New Results About Imaginary Value of the Second Axiomatic Zero of the Zeta Function and the Fifth Bernoulli Number

Ahmed Adnan Tuma; Shakir M. Salman

doi:10.12688/f1000research.172983.1

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Research Article

New Results About Imaginary Value of the Second Axiomatic Zero of the Zeta Function and the Fifth Bernoulli Number

[version 1; peer review: awaiting peer review]

Ahmed Adnan Tuma¹, Shakir M. Salman ²

PUBLISHED 10 Jan 2026

Author details Author details

¹ Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, Iraq
² Mathematics, Department of Mathematics, College of Basic Education, University of Diyala, Baqubah, Iraq

Ahmed Adnan Tuma
Roles: Conceptualization

Shakir M. Salman
Roles: Conceptualization

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

This paper introduces two theorems in number theory about zeta at − 4 and the fifth Bernoulli number B 5 , showing that the zeta function ζ ( − 4 ) has two complex values and that the fifth Bernoulli number is not zero but alsohas two complex values.

The current research concludes that the zeta function ζ ( − 4 ) has two imaginary values that are different from what was previously known to mathematicians, which was believed to be equal to zero, whereas, ζ ( − 4 ) = - i 558 , ζ ( − 4 ) = - i 2232

Therefore, the fifth Bernoulli number is not equal to zero and has two values: B 5 = − 5 i 558 and B 5 = − 5 i 2232 .

Keywords

Zeta function, Bernoulli numbers, trivial zeros.

Corresponding author: Shakir M. Salman

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Tuma AA and Salman SM. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Tuma AA and Salman SM. New Results About Imaginary Value of the Second Axiomatic Zero of the Zeta Function and the Fifth Bernoulli Number [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:46 (https://doi.org/10.12688/f1000research.172983.1) First published: 10 Jan 2026, 15:46 (https://doi.org/10.12688/f1000research.172983.1) Latest published: 10 Jan 2026, 15:46 (https://doi.org/10.12688/f1000research.172983.1)

1. Introduction

In 1713 Swedish mathematician Jacob Bernoulli and Japanese mathematician Seki Takakazu independently discovered numbers now called Bernoulli numbers through their work on the power series of integers $1^{m} + 2^{m} + 3^{m} + \dots + n^{m}$ . After their work, they accidentally discovered the values of these numbers and, through law¹:

S_{m} (n - 1) = \frac{1}{m + 1} \sum_{k = 1}^{m} (\begin{matrix} m + 1 \\ k \end{matrix}) B_{k} n^{m - k + 1}

determined the final values of these numbers, and concluded that all odd Bernoulli numbers are zero except for the first Bernoulli number

B_{1} = \frac{- 1}{2}

, and even Bernoulli numbers have the following values²:

B_{0} = 1, B_{1} = \frac{- 1}{2}, B_{2} = \frac{1}{6}, B_{3} = 0, B_{4} = \frac{- 1}{30}, B_{5} = 0, B_{6} = \frac{1}{42}

In 1734, Swiss mathematician Leonardo Euler formulated the zeta function:

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}},

where,

s \in ℝ

.

Euler defined a zeta function in the domain of real numbers. He succeeded in cleverly linking the zeta function to prime numbers through the law that he derived.

ζ (s) = \prod_{p} \frac{1}{(1 - p^{- s})}

He also succeeded in solving the Basel problem, which was posed in Basel City, and provided proof of the result using the zeta function, where³

ζ (2) = 1 + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + \dots = \frac{π^{2}}{6}

By studying the zeta function and linking it with Bernoulli numbers, Euler succeeded in deducing two of the most important laws. Through this study, he succeeded in linking the Bernoulli numbers with the zeta function.

(1)

ζ (2 k) = \frac{{(- 1)}^{k + 1} \cdot {(2 π)}^{2 k} \cdot B_{2 k}}{2 (2 k)!}; k = 1, 2, 3, \dots

(2)

ζ (- k) = \frac{{(- 1)}^{k} \cdot B_{k + 1}}{k + 1}; k = 0, 1, 2, 3, \dots

From this, he concluded that the zeta function at $- 1$ equals $\frac{- 1}{12}$ .⁴

ζ (- 1) = 1 + 2 + 3 + 4 + \dots = \frac{- 1}{12}

if $k = 1$ then, $ζ (- 1) = \frac{{(- 1)}^{1} \cdot B_{2}}{2} = \frac{- 1 / 6}{2} = \frac{- 1}{12}$

In 1859, German scientist Bernard Riemann extended the range of functions to include imaginary numbers as follows:

ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}},

where,

s \in ℂ

.

Riemann extended the range of the function’s inputs to include imaginary numbers and stated that all non-trivial zeros lie on the critical line, which was named after him for the results he provided on this function. The Riemann conjecture is considered to be an unsolved mathematical problem.⁵

Ramanujan presented a proof in which he proved that the zeta function at the negative number $- 1$ is equal to $\frac{- 1}{12}$ , where

ζ (- 1) = 1 + 2 + 3 + 4 + \dots = \frac{- 1}{12}

Using special sums in a specific manner, as in,⁶ It should also be emphasized that Euler presented a proof in which he showed that the zeta function at $- 1$ equals $\frac{- 1}{12}$ , based on Bernoulli numbers, and Ramanujan proved that the zeta function at $- 1$ equals $\frac{- 1}{12}$ using sums.

Another researcher presented a refutation of what are called axiomatic zeros, whose values depend on Bernoulli numbers⁷ using partial sums. The researcher presented a proof to calculate the value of the zeta function at the number $- 2$ and proved that its value is not equal to zero as Euler extracted it using his second law based on Bernoulli numbers, but rather equals an imaginary value of $\frac{i}{84}$ . The researcher also proved that the third Bernoulli number is not zero based on Euler’s second law, considering that zeta is known and the Bernoulli number is unknown, thus concluding that the third Bernoulli number is equal to $\frac{i}{28}$ .

2. The main results

In this section, two theorems are proved and show that the zeta function at $(- 4)$ has two complex values and that the fifth Bernoulli number $B_{5}$ is not zero but has two complex values $\frac{- 5 i}{558} and \frac{- 5 i}{2232}$ .

Theorem 1:

Zeta function at $- 4$ has two values $\frac{- i}{558} and \frac{- i}{2232}$

Proof:

$ζ (s) = \sum_{n = 1}^{\infty} \frac{1}{n^{s}}$ , where $s \in ℝ$ according to the Euler definition.

ζ (- 4) = \sum_{n = 1}^{\infty} \frac{1}{n^{- 4}} = \sum_{n = 1}^{\infty} n^{4} = 1 + 16 + 81 + 256 + 625 + 1296 + \dots

From,⁸

(3)

\begin{matrix} η (s) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n^{s}}; s = - 4 \\ η (- 4) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n + 1}}{n^{- 4}} = 1 - 16 + 81 - 256 + 625 - 1296 + \dots \\ \begin{matrix} ζ (- 4) - η (- 4) = 32 + 512 + 2592 + \dots = 32 (1 + 16 + 81 + \dots) \\ ζ (- 4) - 32 ζ (- 4) = η (- 4) \\ ζ (- 4) = - \frac{η (- 4)}{31} \end{matrix} \end{matrix}

Now,

η (- 4) = 1 - 16 + 81 - 256 + 625 - 1296 + \dots

1 = [1 {(- 1)}^{1} (- 1)]

1 - 16 = - 15 = {[(1 + 2) (4) + (1 + 2) (1)] [{(- 1)}^{1 + 1} (- 1)]}

1 - 16 + 81 = 66 = {[(1 + 2 + 3) (9) + (1 + 2 + 3) (1 + 1)] [{(- 1)}^{1 + 1 + 1} (- 1)]}

1 - 16 + 81 - 256 = - 190 = {[(1 + 2 + 3 + 4) (16) + (1 + 2 + 3 + 4) (1 + 1 + 1)] [{(- 1)}^{1 + 1 + 1 + 1} (- 1)]}

1 - 16 + 81 - 256 + 625 = 435 = {[(1 + 2 + 3 + 4 + 5) (25) + (1 + 2 + 3 + 4 + 5) (1 + 1 + 1 + 1)] [{(- 1)}^{1 + 1 + 1 + 1 + 1} (- 1)]}

and so on,

(4)

η (- 4) = {(1 + 2 + 3 + \dots) [(1 + 2 + 3 + \dots) + (1 + 2 + \dots)] + [(1 + 2 + 3 + \dots) (1 + 1 + \dots)]} [{(- 1)}^{1 + 1 + 1 + \dots} (- 1)]

η (- 4) = {ζ (- 1) [ζ (- 1) + ζ (- 1)] + ζ (- 1) ζ (0)} [{(- 1)}^{ζ (0)} (- 1)]

η (- 4) = {\frac{- 1}{12} [\frac{- 1}{12} + \frac{- 1}{12}] + \frac{- 1}{12} ∙ \frac{- 1}{2}} [{(- 1)}^{\frac{- 1}{2}} (- 1)]

η (- 4) = \frac{i}{18}

Put in Equation (3) we get

(5)

ζ (- 4) = - \frac{i / 18}{31} = \frac{- i}{558}

By Equation (4):

\begin{array}{l} 1 = [1 {(- 1)}^{1} (- 1)] \\ 1 - 16 = - 15 = {[(1 + 2) (1 + 2 + 1 + 2 - 2) + (1 + 2) (1)] [{(- 1)}^{1 + 1} (- 1)]} \\ 1 - 16 + 81 = 66 = {[(1 + 2 + 3) (1 + 2 + 3 + 1 + 2 + 3 - 3) + (1 + 2 + 3) (1 + 1)] [{(- 1)}^{1 + 1 + 1} (- 1)]} \\ 1 - 16 + 81 - 256 = - 190 = {[(1 + 2 + 3 + 4) (1 + 2 + 3 + 4 + 1 + 2 + 3 + 4 - 4) + (1 + 2 + 3 + 4) (1 + 1 + 1)] [{(- 1)}^{1 + 1 + 1 + 1} (- 1)]} \\ 1 - 16 + 81 - 256 + 625 = 435 = {[(1 + 2 + 3 + 4 + 5) (1 + 2 + 3 + 4 + 5 + 1 + 2 + 3 + 4 + 5 - 5) + (1 + 2 + 3 + 4 + 5) (1 + 1 + 1 + 1)] [{(- 1)}^{1 + 1 + 1 + 1 + 1} (- 1)]} \end{array}

and so on,

(6)

η (- 4) = {(1 + 2 + 3 + \dots) [(1 + 2 + 3 + \dots) + [1 + 2 + 3 + \dots) - (1 + 1 + 1 + \dots)]] + [(1 + 2 + 3 + \dots) (1 + 1 + \dots)]} [{(- 1)}^{1 + 1 + 1 + \dots} (- 1)]

\begin{matrix} η (- 4) = {(ζ (- 1)) [(ζ (- 1)) + [ζ (- 1)) - (ζ (0))]] + [(ζ (- 1)) (ζ (0))]} [{(- 1)}^{ζ (0)} (- 1)] \\ η (- 4) = {(\frac{- 1}{12}) [(\frac{- 1}{12}) + [\frac{- 1}{12} - (\frac{- 1}{2})]] + [(\frac{- 1}{12}) (\frac{- 1}{2})]} [{(- 1)}^{\frac{- 1}{2}} (- 1)] \\ η (- 4) = \frac{i}{72} \end{matrix}

Put in Equation (3) we get

(7)

ζ (- 4) = - \frac{i / 72}{31} = \frac{- i}{2232}

Theorem 2:

The fifth Bernoulli number, $B_{5}$ is not zero but has two values: $\frac{- 5 i}{558} and \frac{- 5 i}{2232} .$

Proof:

Using (2) $ζ (- k) = \frac{{(- 1)}^{k} B_{k + 1}}{k + 1}$ being $ζ (- 4) = \frac{{(- 1)}^{4} B_{5}}{5}$

Since $ζ (- 4) = \frac{- i}{558}$ from $(5)$ , we get, $B_{5} = \frac{- 5 i}{558} .$

And, Since $ζ (- 4) = \frac{- i}{2232}$ from $(7)$ , we get, $B_{5} = \frac{- 5 i}{2232}$

3. Conclusions

We conclude from the current research that the zeta function $ζ (- 4)$ has two imaginary values that are different from what was previously known to mathematicians, which was believed to be equal to zero, whereas,

ζ (- 4) = \frac{- i}{558}, ζ (- 4) = \frac{- i}{2232}

Therefore, the fifth Bernoulli number is not equal to zero and has two values: $B_{5} = \frac{- 5 i}{558}$ and $B_{5} = \frac{- 5 i}{2232}$ .

Data availability statement

No data are associated with this article.

References

1. Larson N: The Bernoulli Numbers: A Brief Primer. Whitman College; 2019. (accessed Aug. 25, 2024). Reference Source
2. Coen LES: Sums of Powers and the Bernoulli Numbers. The Keep, Eastern Illinois University; 1996. Masters Theses.
3. Dwilewicz RJ, Minác J: Values of the Riemann zeta function at integers. Materials Matematics (MM). 2009(6): Page No: 0001–0026. 1887-1097. Reference Source
4. Edwards HM: Riemann’s Zeta Function. London: Academic Press, INC.; 5th ed. 1974.
5. Derbyshir J: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington D.C.: John Henry Press; 2003.
6. Patel GS, Gautam SK: Ramanujan: The New Sum of All Natural Numbers. International Journal for Research in Applied Science and Engineering Technology (IJRASET). Vol. 10(II): p. 1272–1274. 2321-9653. Publisher Full Text Reference Source
7. Ahmed Adnan Tuma: Amazing Results Concerning Zeta Function at (-2) and Bernoulli3^rd Number. Journal of Computational Analysis and Applications (JoCAAA). 2024; 33(05): 342–344. Reference Source
8. Boyadzhiev K, Frontczak R: Series involving Euler’s eta (or Dirichlet eta) function. J. Integer Seq. 2021; 24: A 21.9.1. MR4336093.

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Version 1

VERSION 1 PUBLISHED 10 Jan 2026

Author details Author details

¹ Mathematics, University of Baghdad College of Education for Pure Science Ibn Al-Haitham, Baghdad, Baghdad Governorate, Iraq
² Mathematics, Department of Mathematics, College of Basic Education, University of Diyala, Baqubah, Iraq

Ahmed Adnan Tuma
Roles: Conceptualization

Shakir M. Salman
Roles: Conceptualization

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

Article Versions (1)

version 1

Published: 10 Jan 2026, 15:46

https://doi.org/10.12688/f1000research.172983.1

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© 2026 Tuma AA and Salman SM. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Tuma AA and Salman SM. New Results About Imaginary Value of the Second Axiomatic Zero of the Zeta Function and the Fifth Bernoulli Number [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:46 (https://doi.org/10.12688/f1000research.172983.1)

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[1] 1. Larson N: The Bernoulli Numbers: A Brief Primer. Whitman College; 2019. (accessed Aug. 25, 2024). Reference Source

[2] 2. Coen LES: Sums of Powers and the Bernoulli Numbers. The Keep, Eastern Illinois University; 1996. Masters Theses.

[3] 3. Dwilewicz RJ, Minác J: Values of the Riemann zeta function at integers. Materials Matematics (MM). 2009(6): Page No: 0001–0026. 1887-1097. Reference Source

[4] 4. Edwards HM: Riemann’s Zeta Function. London: Academic Press, INC.; 5th ed. 1974.

[5] 5. Derbyshir J: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Washington D.C.: John Henry Press; 2003.

[6] 6. Patel GS, Gautam SK: Ramanujan: The New Sum of All Natural Numbers. International Journal for Research in Applied Science and Engineering Technology (IJRASET). Vol. 10(II): p. 1272–1274. 2321-9653. Publisher Full Text Reference Source

[7] 7. Ahmed Adnan Tuma: Amazing Results Concerning Zeta Function at (-2) and Bernoulli3^rd Number. Journal of Computational Analysis and Applications (JoCAAA). 2024; 33(05): 342–344. Reference Source

[8] 8. Boyadzhiev K, Frontczak R: Series involving Euler’s eta (or Dirichlet eta) function. J. Integer Seq. 2021; 24: A 21.9.1. MR4336093.

New Results About Imaginary Value of the Second Axiomatic Zero of the Zeta Function and the Fifth Bernoulli Number

Abstract

Keywords

1. Introduction

(1)

(2)

2. The main results

Theorem 1:

Proof:

(3)

(4)

(5)

(6)

(7)

Theorem 2:

Proof:

3. Conclusions

Data availability statement

References

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