Riemann Zeta Function Zeros

Riemann zeta function zeros are the zeros of the Riemann zeta function and come in two different types. So-called "trivial zeros" occur at all negative even integers , , , ..., and "nontrivial zeros" occur at points in the "critical strip" . A nontrivial zero is denoted . The zeros with positive imaginary part, ordered by increasing imaginary part and counted with multiplicity, are commonly denoted (Brent 1979; Edwards 2001, p. 43).

Wiener (1951) showed that the prime number theorem is literally equivalent to the assertion that has no zeros on (Hardy 1999, p. 34; Havil 2003, p. 195). The Riemann hypothesis asserts that the nontrivial zeros of all have real part , a line called the "critical line." This is known to be true for the first zeros.

An attractive poster plotting zeros of the Riemann zeta function on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).

The plots above show the real and imaginary parts of plotted in the complex plane together with the complex modulus of . As can be seen, in the right half-plane, the function is fairly flat, but with a large number of horizontal ridges. It is precisely along these ridges that the nontrivial zeros of lie.

The plot above shows the level curves (red) and (blue) in the complex plane of . Along each curve, only the indicated component of is zero; the other component can vary. The zeros (indicated as black dots) occur where the curves intersect.

The figures above highlight the zeros in the complex plane by plotting (where the zeros are dips) and (where the zeros are peaks).

The above plot shows for between 0 and 60. As can be seen, the first few nontrivial zeros occur at the values given in the following table (Wagon 1991, pp. 361-362 and 367-368; Havil 2003, p. 196; Odlyzko), where the corresponding negative values are also roots. The integers closest to these values are 14, 21, 25, 30, 33, 38, 41, 43, 48, 50, ... (OEIS A002410). The numbers of nontrivial zeros with positive imaginary part less than 10, , , ..., counted with multiplicity, are 0, 29, 649, 10142, 138069, 1747146, ... (OEIS A072080; Odlyzko).

	OEIS
1	A058303	14.134725
2		21.022040
3		25.010858
4		30.424876
5		32.935062
6		37.586178

The so-called xi-function defined by Riemann has precisely the same zeros as the nontrivial zeros of with the additional benefit that is entire and is purely real, making its zeros simpler to locate.

ZetaGrid was a distributed computing project for calculating zeros. It reported 1029.9 billion zeros as of Feb. 18, 2005. Gourdon (2004) used an algorithm of Odlyzko and Schönhage to calculate the first zeros (Pegg 2004, Pegg and Weisstein 2004). The following table lists historical benchmarks in the number of computed zeros (Gourdon 2004).

year		author
1903	15	J. P. Gram
1914	79	R. J. Backlund
1925	138	J. I. Hutchinson
1935		E. C. Titchmarsh
1953		A. M. Turing
1956		D. H. Lehmer
1956		D. H. Lehmer
1958		N. A. Meller
1966		R. S. Lehman
1968		J. B. Rosser, J. M. Yohe, L. Schoenfeld
1977		R. P. Brent
1979		R. P. Brent
1982		R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter
1983		J. van de Lune, H. J. J. te Riele
1986		J. van de Lune, H. J. J. te Riele, D. T. Winter
2001		J. van de Lune (unpublished)
2004		S. Wedeniwski
2004		X. Gourdon and P. Demichel

It is not known whether all values of corresponding to nontrivial zeros are irrational. Wolf (2020) obtained numerical evidence supporting this conjecture by studying continued fractions of high-precision approximations to the first 40000 such values, but this does not prove that these values are irrational.

No zeros with order greater than one are known. While the existence of such zeros would not disprove the Riemann hypothesis, it would cause serious problems for many current computational techniques (Derbyshire 2004, p. 385).

Knausgård (2026) reported unconditional asymptotic lower bounds for distinct and simple nontrivial zeros. Let count zeros with imaginary part in , including multiplicity, and let , , and count distinct zeros, simple zeros, and simple zeros on the critical line, respectively. The bounds are

			(1)
			(2)
			(3)

Knausgård (2026) describes the work as AI-assisted. The Lean development uses three additional axioms recording external computation results, so those finite checks are not verified by Lean's kernel. The external certificates have not been independently rerun here.

A different bound concerns the union of simple zeros and zeros on the critical line. Let count nontrivial zeros with imaginary part in that satisfy at least one of these properties, counted with multiplicity. Cobeli and Zaharescu (2026) proved unconditionally that

(4)

Their argument gives a lower bound approximately 0.88893. This is not a separate bound of for either simple zeros or zeros on the critical line.

Some nontrivial zeros lie extremely close together, a property known as Lehmer's phenomenon.

The Riemann zeta function can be factored over its nontrivial zeros as the Hadamard product

(5)

(Titchmarsh 1987, Voros 1987).

For a positive integer , define the sum over all nontrivial zeros , counted with multiplicity, by

(6)

(Lehmer 1988, Keiper 1992, Finch 2003, p. 168), sometimes also denoted (e.g., Finch 2003, p. 168). For , the sum is interpreted by symmetric height limits as . For , it is absolutely convergent. The functional equation pairs each zero of positive imaginary part with of negative imaginary part, including multiplicity. Thus, with running over the zeros with ,

(7)

This paired sum is absolutely convergent for every and does not require the Riemann hypothesis. Such sums can be computed analytically, and the first few are

					(8)
					(9)
					(10)
					(11)
					(12)
					(13)

where is the Euler-Mascheroni constant, are Stieltjes constants, is the Riemann zeta function, and is Apéry's constant. These values can also be written in terms of the Li constants (Bombieri and Lagarias 1999).

The case

(14)

(OEIS A074760; Edwards 2001, p. 160) is classical and was known to Riemann, who used it in his computation of the roots of (Davenport 1980, pp. 83-84; Edwards 2001, pp. 67 and 159). It is also equal to the constant from Li's criterion.

Assuming the truth of the Riemann hypothesis (so that ), equation (7) can be written for the first few values of in the simple forms

			(15)
			(16)
			(17)
			(18)

and so on.

Zeros of the Riemann zeta function on the critical line are implemented in the Wolfram Language as ZetaZero[k], where a positive integer selects the th zero with positive imaginary part.

Assuming the Riemann hypothesis, Bondarenko and Heap (2026) proved

(19)

Here the positive ordinates are counted with multiplicity. The result gives normalized gaps at most in the infimum limit; it does not assert a strict inequality. Bondarenko and Heap (2026) report ChatGPT assistance with mathematical exploration.

Durkan and Pearce-Crump (2026) extended the half-spacing infimum limit bound unconditionally to every fixed primitive number theoretic character. Under the Riemann hypothesis, they also proved that, for every , a positive proportion of consecutive Riemann zeta function zero gaps have normalized size less than . Under the corresponding hypothesis for a fixed primitive Dirichlet L-series, they obtained

(20)

Here are its positive nontrivial zero ordinates counted with multiplicity. The Riemann zeta function is included as the conductor-one case. Durkan and Pearce-Crump (2026) report ChatGPT assistance with mathematical argument checking and the large-gap optimizations.