Introduction - Lnu.sehomepage.lnu.se/staff/hfrmsi/zeros.pdf · nation of Dirichlet L-functions and...

24
ZEROS OF SYMMETRIC ZETA FUNCTIONS HANS FRISK Abstract. In this paper the zeros of zeta functions, Z (s), which have a symmetrical distribution of non-trivial zeros about the critical line are investigated. Especially, focus will be on a two-parameter family, (ε, α), of such symmetric zeta functions. The condition Z (s) = 0 for Re(s) > 1/2 leads to a function w(s)= ε(s) exp(α(s)) which is studied in detail. The consequences of a violation of the Riemann hypothesis (RH) is considered and a weak form of the RH is proposed. 1. Introduction The connections between number theory and physics have been much studied in recent years [Watkins, 2006]. Two main directions in the research are zeta function techniques and p-adic numbers in high-energy physics [Brekke and Freund, 1993; Elizalde, 1995; Vladimirov, Volovich and Zelenov, 1995; Khrennikov, 1997] and the Riemann zeta function as a source of inspiration in low-energy quantum chaos [Berry and Keating, 1999]. I will concentrate on the zeros of the Riemann zeta function, ζ (s), and the Dirichlet L-functions [Edwards, 1974; Apostol, 1976; Ivic, 1985; Titchmarsh, 1986; Karatsuba and Voronin, 1992; Ten- nenbaum, 1995; Davenport, 2000]. It is well known that the imag- inary parts of the zeros to ζ on the critical line Re(s)=1/2 have some striking similarities with a spectra of a hermitian operator with time reversal symmetry broken [Katz and Sarnak, 1999]. The famous Riemann hypothesis (RH) states that all non-trivial zeros of ζ lie on the critical line and a generalized hypothesis (GRH) states that this is also true for L-functions. These hypotheses are 1

Transcript of Introduction - Lnu.sehomepage.lnu.se/staff/hfrmsi/zeros.pdf · nation of Dirichlet L-functions and...

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ZEROS OF SYMMETRIC ZETA FUNCTIONS

HANS FRISK

Abstract. In this paper the zeros of zeta functions, Z(s),which have a symmetrical distribution of non-trivial zerosabout the critical line are investigated. Especially, focus willbe on a two-parameter family, (ε, α), of such symmetric zetafunctions. The condition Z(s) = 0 for Re(s) > 1/2 leads to afunction w(s) = ε(s) exp(α(s)) which is studied in detail. Theconsequences of a violation of the Riemann hypothesis (RH)is considered and a weak form of the RH is proposed.

1. Introduction

The connections between number theory and physics have beenmuch studied in recent years [Watkins, 2006]. Two main directionsin the research are zeta function techniques and p-adic numbersin high-energy physics [Brekke and Freund, 1993; Elizalde, 1995;Vladimirov, Volovich and Zelenov, 1995; Khrennikov, 1997] andthe Riemann zeta function as a source of inspiration in low-energyquantum chaos [Berry and Keating, 1999].

I will concentrate on the zeros of the Riemann zeta function,ζ(s), and the Dirichlet L-functions [Edwards, 1974; Apostol, 1976;Ivic, 1985; Titchmarsh, 1986; Karatsuba and Voronin, 1992; Ten-nenbaum, 1995; Davenport, 2000]. It is well known that the imag-inary parts of the zeros to ζ on the critical line Re(s) = 1/2 havesome striking similarities with a spectra of a hermitian operatorwith time reversal symmetry broken [Katz and Sarnak, 1999]. Thefamous Riemann hypothesis (RH) states that all non-trivial zerosof ζ lie on the critical line and a generalized hypothesis (GRH)states that this is also true for L-functions. These hypotheses are

1

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2 HANS FRISK

supported by extensive numerical investigations [Odlyzko, 1987;Rumely, 1993].

The main interest in this article is the zeros of linear combina-tions of L-functions and ζ such that the non-trivial zeros are sym-metrically distributed with respect to the critical line, see [Frisk,2001] for some first results on these symmetric zeta functions. In[Bombieri and Hejhal, 1995] rigourous results concerning the dis-tribution of the zeros for linear combinations of L-functions werederived. It was found that under mild assumptions, e. g. GRH,and in the limit of large Im(s) almost all zeros of the linear com-binations are simple and lie on Re(s) = 1/2. In the investigations

below ζ(s)(e−iα + eiα

ps−1/2 ), 0 ≤ α < 2π and p a prime, is included

in the linear combinations. Here and in the following L(s) denoteslinear combinations of Dirichlet L-functions while L(s, χ) are usedfor the basis functions themselves. Only L-functions with evencharacters, χ(−1) = 1, are considered.

The L-functions have no poles in the complex plane and trivialzeros at s = 0,−2,−4, ...... while ζ has a simple pole at s = 1and trivial zeros at s = −2,−4, .... [Apostol, 1976]. This indicatesthat the function ζ(s)/L(s) could be interesting to study since itis has a completely symmetrical distribution of zeros and poles. Isit possible that a zero of ζ with Re(s) > 1/2 in some way requiresthe existence of a pole of L(s)? What kind of differences are therebetween Riemann- and L-zeros existing outside the critical line?Questions like these have been the driving force for the study.

In section 2 the symmetric zeta functions are introduced anda function w(s) = ε(s) exp(iα(s)), closely related to ζ(s)/L(s), isconstructed. In section 3 the numerical results for w(s) are pre-sented and in the final section section the possibility of a violationof the RH is considered.

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 3

2. Symmetric Zeta Functions

Let s = σ + it. For σ > 1, the Riemann zeta function ζ(s) isdefined as

(1) ζ(s) =∞∑

n=1

1

ns.

It is easily seen that the series is uniformly and absolutely conver-gent for σ > 1 [Titchmarsh, 1986]. There are several generalizationsof the Riemann zeta function, but for the purposes of this paperonly two of them are needed. The first is the L-functions which forσ > 1 are defined as

(2) L(s, χ) =∞∑

n=1

χ(n)

ns,

m is a natural number and χ a Dirichlet character modulo m, see[Apostol, 1976] for the definition and basic properties of Dirichletcharacters. Note that if χ is the principal Dirichlet character thenL(s, χ) differs from ζ(s) only by a multiplicative factor.

The second extension is the Hurwitz zeta function which for σ >1 and 0 < α ≤ 1 is defined by the series

(3) ζ(s, α) =∞∑

n=0

1

(n + α)s,

Observe that ζ(s, 1) and ζ(s) coincide. The integral representationof Hurwitz zeta function for σ > 1 is

(4) ζ(s, α) =1

Γ(s)

∫ ∞

0

e−αx

1 − e−xxs−1dx,

where Γ(s) is Euler’s Gamma function. This representation is usedto show that ζ(s, α) admits an analytical continuation to the wholecomplex plane except for a simple pole at s = 1 [Apostol, 1976].

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4 HANS FRISK

The L-functions and the Hurwitz zeta function are closely related.Indeed, we have

L(s, χ) =∞∑

n=1

χ(n)

ns=

m∑l=1

∞∑k=0

χ(km + l)

(km + l)s

=m∑

l=1

χ(l)

ms

∞∑k=0

1

(k + lm

)s=

1

ms

m∑l=1

χ(l)ζ

(s,

l

m.

)(5)

Given a natural number m there are φ(m) distinct Dirichlet char-acters, where φ is the Euler φ-function. For a prime, p, there areexactly p−1 Dirichlet L-functions. In the following is m restrictedto a prime number, p.

ζ(s) and L(s, χ) are examples of symmetric zeta functions, Z(s),for which the non-trivial zeros are located symmetrically aroundthe critical line, i.e.

(6) Z(s) = 0 ⇒ Z(1 − s) = 0.

The natural ansatz for Z(s) is, compare with (5),

(7) Z(s) =1

ps

p∑l=1

clζ

(s,

l

p

).

Here the unknown coefficients, cl, must fulfill cl = cp−l, l = 1, 2, .., p−12

,since only even characters are considered. In the appendix it isshown that the vector space of symmetric zeta functions , over thereal numbers, is spanned by the Dirichlet L-functions and two ad-ditional functions, namely (1 + 1

ps−1/2 ) ζ(s) and (i − ips−1/2 ) ζ(s).

These latter two functions can be combined to

(8)

(e−iα +

eiα

ps−1/2

)ζ(s)

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 5

where 0 ≤ α < 2π. Here it must be pointed out that the func-tions obtained by the ansatz (7) which correspond to the L(s, χ)-functions in (2) differ from them by a phase factor eiδ (eiδτ(χ) =√

pe−iδ where τ(χ) is the Gauss sum [Davenport, 2000]). Despitethis modification I will anyhow call them L-functions below. Thelocation in the s-plane of the zeros of the two factors in (8) are inthe following denoted zα and zR and are called α-zeros and R-zeros,respectively. Using, as above, the notation L(s) for a linear combi-nation of Dirichlet L-functions and introducing the real parameterε ≥ 0, the most general symmetric zeta functions can be expressedin the following way

(9) Z(s) =

(e−iα +

eiα

ps−1/2

)ζ(s) + εL(s).

The rest of this article focus on the location of the zeros of Z(s)in the s-plane, both as a function of ε and of α. For the locationof the zeros of L(s), L-zeros, we use the notation zL. With ε =0 and increasing α the R-zeros are fixed while the α-zeros moveupwards on the critical line. Increasing ε somewhat gives rise to aninteraction between α- and R-zeros near zR. In the other extreme,ε � 1, the zeros of Z(s) oscillate around zL zeros on the criticalline and rotate around L-zeros outside the critical line, clockwisewith increasing α if Re(zL) > 1/2 [Frisk, 2001].

Since Z(s) is symmetric we concentrate in the following on σ ≥1/2. If σ 6= 1/2 the equation Z(s) = 0 determines ε and α uniquely.Solving the equation Z(s) = 0 for ε gives

(10) ε =

(e−iα +

eiα

ps−1/2

) (− ζ(s)

L(s)

).

Now, since ε is real and non-negative, the value of α is also fixed.It is relatively easy to see that by setting

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6 HANS FRISK

b(s) = − ζ(s)

L(s)

a(s) = ps−1/2(11)

the function ε(s)eiα(s), in the following denoted w(s), can be ex-pressed as

(12) w(s) = ε(s)eiα(s) = |b|2

(1 − 1

|a|2)

∣∣∣∣b − b

a

∣∣∣∣

(b − b

a

)∣∣∣∣b − b

a

∣∣∣∣.

Thus, the function w(s) = ε(s)eiα(s) tells us for which ε and α thefunction Z(s) in (9) is zero at a given point s.

At the critical line, s = 12

+ it, the condition Z(s) = 0 leads tothe following relation between ε and α,

(13) ε = −2ζ(s)

L(s)e−iη(s)/2 cos(α − η(s)/2),

where η(s) = Log(p) · Im(s). Since ε is non-negative and , with

our phase convention for L(s), ζ(s)L(s)

e−iη(s)/2 is real it can be seen

from (13) that for a given ε < εmax(t) = 2∣∣∣ ζ(1/2+it)L(1/2+it)

∣∣∣ the function

Z(1/2 + it) is, for a given t-value, zero for two different α-values.For ε=0 the difference between the two angels is π and at ε = εmax

there is only one such α-value.Introducing the polar expresions b(s) = r(s) eiθ(s) and a(s) =

pσ−1/2 eiη(s) it can be seen that the arguments for b(s) and w(s) are

the same if θ(s) = η(s)2

+ kπ2

. For odd integers k (12) reduces to

(14) w(s) = − ζ(s)

L(s)(1 − 1

pσ−1/2),

while for even integers k

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 7

(15) w(s) = − ζ(s)

L(s)(1 +

1

pσ−1/2).

Since ζ(s)L(s)

e−iη(s)/2 is real at the critical line the curves in the right

half-plane where (14) is fulfilled can only reach the critical line atthe location of the R- and L-zeros. Similarly, the curves (15) onlymeet the critical line at points where ε = εmax > 0, see figure 6.

3. Numerical Results For w(s)

From the condition that the symmetric function Z(s) is zero atgiven point s, σ > 1/2, a function w(s) = ε(s)eiα(s) was obtained inthe previous section and here numerical results for w(s) is presentedfor some different primes p and t-regions. If all zeros of ζ(s) andL(s) lie on the critical line the mean spacings for the three types ofzeros (R, L and α) must fulfill the relation 1/DL = 1/Dα + 1/DR

due to conservation of zeros. Here Dα = 2πLog(p)

and DR ≈ 2πLog(t/2π)

.

Thus, DL ≈ Dα if p � t while DL ≈ DR in the other extreme.Most figures presented below show numerical results for the two

interesting limits

(16) limσ→1/2+

|w(σ + it)| = ε+(t)

and

(17) limσ→1/2+

Arg w(σ + it) = α+(t)

The quantities α+(t) and ε+(t) tells us for which α and ε a multiple,usually double, zero is formed at the point s = 1/2 + it. Of courseε+ ≤ εmax, with equality only at local minima and maxima ofεmax(t). Note that ε+ = 0 at the location of the R-zeros since zα

can be located at zR to form a double zero of Z(s) for ε = 0. If

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8 HANS FRISK

L(s) has no multiple zeros on the critical line then ε+(t) will showno singularities. The motion of the zeros on the critical line, forfixed ε and increasing α, is vibrational for ε > ε+ and rotational,i.e. dt

dαalways positive, for ε < ε+.

The function w(s) at the other extreme, σ >> 1, is easy tounderstand, also for large p and t, see figures 1 and 11. Equation

120 125 130 135 140 145 150

0.998

0.999

1.001

1.002

Figure 1. The functions ε(10 + it) (thick) andα(10+ it)/π (fine) for p = 5. The only non-principaleven Dirichlet L-function for this prime is used andε∞ = 1 and α∞ = π. The period of the oscillations is≈ 2π/ Log 2. All figures in this work are made withMathematica

(12) shows that −ζ(s)L(s)

is a very good approximation to w(s) far out

in the right half-plane. Furthermore ζ(s) ≈ 1 + 12s and L(s) ≈

c1 + c22s (c1, c2 ∈ C) in this region so the generic situation is like in

figure 1: ε(σ + it) and α(σ + it) oscillate for fixed σ with periodT ≈ 2π/ Log 2. When σ → ∞ then w(s) → −1/c1. All the curvesε = 1/ |c1| = ε∞ and α = Arg(−1/c1) = α∞ that meet at infinityusually end up at the critical line but ε = ε∞ contours can meetif w(s) has poles (zeros) in the right half-plane. The α = α∞

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 9

contours can end at poles, or possible zeros, in the right half-planebut generally they continue all the way into the critical line. UsingDirichlet L-functions we get ε∞ = 1 and except for the p = 7case below all numerical results are obtained with such functions.From figures 1 and 11 and the discussion above we can understandthat the right-half plane can be divided into regions with ε greateror smaller than ε∞. On the contours with ε = ε∞, dividing theregions, α increases in the clockwise sense around ε > ε∞ regions.

The first results below is for p = 5, for which there is only onenon-principal even Dirichlet L-function [Apostol, 1976], and forlow values of t. Then follows a p = 7 case with L-zeros outside thecritical line. Somewhat larger t and p are investigated at the end.

3.1. p=5 and small t. In figure 2 the function εmax(t) is com-pared to a numerical approximation of the limit ε+(t) in the specificcase 114 ≤ t ≤ 121. This figure should be studied together withfigure 3 which shows a numerical illustration of α+(t) in the sameregion. Comparing the two figures we can observe that α+(t) haslocal extreme values at the location of the L-zeros. A zero of Z(s)at zL on the critical line can occur only if an α-zero also is locatedthere, we assume ζ(zL) 6= 0. This happens for just two α-values,We realize now that α+(zL) must be equal to one of these two α-values, denote it αL. In an interval close to the L-zero α+(t) ≤ αL

if the zero move, with increasing α and constant ε, in to the criticalline. From the discussion above, concerning figures 1 and 11, wecan draw the conclusion that ε+(t) is then an increasing functionin the interval. The local minima can be explained in a similarmanner. Extreme values of α+(t) also occur at location of triplezeros of Z, as for t ≈ 115 in figure 3. This occurs at the pointswhere ε+(t) > 0 and has local extreme values with ε+ 6= εmax. Thetriple zero at t ≈ 115 is due to the fact that only one L-zero islying between the two nearest R-zeros, see figure 2, which impliesa triple zero of Z(s) for the ε+-value which is the local maxima.

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10 HANS FRISK

114 115 116 117 118 119 120 121

2

4

6

8

10

12

14

Figure 2. The function εmax(t) (dashed) is com-pared to ε(0.505+ it) (solid) for p = 5 and 114 ≤ t ≤121.

The two figures also illustrates the statements made in connec-tion with (14,15): α+(t) cross a short-dashed line only at the lo-cation of R- (with positive slope) or L-zeros (generally a local ex-treme value) and it cross a long-dashed line only at points whereε+ = εmax 6= 0.

3.2. p=7 and small t. For p=7 there are two non-principal evenDirichlet L-functions. Linear combinations L(s) of these two canhave zeros outside the critical line. In figures 4-6 the linear com-bination has purely imaginary coefficients in the Dirichlet seriesand then such zeros are abundant. It can then happen, as in fig-ure 4, that εmax(t) has a local maxima and for this particular εand t the zero of Z(s), circulating around the L-zero, reach thecritical line. Consequently εmax = ε+ at this local maxima. Inthe corresponding α+(t) plot in figure 5 the presence of the L-zerooutside the critical line gives rise to a strong increase of α+ in the

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 11

114 115 116 117 118 119 120 121

1

1.5

2

2.5

3

3.5

4

4.5

Figure 3. The function α(0.505 + it) for p = 5 and

114 ≤ t ≤ 121 (solid line). The lines are α = Log(p)·t2

+kπ2

, short-dashed for odd integers k and long-dashedfor even k. The function α+(t) has local extremevalues at the location of L-zeros and at the triplezero of Z(s) at t ≈ 115.

region t=234.6-236.4. Note that the slope of α+(t) at the locationof R-zeros is always positive which means that contours with con-stant α, originating from a L-zero with σL > 1/2 and approachingthe critical line, can pass a R-zero smoothly. In connection with(14,15) it was pointed out that along certain curves in the s-planethe functions w(s) and b(s) only differ by a real factor, 1 ± 1

pσ−1/2 .

Such curves are shown in figure 6 for the same L(s) as in figures4 and 5. Two L-zeros with σL > 1/2, where four of these curvesmeet, can be found in the figure. Curves of this type can only meetthe critical line at the location of R- and L-zeros and points whereεmax = ε+ > 0. Far out in the s-plane, σ � 1, the curves arealmost horizontal with spacings ∆t ≈ π/ Log(7).

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12 HANS FRISK

233 234 235 236 237

5

10

15

20

25

Figure 4. ε(0.505+it) (solid) and εmax(t) (dashed)for p=7 and 233 ≤ t ≤ 237. A linear combination ofDirichlet L-functions , with purely imaginary coeffi-cients, is used such that w(s) has poles for σ > 1/2.Local maxima for εmax, e.g. at t ≈ 234.9, then occur.

How are the poles of w(s) distributed? A systematic study inthe region 200 < t < 500 shows that there are 54 zeros of L(s) inthe right-half plane. Only three of them are located in a ε < ε∞region. Around zL = 1.05 + i202.4, 1.08 + i386.9 and 0.89 + i449.1there are just small islands with ε > ε∞ but the remaining 51 zeroslie in ε > ε∞ area. In section 4 this remarkable fact is discussedand there it is also shown that the permanent pole at s = 1 alwayslies in a ε > ε∞ region if Dirichlet L-functions are used.

3.3. p=5 and large t. In a region where t � p the spacingsbetween R- or Dirichlet L-zeros on the critical line are much smallerthan the spacings between the α-zeros. Figures 7 and 8 illustratessuch a case. In between the two lowest R-zeros there are three L-zeros and in this region of the (t, p)-plane this means a strong dropin α+(t). Note that if the RH is violated two R-zeros are missing on

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 13

233 234 235 236 237

1

2

3

4

5

6

Figure 5. The function α(0.505 + it) for p=7 and233 ≤ t ≤ 237. A pair of L-zeros outside the criti-cal line give rise to the increase of α+ in the regionaround t=235. The same L(s) as in figure 4 is used.

the critical line and the ε+(t) and α+(t) curves could then look likethe curves in the left part of figures 7 and 8. The three L-zeros arefollowed by a long sequence of consecutive R- and L-zeros, givingrise to an average increase of Log(5)t/2 in α+(t). In between twoR-zeros in this chain there is a triple zero of Z(s).

3.4. p=9901 and small t. Let us conclude this section with theinteresting, and not much explored, case p � t. In figures 9 and 10an illustration is shown with p = 9901 and 32 ≤ t ≤ 38. L(s, χ) isin this case the non-principal Dirichlet L-function with real char-acters. Compared to figures 7 and 8 the role of the R- and α-zerosis now shifted. The spacings between the L-zeros on the criticalline are now of the same size as Dα. An interesting difference infigure 9, compared to figures 2,4 and 7, is that the local minima ofεmax(t) now also can be minima of ε+(t), there are four such casesin figure 9. They occur for L-spacings larger than Dα since there is

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14 HANS FRISK

0.5 1 1.5 2 2.5 3233

234

235

236

237

Figure 6. Curves in the right half-plane where(14,15) is fulfilled. The same linear combination forL(s) as in figure 4 and 5 has been used. See figure4 for the location of R- and L-zeros on the criticalline and also for the points where εmax = ε+ 6= 0.Note the two L-zeros at s ≈ 1.3 + i234.9 and s ≈1.9 + i236.7. The four curves to the right continueout in the right half-plane and for large σ values theyare separated by ∆t ≈ π/ Log(7).

then an interval in α where two α-zeros are located in between theL-zeros. Increasing ε, keeping α fixed, these two zeros can start tointeract and form a multiple zero on the critical line. In figure 10

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 15

20005 20006 20007 20008 20009 20010

2.5

5

7.5

10

12.5

15

17.5

20

Figure 7. ε(0.505+it) (solid) and εmax(t) (dashed)p=5 and 20005 ≤ t ≤ 20010. In this region themean spacings between the different types of zerosare DL ≈0.65, DR ≈0.78 and Dα=3.90. There arethree L-zeros between the two lowest R-zeros. ε+(t)is small on this scale in the right part of the figure.

the slope of α+(t) is positive when cutting long-dashed lines at thelocations of these minima of ε+(t). The topological structure in thevicinity of these minima is thus the same as close to a R-zero. Aschematic picture of how these structures appear is shown in figure11.

4. remarks and speculations

From the knowledge of w(s) presented in the previous sections,is it possible to say something about the RH or the GRH? Let usconclude with some remarks and speculations about this question.

If ζ(zR) = 0, σR > 1/2, not all L(s, χ) can be zero at zR. This fol-lows simply from the construction in the appendix since the vectorF (zR) must then be zero due to the ortogonality of the eigenvectors

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16 HANS FRISK

20005 20006 20007 20008 20009 20010

1

2

3

4

5

6

Figure 8. α(0.505 + it) corresponding to figure 7.Since all spacings between L-zeros are small com-pared to Dα the lowest three L-zeros gives rise toa strong decrease in α. The triple-zero of Z(s) andthe L-zero close to t=20006 lie close so the two localextreme values of α+(t) are hardly visible. Abovet=20006.6 there are five local minima and five localmaxima due to the L-zeros and the triple zeros ofZ(s).

to the matrix W Tp but this can certainly not occur for large primes

since for example F1(zR)/pzR → 1 when p → ∞.If we compare a L-zero with a R-zero in the right half-plane

some differences can be noted. Firstly, when for increasing ε twomirror-zeros, circulating around each zR, meet at the critical linea third zero must be involved to form a triple zero, in contrastwith the L-case. This follows from continuity and (13). Secondly,contours with constant α, originating from a R-zero outside thecritical line, that approach the critical line can not generally passsmoothly the L-zeros, remember α+(t) has generally local extreme

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 17

32 33 34 35 36 37 38

2

4

6

8

10

12

14

Figure 9. ε(0.505+it) (solid) and εmax(t) (dashed)in the region 32 ≤ t ≤ 38 and with p=9901.L(s) is here the non-principal Dirichlet L-functionwith real characters. The mean spacings areDL ≈0.57,Dα=0.68 and DR ≈3.66. A differencecompared to figures 2,4 and 7 is that local minimaof εmax(t) here can be local minima also for ε+(t) (att ≈ 32.2, 33.7, 35.1, 36.7).

values there. On the other hand α-contours coming from a L-zerocan pass smoothly the R-zeros due to the upward moving α-zerosand it can be seen in figure 5, as in figs. 3, 8 and 10, that theslope of α+(t) is positive at the location of the R-zeros. A thirddifference occurs in connection with figure 6. How will the fourcurves originating from a R-zero in the right half-plane behave?My claim is that, at least for large p, all four curves will go into thecritical line. This is again in contrast with the corresponding L-case where all four curves can go off to infinity. The reason is thatthere are two type of curves, see (14) and (15), and for large σ theyalternate, with distance ∆t ≈ π

Log(p)between them. If there is then

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18 HANS FRISK

32 33 34 35 36 37 38

1

2

3

4

5

6

Figure 10. α(0.505 + it) corresponding to figure 9.The spacings between L-zeros are of the same sizeas Dα. Note that the slope is positive at the long-dashed lines for four cases. This is were both εmax(t)and ε+(t) have local minima different from zero (seefigure 9).

a long chain of alternating L-zeros and points where εmax = ε+ onthe critical line there is no possiblility for the curves from a R-zeroto go off to infinity. If these observations give some restrictions onR-zeros existing outside the critical axis is an open question.

However, most interesting seems the numerical observation men-tioned in section 3.2: Most poles of w(s) are located in ε > ε∞regions. Furthermore, the permanent pole at s = 1 of b(s) is atleast for Dirichlet L-functions, which we assume from now one, in

a |b(s)| > 1 region since |1 + 1−χ(p)pσ | ≥ 1 for all primes. The last

inequality tells us that the ratio of the two Euler products is alwaysgreater than one on the real axis, i.e. |b(σ)| is monotone on the in-terval σ > 1, so the pole can not be located in a |b(s)| < 1 region.Using (12) we can conclude that the pole must at least for real

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 19

Figure 11. A very schematic picture of a regionin the right-half plane where ε ≥ ε∞. Some con-tours with constant ε are shown and the arrows indi-cate the direction where α increases, see also figure1. Four L-zeros at the critical line are also shownand the spacing between L2 and L3 is assumed tobe larger than Dα. For a specific ε two contours canmeet at a double-zero and form the shaded region.This is the mechanism behind the four minima ofε+(t) seen in figure 9. In section 4 the possiblity ofa R-zero in the shaded region is discussed.

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20 HANS FRISK

characters also be located in a ε > 1 region. The obvious questionis now: If the RH is not valid can then these exotic R-zeros outsidethe critical axis really be located in a ε > 1 region? The importantpoint is here that in these regions one can say that the exotic R-zero is, so to speak, created from the L-zeros on the critical line atsome specific ε > 1, like the creation of the shaded region in figure11, and at the same time zeros must be located at the L-zeros fortwo different α-values during each period, α = Log p · tL/2 ± π

2(mod 2π). The difference between this case and pole in a ε < 1region, like the three cases mentioned in section 3.2, is that withtwo L-zeros off the critical line there are less constraints on themotion of the zeros. One possible way of creating an exotic R-zerois shown in figure 11. Since at least four zeros must take part in theprocess one could think of a possibility of a R-zero in the shadedregion. However, if p � t there seems to be no room for an increaseof 2π in α around the shaded region due to the above mentionedconstraints and since the spacings between the L-zeros are close toDα. On the other hand, if p � t four consecutive L-zeros on thecritical axis is a rare event so even here more zeros, over a longert-distance, must participate in the formation of the exotic R-zero.Another mechanism for the formation of a R-zero in a ε > 1 regionis that along one specific ε contour, like in figure 11, the increase ofα is larger than 2π and at some point it cross itself to form a zerocirculating around the location of the exotic R-zero. Here two zerosare outside the critical line during one period and it is unclear howthe constraints then can be fulfilled. Further research is certainlyneeded here. The goal for such a project could be to try to provea weak form of the RH: Zeros of ζ(s) outside the critical line mustbe located in ε < 1 regions if all L-zeros have σL=1/2.

This article was written under a partial support of the project”Mathematical Modeling” at the University of Vaxjo. I am gratefulto my student Henrik Larsson for assistance with the figures.

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 21

5. Appendix

In this appendix the ideas leading to the studied function w(s)in (12) are developed. The functional equation of Hurwitz zetafunction is used to construct the symmetric function Z(s) in (9).It states [Apostol, 1976] that if 1 ≤ l ≤ p then for all s ∈ C wehave

(18) ζ

(1 − s,

l

p

)=

2Γ(s)

(2πp)s

p∑r=1

cos

(πs

2− 2πrl

p

(s,

r

p

).

By replacing l → p − l above also the following equations for 0 ≤l < p can be obtained

(19) ζ

(1 − s,

p − l

p

)=

2Γ(s)

(2πp)s

p∑r=1

cos

(πs

2+

2πrl

p

(s,

r

p

).

Adding these equations together and with Fk(1−s) = ζ(1 − s, k

p

)+

ζ(1 − s, p−k

p

)for 1 ≤ k ≤ (p−1)/2 as well as ρ(s) = 4Γ(s) cos

(πs2

)/(2πp)s

we get

(20) Fk(1 − s) = ρ(s)

(p−1)/2∑

r=1

cos

(2πrk

p

)Fr(s) + ζ(s)

.

Note that the functional equation of Hurwitz zeta function also isvalid when l = p and in this case it becomes

(21) ζ(1 − s) = ρ(s)

(p−1)/2∑

r=1

1

2Fr(s) +

1

2ζ(s)

.

Let

F (1 − s) =(F1(1 − s), F2(1 − s), ..., F (p−1)

2

(1 − s), ζ(1 − s))T

We can now rewrite equations (20) and (21) as a new functionalequation which takes care of all (p + 1)/2 functional equations at

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22 HANS FRISK

once, namely

(22) F (1 − s) = ρ(s)WpF (s)

where Wp is a ((p + 1)/2) × ((p + 1)/2) matrix having the form

(23) Wp =

1Mp 1

. . ....

1/2 1/2 · · · 1/2

and (Mp)r,k = cos(

2πrkp

), 1 ≤ r, k ≤ (p−1)/2. In exactly the same

way it can be shown that

(24) F (s) = µ(s)WpF (1 − s)

where µ(s) = 4Γ(1− s) sin(

πs2

)/(2πp)1−s. Note that Mp is a sym-

metric matrix. Introducing now a complex column vector, c, whosecomponents are the coefficients of the Dirichlet series, see (7), thefunctional equation takes the form

(25) F T (1 − s)c = ρ(s)F T (s)W Tp c.

If the l.h.s. of (25) vanishes at some point 1 − s then the sym-metry condition (6) requires that also F T (s)c vanishes. Thus thesymmetry condition for the coefficients becomes

(26) W Tp c = λc.

for some λ ∈ C. However, combining (22) and (24) gives that(W T

p )2 is equal to p/4 times the identity matrix [Apostol, 1976] so

there is only two possibilities for λ, λ = ±√

p

2. Since the components

of c are complex numbers we write c in the form c = x + iy. From(26) we see that the vectors x can be choosen as the eigenvectorsto W T

p associated with the eigenvalue√

p/2 while y then are theeigenvectors associated the the eigenvalue −√

p/2. By inspectionof the matrix W T

p , and using the formula 1/2+cos(2πp

)+ cos(4πp

)+

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ZEROS OF SYMMETRIC ZETA FUNCTIONS 23

...... cos( (p−1)πp

) = 0, we find that one pair of eigenvectors, x and y,are {

x =(1, · · · , 1 +

√p)

y =(1, · · · , 1 −√

p).

A linear combination, cosα x− i sinα y, of these eigenvectors givesthe coefficients for the following function(

e−iα +eiα

ps− 12

)ζ(s).

We know that the Dirichlet L-functions are symmetric zeta fuctionsso they must therefore be obtained from the other eigenvectors ofW T

p , and in fact from the eigenvectors to Mp. By comparing thenumber of even Dirichlet characters with the matrix dimension itis found that no other symmetric zeta functions exist. Note that(1,1,...1,0) is an eigenvector to Mp, with eigenvalue -1/2, but notto W T

p . Note also that from each pair, x and y, two L-functionswith coefficients, x ± iy, can be formed.

References

T. M. Apostol, ”Introduction to Analytic Number Theory”,Springer-Verlag, 1976.

M.Berry and J. Keating, Siam Review 41, (1999), 236-266.E. Bombieri and D. A. Hejhal, Duke Math J 80, (1995), 821 - 862.L. Brekke and P. G. O. Freund, Phys. Rep. 233 (1993) 1-66.H Davenport, ”Multiplicative Number Theory”, third edition ,

Springer-Verlag 2000.H. M. Edwards, ”Riemann’s Zeta Function”, Academic Press, 1974.E. Elizalde, ”Ten Physical Applications of Spectral Zeta Func-

tions”, Springer -Verlag, 1995.H. Frisk and S de Gosson, http://arXiv.org/abs/math-ph/

?0102007.A. Ivic, ”The Riemann Zeta-Function”, John Wiley and Sons, 1985.A. A. Karatsuba and S. M. Voronin, ”The Riemann Zeta-

Function”, Walter de Gruyter, 1992.

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24 HANS FRISK

N. M. Katz and P. Sarnak, Bull. of the AMS 36, (1999), 1-26.A. Khrennikov, ”Non-Archimedan Analysis: Quantum Paradoxes,

Dynamical Systems and Biological Models”, Kluwer AcademicPublishers, 1997.

A. M. Odlyzko, Math. Comp. 48, (1987).R. Rumely, Math Comp 61, (1993), 415-440.G. Tennenbaum, ”Introduction to Analytic and Probabilistic Num-

ber Theory”, Cambridge Univ. Press, 1995.E. C. Titchmarsh, ”The Theory of the Riemann Zeta-Function”,

Oxford: The Clarendon Press, 1986.V. S. Vladimirov, I. V. Volovich and E. I. Zelenov, ”P-adic Analysis

and Mathematical Physics”, World Scientific, 1995.Watkins , see http://www.maths.ex.ac.uk/ mwatkins/zeta/physcics.htm

Vaxjo University, Department of Mathematics, 351 95 Vaxjo,Sweden