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Further Insights into the Mysteries of the Values of Zeta Functions at Integers

  • Ján Mináč , Tung T. Nguyen and Nguyễn Duy Tân EMAIL logo
Published/Copyright: August 4, 2023
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 73 Issue 4

ABSTRACT

We present a remarkably simple and surprisingly natural interpretation of the values of zeta functions at negative integers and zero. Namely, we are able to relate these values to areas related to partial sums of powers. We apply these results to further interpretations of values of L-functions at negative integers. We hint in a very brief way at some expected connections of this work with other current efforts to understand the mysteries of the values of zeta functions at integers.

2020 Mathematics Subject Classification: Primary 11M35; 11M06; 11B68
Keywords: Hurwitz zeta functions; Bernoulli polynomials; generalized Bernoulli numbers; L-functions

(Communicated by Milan Paštéka)

To the memory of Goro Shimura with gratitude and admiration

Funding statement: J. M. is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant R0370A01. He gratefully acknowledges the Western University Faculty of Science Distinguished Professorship in 2020–2021 and support of Western Academy for Advanced Research as Western Fellow.

Funding statement: N. D. T. is partially supported by the Ministry of Education and Training of Vietnam, grant B2022-CTT-01.

Appendix A. Some formulas and figures

In this appendix, we provide the formulas of Sn (x) for small n and their graphs on the interval [0, 1]. First, we have the following formulas for Sn (x) with 1 ≤ n ≤ 6.

S1(x)=(x−1)x2,S2(x)=(x−1)x(2x−1)6,S3(x)=(x(x−1))24,S4(x)=16(x6−3x5+52x4−12x2),S5(x)=16( x6−3x5+52x4−12x2,S6(x)=17(x7−72x6+72x5−76x3+16x).

We plot the graphs of Sn (x) for n ∈ {1, 3, 5}.

Here are the graphs of Sn (x) for n ∈ {2, 4, 6}.

Finally, we plot the region bounded by Sn (x) and y = 0 over [ 0,  12 ] .

Next, we compute certain generalized Bernoulli numbers Bn,χ where χ is a quadratic character of conductor p. Concretely, this is the character given by Legendre symbol χ(a)=(ap) , a∈ℤ . First, we provide a table of Bn,χ for p ≤ 11.

n Bn B n,χ 3 B n,χ 4 B n,χ 5 B n,χ 7 B n,χ 11
0 1 0 0 0 0 0
1 -1/2 -1/3 -1/2 0 -1 -1
2 1/6 0 0 4/5 0 0
3 0 2/3 3/2 0 48/7 18
4 -1/30 0 0 -8 0 0
5 0 -10/3 -25/2 0 -160 -12750/11
6 1/42. 0 0 804/5 0 0
7 0 98/3 427/2 0 8176 152082
8 -1/30 0 0 -5776 0 0.
9 0 -1618/3 -12465/2 0 -5086656/7 -33743250
10 5/66 0 0 1651004/5 0 0
11 0 40634/3 555731/2 0 99070928 11392546506
12 -691/2730 0 0 -27622104 0 0

Next, we provide a table of Bn,χ for 13 ≤ p ≤ 23.

n B n,χ 13 B n,χ 17 B n,χ 19 B n,χ 23
0 0 0 0 0
1 0 0 -1 -3
2 4 8 0 0
3 0 0 66 144
4 -232 -656 0 0
5 0 0 -13450 -34080
6 401556/13 138984 0 0
7 0 0 5303074 18665136
8 -7482704 -958428704/17 0 0
9 0 0 -66751985430/19 -17895000384
10 2890943420 37040430040 0 0
11 0 0 3539203405562 605747775717744/23
12 -1634752049016 -35766492971568 0 0

While working on these numerical data, we observed that in our examples, it is always the case that υp(Bp−12,χ)=−1 . Using the work of Ernvall (see [15]), we are able to prove a more general statement.

Remark A.1

After finding a direct proof and searching further the literature, we also found Carlitz’s result (see [10: Theorem 3]) that implies the above observation. The announcement of results in [10] is in [9]. Observe however that there is a misprint in Theorem 1, line 4 in [9]. The relevant part of [10: Theorem 3] states in our special case where values of character are just 1 or −1 the following: “Let g be a primitive root mod p. If the conductor f of the primitive character mod f is a prime number p, p > 2, then Bn,χn is integer unless p and 1 – χ(g)g n are not coprime in which case …”. This statement does imply that υp(Bp−12,χ)=−1 and also the more general version of our Proposition A.1. In this case (see [10: Theorem 3] for the relevant notations) ν = 0 and using that g is a primitive root mod p, we see that that gcd(p, 1 –χ(g)g n ) = gcd(p, 1 + gn ) > 1 iff n≡p−12 (mod (p – 1)), and therefore we have that pBp−12,χ≡−1 (mod p). For the reader’s convenience, here we provide a direct short proof of this very interesting statement.

Proposition A.2

Let χ be a quadratic character of conductor p where p is an odd prime number and n ≥ 0. Then:

  1. if n≡p−12 (mod (p – 1)), then Bn,χ∈ℤ ;

  2. if n≡p−12 (mod (p −1)), then Bn,χ=ap , where a∈ℤ and a ≡ −1 (mod p).

Proof

By [15: Theorem 1.2], we know that if n ≢ δχ (mod 2) then Bn,χ = 0. Furthermore, we remark that for all odd primes p

p−12≡δχ(mod 2).

Therefore, it is is sufficient to prove this proposition when n ≡ δχ (mod 2).

By [15: Theorem 1.4], we know that pBn,χ is p-integral. Additionally, by [15: Theorem 1.5], Bn,χ is q-integral for all primes q ≠ p. Combining these two facts, we can conclude that pBn,χ∈ℤ for all n ≥ 0. Furthermore, by [15: Theorem 1.6], we know that

p2Bn,χ≡∑a=1p2χ(a)an(mod p2).

We have the following congruences modulo p 2

∑a=1p2χ(a)an=∑a=1p(∑k=0p−1χ(kp+a)(kp+a)n)=∑a=1pχ(a)∑k=0p−1(an+n(kp)an−1)=∑a=1pχ(a)(pan+npan−1∑k=1p−1k)=p∑a=1pχ(a)an+np2(p−1)2 ∑a=1pχ(a)an−1=p∑1pχ(a)an(mod p2).

The second congruence comes from the expansion of (a + kp) n leaving out terms which are divisible by p 2. The last congruence comes from the identity

∑k=1p−1k=p(p−1)2.

We have the following well-known simple lemma.

Lemma A.3

Let r be a natural number. Then

∑a=1par={ 0if p−1∤r−1if p−1|r .

Using this lemma and the fact that χ(a)≡ap−12 (mod p), we have the following congruences modulo p

∑a=1pχ(a)an≡∑a=1pan+p−12≡{ −1if n≡p−12(mod (p−1))0else.

Consequently, we have the following congruences modulo p 2

p2Bn,χ≡{ −pif n≡p−12(mod (p−1))0else.

Combining this congruence and the fact that pBn,χ∈ℤ , the proposition follows easily.

Acknowledgements

Over the years after publication of the short note [30], Mináč received a number of encouraging correspondences and discussions concerning further generalizations and explorations of possible values of other zeta functions. We are extremely grateful for our correspondents including but not limited to S. Chebolu, R. Dwilewicz, G. Everest, E. Frenkel, A. Granville, J. Merzel, L. Muller, Š. Porubský, P. Ribenboim, Ch. Rottger, A. Schultz, M. Z. Spivey, B. Sury. Further, we are grateful to K. Dilcher and R. Ernvall for helping us to obtain some related references including the very nice R. Ernvall’s thesis [15]. We are also grateful to an anonymous referee for his/her careful reading of the manusript, and for providing us with helpful comments and valuable suggestions. Last but not least, we are grateful to Leslie Hallock for her careful proofread.

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Received: 2022-07-20
Accepted: 2022-09-06
Published Online: 2023-08-04

© 2023 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. A Note on Special Subsets of the Rudin-Frolík Order for Regulars
  2. The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
  3. The Deranged Bell Numbers
  4. On Index and Monogenity of Certain Number Fields Defined by Trinomials
  5. The k-Generalized Lucas Numbers Close to a Power of 2
  6. Shifted Power of a Polynomial with Integral Roots
  7. Further Insights into the Mysteries of the Values of Zeta Functions at Integers
  8. Memoryless Properties on Time Scales
  9. A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
  10. Besov and Triebel-Lizorkin Capacity in Metric Spaces
  11. Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
  12. An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
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  14. A New Series Space Derived by Absolute Generalized Nörlund Means
  15. Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
  16. The Uniform Effros Property and Local Homogeneity
  17. Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
  18. The Unit-Gompertz Quantile Regression Model for the Bounded Responses
  19. An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
  20. Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
https://doi.org/10.1515/ms-2023-0066
Keywords for this article
Hurwitz zeta functions; Bernoulli polynomials; generalized Bernoulli numbers; L-functions

Articles in the same Issue

  1. A Note on Special Subsets of the Rudin-Frolík Order for Regulars
  2. The 2-Class Group of Certain Families of Imaginary Triquadratic Fields
  3. The Deranged Bell Numbers
  4. On Index and Monogenity of Certain Number Fields Defined by Trinomials
  5. The k-Generalized Lucas Numbers Close to a Power of 2
  6. Shifted Power of a Polynomial with Integral Roots
  7. Further Insights into the Mysteries of the Values of Zeta Functions at Integers
  8. Memoryless Properties on Time Scales
  9. A Study of the Higher-Order Schwarzian Derivatives of Hirotaka Tamanoi
  10. Besov and Triebel-Lizorkin Capacity in Metric Spaces
  11. Oscillation of Odd Order Linear Differential Equations with Deviating Arguments with Dominating Delay Part
  12. An Elliptic Type Inclusion Problem on the Heisenberg Lie Group
  13. Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator
  14. A New Series Space Derived by Absolute Generalized Nörlund Means
  15. Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors
  16. The Uniform Effros Property and Local Homogeneity
  17. Limit Theorems for Weighted Sums of Asymptotically Negatively Associated Random Variables Under Some General Conditions
  18. The Unit-Gompertz Quantile Regression Model for the Bounded Responses
  19. An Extended Gamma-Lindley Model and Inference for the Prediction of Covid-19 in Tunisia
  20. Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals
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