), 98–122. Section 5, equations (64)–(65). https://arxiv.org/abs/math/0412039v4 https://doi.org/10.1016/j.jnt.2005.08.010

[2] M. Suzuki. An additional writing on Lagarias–Suzuki (2006). Author-hosted academic note, catalogued 2014. Archived author PDF: https://web.archive.org/web/20170807231019/http://www.math.titech.ac.jp:80/~msuzuki/add_ls06.pdf

[3] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds. NIST Digital Library of Mathematical Functions. Sections 5.11, 10.25, 10.27, and 10.43. https://dlmf.nist.gov/5.11 https://dlmf.nist.gov/10.25.E2 https://dlmf.nist.gov/10.27.E4 https://dlmf.nist.gov/10.43.E19